Result for (x1 / (128*128*128) * y1 + x2 / (128*128*128) * y2) / (y1 + y2) * 128*128*128

Alternative 1: 99.7% accurate, 1.2× speedup?

\[\left(\left(\mathsf{fma}\left(\frac{y2 \cdot 4.76837158203125 \cdot 10^{-7}}{y1 + y2}, x2, \left(x1 \cdot 4.76837158203125 \cdot 10^{-7}\right) \cdot \frac{y1}{y1 + y2}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]

(FPCore (x1 y1 x2 y2)
  :precision binary64
  (*
 (*
  (*
   (fma
    (/ (* y2 4.76837158203125e-7) (+ y1 y2))
    x2
    (* (* x1 4.76837158203125e-7) (/ y1 (+ y1 y2))))
   128.0)
  128.0)
 128.0))

double code(double x1, double y1, double x2, double y2) {
	return ((fma(((y2 * 4.76837158203125e-7) / (y1 + y2)), x2, ((x1 * 4.76837158203125e-7) * (y1 / (y1 + y2)))) * 128.0) * 128.0) * 128.0;
}

function code(x1, y1, x2, y2)
	return Float64(Float64(Float64(fma(Float64(Float64(y2 * 4.76837158203125e-7) / Float64(y1 + y2)), x2, Float64(Float64(x1 * 4.76837158203125e-7) * Float64(y1 / Float64(y1 + y2)))) * 128.0) * 128.0) * 128.0)
end

code[x1_, y1_, x2_, y2_] := N[(N[(N[(N[(N[(N[(y2 * 4.76837158203125e-7), $MachinePrecision] / N[(y1 + y2), $MachinePrecision]), $MachinePrecision] * x2 + N[(N[(x1 * 4.76837158203125e-7), $MachinePrecision] * N[(y1 / N[(y1 + y2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 128.0), $MachinePrecision] * 128.0), $MachinePrecision] * 128.0), $MachinePrecision]

\left(\left(\mathsf{fma}\left(\frac{y2 \cdot 4.76837158203125 \cdot 10^{-7}}{y1 + y2}, x2, \left(x1 \cdot 4.76837158203125 \cdot 10^{-7}\right) \cdot \frac{y1}{y1 + y2}\right) \cdot 128\right) \cdot 128\right) \cdot 128

Derivation

Initial program 77.8%
\[\left(\left(\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2} \cdot 128\right) \cdot 128\right) \cdot 128 \]
Step-by-step derivation
1. lift-/.f64N/A
  \[\leadsto \left(\left(\color{blue}{\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2}} \cdot 128\right) \cdot 128\right) \cdot 128 \]
2. mult-flipN/A
  \[\leadsto \left(\left(\color{blue}{\left(\left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2\right) \cdot \frac{1}{y1 + y2}\right)} \cdot 128\right) \cdot 128\right) \cdot 128 \]
3. *-commutativeN/A
  \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{y1 + y2} \cdot \left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2\right)\right)} \cdot 128\right) \cdot 128\right) \cdot 128 \]
4. lift-+.f64N/A
  \[\leadsto \left(\left(\left(\frac{1}{y1 + y2} \cdot \color{blue}{\left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2\right)}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
5. +-commutativeN/A
  \[\leadsto \left(\left(\left(\frac{1}{y1 + y2} \cdot \color{blue}{\left(\frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2 + \frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1\right)}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
6. distribute-lft-inN/A
  \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{y1 + y2} \cdot \left(\frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2\right) + \frac{1}{y1 + y2} \cdot \left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1\right)\right)} \cdot 128\right) \cdot 128\right) \cdot 128 \]
7. *-commutativeN/A
  \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2\right) \cdot \frac{1}{y1 + y2}} + \frac{1}{y1 + y2} \cdot \left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1\right)\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
8. mult-flipN/A
  \[\leadsto \left(\left(\left(\color{blue}{\frac{\frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2}} + \frac{1}{y1 + y2} \cdot \left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1\right)\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
9. lift-*.f64N/A
  \[\leadsto \left(\left(\left(\frac{\color{blue}{\frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}}{y1 + y2} + \frac{1}{y1 + y2} \cdot \left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1\right)\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
10. associate-/l*N/A
  \[\leadsto \left(\left(\left(\color{blue}{\frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot \frac{y2}{y1 + y2}} + \frac{1}{y1 + y2} \cdot \left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1\right)\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
11. *-commutativeN/A
  \[\leadsto \left(\left(\left(\color{blue}{\frac{y2}{y1 + y2} \cdot \frac{x2}{\left(128 \cdot 128\right) \cdot 128}} + \frac{1}{y1 + y2} \cdot \left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1\right)\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
12. *-commutativeN/A
  \[\leadsto \left(\left(\left(\frac{y2}{y1 + y2} \cdot \frac{x2}{\left(128 \cdot 128\right) \cdot 128} + \color{blue}{\left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1\right) \cdot \frac{1}{y1 + y2}}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
13. mult-flipN/A
  \[\leadsto \left(\left(\left(\frac{y2}{y1 + y2} \cdot \frac{x2}{\left(128 \cdot 128\right) \cdot 128} + \color{blue}{\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1}{y1 + y2}}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
Applied rewrites99.6%
\[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\frac{y2}{y2 + y1}, 4.76837158203125 \cdot 10^{-7} \cdot x2, \left(4.76837158203125 \cdot 10^{-7} \cdot y1\right) \cdot \frac{x1}{y2 + y1}\right)} \cdot 128\right) \cdot 128\right) \cdot 128 \]
Step-by-step derivation
1. lift-fma.f64N/A
  \[\leadsto \left(\left(\color{blue}{\left(\frac{y2}{y2 + y1} \cdot \left(\frac{1}{2097152} \cdot x2\right) + \left(\frac{1}{2097152} \cdot y1\right) \cdot \frac{x1}{y2 + y1}\right)} \cdot 128\right) \cdot 128\right) \cdot 128 \]
2. lift-*.f64N/A
  \[\leadsto \left(\left(\left(\frac{y2}{y2 + y1} \cdot \color{blue}{\left(\frac{1}{2097152} \cdot x2\right)} + \left(\frac{1}{2097152} \cdot y1\right) \cdot \frac{x1}{y2 + y1}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
3. associate-*r*N/A
  \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{y2}{y2 + y1} \cdot \frac{1}{2097152}\right) \cdot x2} + \left(\frac{1}{2097152} \cdot y1\right) \cdot \frac{x1}{y2 + y1}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
4. lower-fma.f64N/A
  \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\frac{y2}{y2 + y1} \cdot \frac{1}{2097152}, x2, \left(\frac{1}{2097152} \cdot y1\right) \cdot \frac{x1}{y2 + y1}\right)} \cdot 128\right) \cdot 128\right) \cdot 128 \]
5. lift-/.f64N/A
  \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\frac{y2}{y2 + y1}} \cdot \frac{1}{2097152}, x2, \left(\frac{1}{2097152} \cdot y1\right) \cdot \frac{x1}{y2 + y1}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
6. lift-+.f64N/A
  \[\leadsto \left(\left(\mathsf{fma}\left(\frac{y2}{\color{blue}{y2 + y1}} \cdot \frac{1}{2097152}, x2, \left(\frac{1}{2097152} \cdot y1\right) \cdot \frac{x1}{y2 + y1}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
7. +-commutativeN/A
  \[\leadsto \left(\left(\mathsf{fma}\left(\frac{y2}{\color{blue}{y1 + y2}} \cdot \frac{1}{2097152}, x2, \left(\frac{1}{2097152} \cdot y1\right) \cdot \frac{x1}{y2 + y1}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
8. lift-+.f64N/A
  \[\leadsto \left(\left(\mathsf{fma}\left(\frac{y2}{\color{blue}{y1 + y2}} \cdot \frac{1}{2097152}, x2, \left(\frac{1}{2097152} \cdot y1\right) \cdot \frac{x1}{y2 + y1}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
9. associate-*l/N/A
  \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\frac{y2 \cdot \frac{1}{2097152}}{y1 + y2}}, x2, \left(\frac{1}{2097152} \cdot y1\right) \cdot \frac{x1}{y2 + y1}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
10. lower-/.f64N/A
  \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{\frac{y2 \cdot \frac{1}{2097152}}{y1 + y2}}, x2, \left(\frac{1}{2097152} \cdot y1\right) \cdot \frac{x1}{y2 + y1}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
11. lower-*.f6499.6%
  \[\leadsto \left(\left(\mathsf{fma}\left(\frac{\color{blue}{y2 \cdot 4.76837158203125 \cdot 10^{-7}}}{y1 + y2}, x2, \left(4.76837158203125 \cdot 10^{-7} \cdot y1\right) \cdot \frac{x1}{y2 + y1}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
12. lift-*.f64N/A
  \[\leadsto \left(\left(\mathsf{fma}\left(\frac{y2 \cdot \frac{1}{2097152}}{y1 + y2}, x2, \color{blue}{\left(\frac{1}{2097152} \cdot y1\right) \cdot \frac{x1}{y2 + y1}}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
13. lift-*.f64N/A
  \[\leadsto \left(\left(\mathsf{fma}\left(\frac{y2 \cdot \frac{1}{2097152}}{y1 + y2}, x2, \color{blue}{\left(\frac{1}{2097152} \cdot y1\right)} \cdot \frac{x1}{y2 + y1}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
14. *-commutativeN/A
  \[\leadsto \left(\left(\mathsf{fma}\left(\frac{y2 \cdot \frac{1}{2097152}}{y1 + y2}, x2, \color{blue}{\left(y1 \cdot \frac{1}{2097152}\right)} \cdot \frac{x1}{y2 + y1}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
15. metadata-evalN/A
  \[\leadsto \left(\left(\mathsf{fma}\left(\frac{y2 \cdot \frac{1}{2097152}}{y1 + y2}, x2, \left(y1 \cdot \color{blue}{\frac{1}{2097152}}\right) \cdot \frac{x1}{y2 + y1}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
16. metadata-evalN/A
  \[\leadsto \left(\left(\mathsf{fma}\left(\frac{y2 \cdot \frac{1}{2097152}}{y1 + y2}, x2, \left(y1 \cdot \frac{1}{\color{blue}{16384 \cdot 128}}\right) \cdot \frac{x1}{y2 + y1}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
17. metadata-evalN/A
  \[\leadsto \left(\left(\mathsf{fma}\left(\frac{y2 \cdot \frac{1}{2097152}}{y1 + y2}, x2, \left(y1 \cdot \frac{1}{\color{blue}{\left(128 \cdot 128\right)} \cdot 128}\right) \cdot \frac{x1}{y2 + y1}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
18. mult-flip-revN/A
  \[\leadsto \left(\left(\mathsf{fma}\left(\frac{y2 \cdot \frac{1}{2097152}}{y1 + y2}, x2, \color{blue}{\frac{y1}{\left(128 \cdot 128\right) \cdot 128}} \cdot \frac{x1}{y2 + y1}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
19. lift-/.f64N/A
  \[\leadsto \left(\left(\mathsf{fma}\left(\frac{y2 \cdot \frac{1}{2097152}}{y1 + y2}, x2, \frac{y1}{\left(128 \cdot 128\right) \cdot 128} \cdot \color{blue}{\frac{x1}{y2 + y1}}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
20. lift-+.f64N/A
  \[\leadsto \left(\left(\mathsf{fma}\left(\frac{y2 \cdot \frac{1}{2097152}}{y1 + y2}, x2, \frac{y1}{\left(128 \cdot 128\right) \cdot 128} \cdot \frac{x1}{\color{blue}{y2 + y1}}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
21. +-commutativeN/A
  \[\leadsto \left(\left(\mathsf{fma}\left(\frac{y2 \cdot \frac{1}{2097152}}{y1 + y2}, x2, \frac{y1}{\left(128 \cdot 128\right) \cdot 128} \cdot \frac{x1}{\color{blue}{y1 + y2}}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
22. lift-+.f64N/A
  \[\leadsto \left(\left(\mathsf{fma}\left(\frac{y2 \cdot \frac{1}{2097152}}{y1 + y2}, x2, \frac{y1}{\left(128 \cdot 128\right) \cdot 128} \cdot \frac{x1}{\color{blue}{y1 + y2}}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
23. frac-timesN/A
  \[\leadsto \left(\left(\mathsf{fma}\left(\frac{y2 \cdot \frac{1}{2097152}}{y1 + y2}, x2, \color{blue}{\frac{y1 \cdot x1}{\left(\left(128 \cdot 128\right) \cdot 128\right) \cdot \left(y1 + y2\right)}}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
Applied rewrites99.7%
\[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\frac{y2 \cdot 4.76837158203125 \cdot 10^{-7}}{y1 + y2}, x2, \left(x1 \cdot 4.76837158203125 \cdot 10^{-7}\right) \cdot \frac{y1}{y1 + y2}\right)} \cdot 128\right) \cdot 128\right) \cdot 128 \]
Add Preprocessing

Alternative 2: 99.6% accurate, 1.2× speedup?

\[\left(\left(\mathsf{fma}\left(\frac{y2}{y2 + y1}, 4.76837158203125 \cdot 10^{-7} \cdot x2, \left(4.76837158203125 \cdot 10^{-7} \cdot y1\right) \cdot \frac{x1}{y2 + y1}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]

(FPCore (x1 y1 x2 y2)
  :precision binary64
  (*
 (*
  (*
   (fma
    (/ y2 (+ y2 y1))
    (* 4.76837158203125e-7 x2)
    (* (* 4.76837158203125e-7 y1) (/ x1 (+ y2 y1))))
   128.0)
  128.0)
 128.0))

double code(double x1, double y1, double x2, double y2) {
	return ((fma((y2 / (y2 + y1)), (4.76837158203125e-7 * x2), ((4.76837158203125e-7 * y1) * (x1 / (y2 + y1)))) * 128.0) * 128.0) * 128.0;
}

function code(x1, y1, x2, y2)
	return Float64(Float64(Float64(fma(Float64(y2 / Float64(y2 + y1)), Float64(4.76837158203125e-7 * x2), Float64(Float64(4.76837158203125e-7 * y1) * Float64(x1 / Float64(y2 + y1)))) * 128.0) * 128.0) * 128.0)
end

code[x1_, y1_, x2_, y2_] := N[(N[(N[(N[(N[(y2 / N[(y2 + y1), $MachinePrecision]), $MachinePrecision] * N[(4.76837158203125e-7 * x2), $MachinePrecision] + N[(N[(4.76837158203125e-7 * y1), $MachinePrecision] * N[(x1 / N[(y2 + y1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 128.0), $MachinePrecision] * 128.0), $MachinePrecision] * 128.0), $MachinePrecision]

\left(\left(\mathsf{fma}\left(\frac{y2}{y2 + y1}, 4.76837158203125 \cdot 10^{-7} \cdot x2, \left(4.76837158203125 \cdot 10^{-7} \cdot y1\right) \cdot \frac{x1}{y2 + y1}\right) \cdot 128\right) \cdot 128\right) \cdot 128

Derivation

Initial program 77.8%
\[\left(\left(\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2} \cdot 128\right) \cdot 128\right) \cdot 128 \]
Step-by-step derivation
1. lift-/.f64N/A
  \[\leadsto \left(\left(\color{blue}{\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2}} \cdot 128\right) \cdot 128\right) \cdot 128 \]
2. mult-flipN/A
  \[\leadsto \left(\left(\color{blue}{\left(\left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2\right) \cdot \frac{1}{y1 + y2}\right)} \cdot 128\right) \cdot 128\right) \cdot 128 \]
3. *-commutativeN/A
  \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{y1 + y2} \cdot \left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2\right)\right)} \cdot 128\right) \cdot 128\right) \cdot 128 \]
4. lift-+.f64N/A
  \[\leadsto \left(\left(\left(\frac{1}{y1 + y2} \cdot \color{blue}{\left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2\right)}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
5. +-commutativeN/A
  \[\leadsto \left(\left(\left(\frac{1}{y1 + y2} \cdot \color{blue}{\left(\frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2 + \frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1\right)}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
6. distribute-lft-inN/A
  \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{y1 + y2} \cdot \left(\frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2\right) + \frac{1}{y1 + y2} \cdot \left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1\right)\right)} \cdot 128\right) \cdot 128\right) \cdot 128 \]
7. *-commutativeN/A
  \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2\right) \cdot \frac{1}{y1 + y2}} + \frac{1}{y1 + y2} \cdot \left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1\right)\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
8. mult-flipN/A
  \[\leadsto \left(\left(\left(\color{blue}{\frac{\frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2}} + \frac{1}{y1 + y2} \cdot \left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1\right)\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
9. lift-*.f64N/A
  \[\leadsto \left(\left(\left(\frac{\color{blue}{\frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}}{y1 + y2} + \frac{1}{y1 + y2} \cdot \left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1\right)\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
10. associate-/l*N/A
  \[\leadsto \left(\left(\left(\color{blue}{\frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot \frac{y2}{y1 + y2}} + \frac{1}{y1 + y2} \cdot \left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1\right)\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
11. *-commutativeN/A
  \[\leadsto \left(\left(\left(\color{blue}{\frac{y2}{y1 + y2} \cdot \frac{x2}{\left(128 \cdot 128\right) \cdot 128}} + \frac{1}{y1 + y2} \cdot \left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1\right)\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
12. *-commutativeN/A
  \[\leadsto \left(\left(\left(\frac{y2}{y1 + y2} \cdot \frac{x2}{\left(128 \cdot 128\right) \cdot 128} + \color{blue}{\left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1\right) \cdot \frac{1}{y1 + y2}}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
13. mult-flipN/A
  \[\leadsto \left(\left(\left(\frac{y2}{y1 + y2} \cdot \frac{x2}{\left(128 \cdot 128\right) \cdot 128} + \color{blue}{\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1}{y1 + y2}}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
Applied rewrites99.6%
\[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\frac{y2}{y2 + y1}, 4.76837158203125 \cdot 10^{-7} \cdot x2, \left(4.76837158203125 \cdot 10^{-7} \cdot y1\right) \cdot \frac{x1}{y2 + y1}\right)} \cdot 128\right) \cdot 128\right) \cdot 128 \]
Add Preprocessing

Alternative 3: 99.6% accurate, 1.2× speedup?

\[\left(\left(\mathsf{fma}\left(\frac{y1}{y2 + y1}, 4.76837158203125 \cdot 10^{-7} \cdot x1, \left(4.76837158203125 \cdot 10^{-7} \cdot y2\right) \cdot \frac{x2}{y2 + y1}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]

(FPCore (x1 y1 x2 y2)
  :precision binary64
  (*
 (*
  (*
   (fma
    (/ y1 (+ y2 y1))
    (* 4.76837158203125e-7 x1)
    (* (* 4.76837158203125e-7 y2) (/ x2 (+ y2 y1))))
   128.0)
  128.0)
 128.0))

double code(double x1, double y1, double x2, double y2) {
	return ((fma((y1 / (y2 + y1)), (4.76837158203125e-7 * x1), ((4.76837158203125e-7 * y2) * (x2 / (y2 + y1)))) * 128.0) * 128.0) * 128.0;
}

function code(x1, y1, x2, y2)
	return Float64(Float64(Float64(fma(Float64(y1 / Float64(y2 + y1)), Float64(4.76837158203125e-7 * x1), Float64(Float64(4.76837158203125e-7 * y2) * Float64(x2 / Float64(y2 + y1)))) * 128.0) * 128.0) * 128.0)
end

code[x1_, y1_, x2_, y2_] := N[(N[(N[(N[(N[(y1 / N[(y2 + y1), $MachinePrecision]), $MachinePrecision] * N[(4.76837158203125e-7 * x1), $MachinePrecision] + N[(N[(4.76837158203125e-7 * y2), $MachinePrecision] * N[(x2 / N[(y2 + y1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 128.0), $MachinePrecision] * 128.0), $MachinePrecision] * 128.0), $MachinePrecision]

\left(\left(\mathsf{fma}\left(\frac{y1}{y2 + y1}, 4.76837158203125 \cdot 10^{-7} \cdot x1, \left(4.76837158203125 \cdot 10^{-7} \cdot y2\right) \cdot \frac{x2}{y2 + y1}\right) \cdot 128\right) \cdot 128\right) \cdot 128

Derivation

Initial program 77.8%
\[\left(\left(\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2} \cdot 128\right) \cdot 128\right) \cdot 128 \]
Step-by-step derivation
1. lift-/.f64N/A
  \[\leadsto \left(\left(\color{blue}{\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2}} \cdot 128\right) \cdot 128\right) \cdot 128 \]
2. lift-+.f64N/A
  \[\leadsto \left(\left(\frac{\color{blue}{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}}{y1 + y2} \cdot 128\right) \cdot 128\right) \cdot 128 \]
3. div-addN/A
  \[\leadsto \left(\left(\color{blue}{\left(\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1}{y1 + y2} + \frac{\frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2}\right)} \cdot 128\right) \cdot 128\right) \cdot 128 \]
4. mult-flipN/A
  \[\leadsto \left(\left(\left(\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1}{y1 + y2} + \color{blue}{\left(\frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2\right) \cdot \frac{1}{y1 + y2}}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
5. lift-*.f64N/A
  \[\leadsto \left(\left(\left(\frac{\color{blue}{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1}}{y1 + y2} + \left(\frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2\right) \cdot \frac{1}{y1 + y2}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
6. associate-/l*N/A
  \[\leadsto \left(\left(\left(\color{blue}{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot \frac{y1}{y1 + y2}} + \left(\frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2\right) \cdot \frac{1}{y1 + y2}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
7. *-commutativeN/A
  \[\leadsto \left(\left(\left(\color{blue}{\frac{y1}{y1 + y2} \cdot \frac{x1}{\left(128 \cdot 128\right) \cdot 128}} + \left(\frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2\right) \cdot \frac{1}{y1 + y2}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
8. mult-flipN/A
  \[\leadsto \left(\left(\left(\frac{y1}{y1 + y2} \cdot \frac{x1}{\left(128 \cdot 128\right) \cdot 128} + \color{blue}{\frac{\frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2}}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
9. lower-fma.f64N/A
  \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\frac{y1}{y1 + y2}, \frac{x1}{\left(128 \cdot 128\right) \cdot 128}, \frac{\frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2}\right)} \cdot 128\right) \cdot 128\right) \cdot 128 \]
Applied rewrites99.6%
\[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\frac{y1}{y2 + y1}, 4.76837158203125 \cdot 10^{-7} \cdot x1, \left(4.76837158203125 \cdot 10^{-7} \cdot y2\right) \cdot \frac{x2}{y2 + y1}\right)} \cdot 128\right) \cdot 128\right) \cdot 128 \]
Add Preprocessing

Alternative 4: 91.8% accurate, 0.6× speedup?

\[\begin{array}{l} t_0 := \left(128 \cdot 128\right) \cdot 128\\ \mathbf{if}\;\left(\left(\frac{\frac{x1}{t\_0} \cdot y1 + \frac{x2}{t\_0} \cdot y2}{y1 + y2} \cdot 128\right) \cdot 128\right) \cdot 128 \leq 0:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(1, 4.76837158203125 \cdot 10^{-7} \cdot x2, \left(4.76837158203125 \cdot 10^{-7} \cdot y1\right) \cdot \frac{x1}{y2 + y1}\right) \cdot 128\right) \cdot 128\right) \cdot 128\\ \mathbf{else}:\\ \;\;\;\;\frac{x1 \cdot y1}{y1 + y2} + \frac{x2 \cdot y2}{y1 + y2}\\ \end{array} \]

(FPCore (x1 y1 x2 y2)
  :precision binary64
  (let* ((t_0 (* (* 128.0 128.0) 128.0)))
  (if (<=
       (*
        (*
         (*
          (/ (+ (* (/ x1 t_0) y1) (* (/ x2 t_0) y2)) (+ y1 y2))
          128.0)
         128.0)
        128.0)
       0.0)
    (*
     (*
      (*
       (fma
        1.0
        (* 4.76837158203125e-7 x2)
        (* (* 4.76837158203125e-7 y1) (/ x1 (+ y2 y1))))
       128.0)
      128.0)
     128.0)
    (+ (/ (* x1 y1) (+ y1 y2)) (/ (* x2 y2) (+ y1 y2))))))

double code(double x1, double y1, double x2, double y2) {
	double t_0 = (128.0 * 128.0) * 128.0;
	double tmp;
	if ((((((((x1 / t_0) * y1) + ((x2 / t_0) * y2)) / (y1 + y2)) * 128.0) * 128.0) * 128.0) <= 0.0) {
		tmp = ((fma(1.0, (4.76837158203125e-7 * x2), ((4.76837158203125e-7 * y1) * (x1 / (y2 + y1)))) * 128.0) * 128.0) * 128.0;
	} else {
		tmp = ((x1 * y1) / (y1 + y2)) + ((x2 * y2) / (y1 + y2));
	}
	return tmp;
}

function code(x1, y1, x2, y2)
	t_0 = Float64(Float64(128.0 * 128.0) * 128.0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(Float64(x1 / t_0) * y1) + Float64(Float64(x2 / t_0) * y2)) / Float64(y1 + y2)) * 128.0) * 128.0) * 128.0) <= 0.0)
		tmp = Float64(Float64(Float64(fma(1.0, Float64(4.76837158203125e-7 * x2), Float64(Float64(4.76837158203125e-7 * y1) * Float64(x1 / Float64(y2 + y1)))) * 128.0) * 128.0) * 128.0);
	else
		tmp = Float64(Float64(Float64(x1 * y1) / Float64(y1 + y2)) + Float64(Float64(x2 * y2) / Float64(y1 + y2)));
	end
	return tmp
end

code[x1_, y1_, x2_, y2_] := Block[{t$95$0 = N[(N[(128.0 * 128.0), $MachinePrecision] * 128.0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(N[(x1 / t$95$0), $MachinePrecision] * y1), $MachinePrecision] + N[(N[(x2 / t$95$0), $MachinePrecision] * y2), $MachinePrecision]), $MachinePrecision] / N[(y1 + y2), $MachinePrecision]), $MachinePrecision] * 128.0), $MachinePrecision] * 128.0), $MachinePrecision] * 128.0), $MachinePrecision], 0.0], N[(N[(N[(N[(1.0 * N[(4.76837158203125e-7 * x2), $MachinePrecision] + N[(N[(4.76837158203125e-7 * y1), $MachinePrecision] * N[(x1 / N[(y2 + y1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 128.0), $MachinePrecision] * 128.0), $MachinePrecision] * 128.0), $MachinePrecision], N[(N[(N[(x1 * y1), $MachinePrecision] / N[(y1 + y2), $MachinePrecision]), $MachinePrecision] + N[(N[(x2 * y2), $MachinePrecision] / N[(y1 + y2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

\begin{array}{l}
t_0 := \left(128 \cdot 128\right) \cdot 128\\
\mathbf{if}\;\left(\left(\frac{\frac{x1}{t\_0} \cdot y1 + \frac{x2}{t\_0} \cdot y2}{y1 + y2} \cdot 128\right) \cdot 128\right) \cdot 128 \leq 0:\\
\;\;\;\;\left(\left(\mathsf{fma}\left(1, 4.76837158203125 \cdot 10^{-7} \cdot x2, \left(4.76837158203125 \cdot 10^{-7} \cdot y1\right) \cdot \frac{x1}{y2 + y1}\right) \cdot 128\right) \cdot 128\right) \cdot 128\\

\mathbf{else}:\\
\;\;\;\;\frac{x1 \cdot y1}{y1 + y2} + \frac{x2 \cdot y2}{y1 + y2}\\


\end{array}

Derivation

Split input into 2 regimes
if (*.f64 (*.f64 (*.f64 (/.f64 (+.f64 (*.f64 (/.f64 x1 (*.f64 (*.f64 #s(literal 128 binary64) #s(literal 128 binary64)) #s(literal 128 binary64))) y1) (*.f64 (/.f64 x2 (*.f64 (*.f64 #s(literal 128 binary64) #s(literal 128 binary64)) #s(literal 128 binary64))) y2)) (+.f64 y1 y2)) #s(literal 128 binary64)) #s(literal 128 binary64)) #s(literal 128 binary64)) < 0.0
1. Initial program 77.8%
  \[\left(\left(\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2} \cdot 128\right) \cdot 128\right) \cdot 128 \]
2. Step-by-step derivation
  1. lift-/.f64N/A
    \[\leadsto \left(\left(\color{blue}{\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2}} \cdot 128\right) \cdot 128\right) \cdot 128 \]
  2. mult-flipN/A
    \[\leadsto \left(\left(\color{blue}{\left(\left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2\right) \cdot \frac{1}{y1 + y2}\right)} \cdot 128\right) \cdot 128\right) \cdot 128 \]
  3. *-commutativeN/A
    \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{y1 + y2} \cdot \left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2\right)\right)} \cdot 128\right) \cdot 128\right) \cdot 128 \]
  4. lift-+.f64N/A
    \[\leadsto \left(\left(\left(\frac{1}{y1 + y2} \cdot \color{blue}{\left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2\right)}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
  5. +-commutativeN/A
    \[\leadsto \left(\left(\left(\frac{1}{y1 + y2} \cdot \color{blue}{\left(\frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2 + \frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1\right)}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
  6. distribute-lft-inN/A
    \[\leadsto \left(\left(\color{blue}{\left(\frac{1}{y1 + y2} \cdot \left(\frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2\right) + \frac{1}{y1 + y2} \cdot \left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1\right)\right)} \cdot 128\right) \cdot 128\right) \cdot 128 \]
  7. *-commutativeN/A
    \[\leadsto \left(\left(\left(\color{blue}{\left(\frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2\right) \cdot \frac{1}{y1 + y2}} + \frac{1}{y1 + y2} \cdot \left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1\right)\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
  8. mult-flipN/A
    \[\leadsto \left(\left(\left(\color{blue}{\frac{\frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2}} + \frac{1}{y1 + y2} \cdot \left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1\right)\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
  9. lift-*.f64N/A
    \[\leadsto \left(\left(\left(\frac{\color{blue}{\frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}}{y1 + y2} + \frac{1}{y1 + y2} \cdot \left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1\right)\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
  10. associate-/l*N/A
    \[\leadsto \left(\left(\left(\color{blue}{\frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot \frac{y2}{y1 + y2}} + \frac{1}{y1 + y2} \cdot \left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1\right)\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
  11. *-commutativeN/A
    \[\leadsto \left(\left(\left(\color{blue}{\frac{y2}{y1 + y2} \cdot \frac{x2}{\left(128 \cdot 128\right) \cdot 128}} + \frac{1}{y1 + y2} \cdot \left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1\right)\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
  12. *-commutativeN/A
    \[\leadsto \left(\left(\left(\frac{y2}{y1 + y2} \cdot \frac{x2}{\left(128 \cdot 128\right) \cdot 128} + \color{blue}{\left(\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1\right) \cdot \frac{1}{y1 + y2}}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
  13. mult-flipN/A
    \[\leadsto \left(\left(\left(\frac{y2}{y1 + y2} \cdot \frac{x2}{\left(128 \cdot 128\right) \cdot 128} + \color{blue}{\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1}{y1 + y2}}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
3. Applied rewrites99.6%
  \[\leadsto \left(\left(\color{blue}{\mathsf{fma}\left(\frac{y2}{y2 + y1}, 4.76837158203125 \cdot 10^{-7} \cdot x2, \left(4.76837158203125 \cdot 10^{-7} \cdot y1\right) \cdot \frac{x1}{y2 + y1}\right)} \cdot 128\right) \cdot 128\right) \cdot 128 \]
4. Taylor expanded in y1 around 0
  \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{1}, 4.76837158203125 \cdot 10^{-7} \cdot x2, \left(4.76837158203125 \cdot 10^{-7} \cdot y1\right) \cdot \frac{x1}{y2 + y1}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
5. Step-by-step derivation
6. Applied rewrites74.8%
  \[\leadsto \left(\left(\mathsf{fma}\left(\color{blue}{1}, 4.76837158203125 \cdot 10^{-7} \cdot x2, \left(4.76837158203125 \cdot 10^{-7} \cdot y1\right) \cdot \frac{x1}{y2 + y1}\right) \cdot 128\right) \cdot 128\right) \cdot 128 \]
if 0.0 < (*.f64 (*.f64 (*.f64 (/.f64 (+.f64 (*.f64 (/.f64 x1 (*.f64 (*.f64 #s(literal 128 binary64) #s(literal 128 binary64)) #s(literal 128 binary64))) y1) (*.f64 (/.f64 x2 (*.f64 (*.f64 #s(literal 128 binary64) #s(literal 128 binary64)) #s(literal 128 binary64))) y2)) (+.f64 y1 y2)) #s(literal 128 binary64)) #s(literal 128 binary64)) #s(literal 128 binary64))
1. Initial program 77.8%
  \[\left(\left(\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2} \cdot 128\right) \cdot 128\right) \cdot 128 \]
2. Taylor expanded in x1 around 0
  \[\leadsto \color{blue}{\frac{x1 \cdot y1}{y1 + y2} + \frac{x2 \cdot y2}{y1 + y2}} \]
3. Step-by-step derivation
  1. lower-+.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{y1 + y2} + \color{blue}{\frac{x2 \cdot y2}{y1 + y2}} \]
  2. lower-/.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{y1 + y2} + \frac{\color{blue}{x2 \cdot y2}}{y1 + y2} \]
  3. lower-*.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{y1 + y2} + \frac{\color{blue}{x2} \cdot y2}{y1 + y2} \]
  4. lower-+.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{y1 + y2} + \frac{x2 \cdot \color{blue}{y2}}{y1 + y2} \]
  5. lower-/.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{y1 + y2} + \frac{x2 \cdot y2}{\color{blue}{y1 + y2}} \]
  6. lower-*.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{y1 + y2} + \frac{x2 \cdot y2}{\color{blue}{y1} + y2} \]
  7. lower-+.f6479.5%
    \[\leadsto \frac{x1 \cdot y1}{y1 + y2} + \frac{x2 \cdot y2}{y1 + \color{blue}{y2}} \]
4. Applied rewrites79.5%
  \[\leadsto \color{blue}{\frac{x1 \cdot y1}{y1 + y2} + \frac{x2 \cdot y2}{y1 + y2}} \]
Recombined 2 regimes into one program.
Add Preprocessing

Alternative 5: 87.7% accurate, 0.6× speedup?

\[\begin{array}{l} t_0 := \left(128 \cdot 128\right) \cdot 128\\ \mathbf{if}\;\left(\left(\frac{\frac{x1}{t\_0} \cdot y1 + \frac{x2}{t\_0} \cdot y2}{y1 + y2} \cdot 128\right) \cdot 128\right) \cdot 128 \leq 2 \cdot 10^{-265}:\\ \;\;\;\;\frac{y1}{y1 + y2} \cdot x1\\ \mathbf{else}:\\ \;\;\;\;\frac{x1 \cdot y1}{y1 + y2} + \frac{x2 \cdot y2}{y1 + y2}\\ \end{array} \]

(FPCore (x1 y1 x2 y2)
  :precision binary64
  (let* ((t_0 (* (* 128.0 128.0) 128.0)))
  (if (<=
       (*
        (*
         (*
          (/ (+ (* (/ x1 t_0) y1) (* (/ x2 t_0) y2)) (+ y1 y2))
          128.0)
         128.0)
        128.0)
       2e-265)
    (* (/ y1 (+ y1 y2)) x1)
    (+ (/ (* x1 y1) (+ y1 y2)) (/ (* x2 y2) (+ y1 y2))))))

double code(double x1, double y1, double x2, double y2) {
	double t_0 = (128.0 * 128.0) * 128.0;
	double tmp;
	if ((((((((x1 / t_0) * y1) + ((x2 / t_0) * y2)) / (y1 + y2)) * 128.0) * 128.0) * 128.0) <= 2e-265) {
		tmp = (y1 / (y1 + y2)) * x1;
	} else {
		tmp = ((x1 * y1) / (y1 + y2)) + ((x2 * y2) / (y1 + y2));
	}
	return tmp;
}

real(8) function code(x1, y1, x2, y2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: y1
    real(8), intent (in) :: x2
    real(8), intent (in) :: y2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (128.0d0 * 128.0d0) * 128.0d0
    if ((((((((x1 / t_0) * y1) + ((x2 / t_0) * y2)) / (y1 + y2)) * 128.0d0) * 128.0d0) * 128.0d0) <= 2d-265) then
        tmp = (y1 / (y1 + y2)) * x1
    else
        tmp = ((x1 * y1) / (y1 + y2)) + ((x2 * y2) / (y1 + y2))
    end if
    code = tmp
end function

public static double code(double x1, double y1, double x2, double y2) {
	double t_0 = (128.0 * 128.0) * 128.0;
	double tmp;
	if ((((((((x1 / t_0) * y1) + ((x2 / t_0) * y2)) / (y1 + y2)) * 128.0) * 128.0) * 128.0) <= 2e-265) {
		tmp = (y1 / (y1 + y2)) * x1;
	} else {
		tmp = ((x1 * y1) / (y1 + y2)) + ((x2 * y2) / (y1 + y2));
	}
	return tmp;
}

def code(x1, y1, x2, y2):
	t_0 = (128.0 * 128.0) * 128.0
	tmp = 0
	if (((((((x1 / t_0) * y1) + ((x2 / t_0) * y2)) / (y1 + y2)) * 128.0) * 128.0) * 128.0) <= 2e-265:
		tmp = (y1 / (y1 + y2)) * x1
	else:
		tmp = ((x1 * y1) / (y1 + y2)) + ((x2 * y2) / (y1 + y2))
	return tmp

function code(x1, y1, x2, y2)
	t_0 = Float64(Float64(128.0 * 128.0) * 128.0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(Float64(x1 / t_0) * y1) + Float64(Float64(x2 / t_0) * y2)) / Float64(y1 + y2)) * 128.0) * 128.0) * 128.0) <= 2e-265)
		tmp = Float64(Float64(y1 / Float64(y1 + y2)) * x1);
	else
		tmp = Float64(Float64(Float64(x1 * y1) / Float64(y1 + y2)) + Float64(Float64(x2 * y2) / Float64(y1 + y2)));
	end
	return tmp
end

function tmp_2 = code(x1, y1, x2, y2)
	t_0 = (128.0 * 128.0) * 128.0;
	tmp = 0.0;
	if ((((((((x1 / t_0) * y1) + ((x2 / t_0) * y2)) / (y1 + y2)) * 128.0) * 128.0) * 128.0) <= 2e-265)
		tmp = (y1 / (y1 + y2)) * x1;
	else
		tmp = ((x1 * y1) / (y1 + y2)) + ((x2 * y2) / (y1 + y2));
	end
	tmp_2 = tmp;
end

code[x1_, y1_, x2_, y2_] := Block[{t$95$0 = N[(N[(128.0 * 128.0), $MachinePrecision] * 128.0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(N[(x1 / t$95$0), $MachinePrecision] * y1), $MachinePrecision] + N[(N[(x2 / t$95$0), $MachinePrecision] * y2), $MachinePrecision]), $MachinePrecision] / N[(y1 + y2), $MachinePrecision]), $MachinePrecision] * 128.0), $MachinePrecision] * 128.0), $MachinePrecision] * 128.0), $MachinePrecision], 2e-265], N[(N[(y1 / N[(y1 + y2), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision], N[(N[(N[(x1 * y1), $MachinePrecision] / N[(y1 + y2), $MachinePrecision]), $MachinePrecision] + N[(N[(x2 * y2), $MachinePrecision] / N[(y1 + y2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

\begin{array}{l}
t_0 := \left(128 \cdot 128\right) \cdot 128\\
\mathbf{if}\;\left(\left(\frac{\frac{x1}{t\_0} \cdot y1 + \frac{x2}{t\_0} \cdot y2}{y1 + y2} \cdot 128\right) \cdot 128\right) \cdot 128 \leq 2 \cdot 10^{-265}:\\
\;\;\;\;\frac{y1}{y1 + y2} \cdot x1\\

\mathbf{else}:\\
\;\;\;\;\frac{x1 \cdot y1}{y1 + y2} + \frac{x2 \cdot y2}{y1 + y2}\\


\end{array}

Derivation

Split input into 2 regimes
if (*.f64 (*.f64 (*.f64 (/.f64 (+.f64 (*.f64 (/.f64 x1 (*.f64 (*.f64 #s(literal 128 binary64) #s(literal 128 binary64)) #s(literal 128 binary64))) y1) (*.f64 (/.f64 x2 (*.f64 (*.f64 #s(literal 128 binary64) #s(literal 128 binary64)) #s(literal 128 binary64))) y2)) (+.f64 y1 y2)) #s(literal 128 binary64)) #s(literal 128 binary64)) #s(literal 128 binary64)) < 2e-265
1. Initial program 77.8%
  \[\left(\left(\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2} \cdot 128\right) \cdot 128\right) \cdot 128 \]
2. Taylor expanded in x1 around inf
  \[\leadsto \color{blue}{\frac{x1 \cdot y1}{y1 + y2}} \]
3. Step-by-step derivation
  1. lower-/.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{\color{blue}{y1 + y2}} \]
  2. lower-*.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{\color{blue}{y1} + y2} \]
  3. lower-+.f6442.1%
    \[\leadsto \frac{x1 \cdot y1}{y1 + \color{blue}{y2}} \]
4. Applied rewrites42.1%
  \[\leadsto \color{blue}{\frac{x1 \cdot y1}{y1 + y2}} \]
5. Step-by-step derivation
  1. lift-/.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{\color{blue}{y1 + y2}} \]
  2. lift-*.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{\color{blue}{y1} + y2} \]
  3. associate-/l*N/A
    \[\leadsto x1 \cdot \color{blue}{\frac{y1}{y1 + y2}} \]
  4. lift-/.f64N/A
    \[\leadsto x1 \cdot \frac{y1}{\color{blue}{y1 + y2}} \]
  5. *-commutativeN/A
    \[\leadsto \frac{y1}{y1 + y2} \cdot \color{blue}{x1} \]
  6. lower-*.f6451.6%
    \[\leadsto \frac{y1}{y1 + y2} \cdot \color{blue}{x1} \]
6. Applied rewrites51.6%
  \[\leadsto \frac{y1}{y1 + y2} \cdot \color{blue}{x1} \]
if 2e-265 < (*.f64 (*.f64 (*.f64 (/.f64 (+.f64 (*.f64 (/.f64 x1 (*.f64 (*.f64 #s(literal 128 binary64) #s(literal 128 binary64)) #s(literal 128 binary64))) y1) (*.f64 (/.f64 x2 (*.f64 (*.f64 #s(literal 128 binary64) #s(literal 128 binary64)) #s(literal 128 binary64))) y2)) (+.f64 y1 y2)) #s(literal 128 binary64)) #s(literal 128 binary64)) #s(literal 128 binary64))
1. Initial program 77.8%
  \[\left(\left(\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2} \cdot 128\right) \cdot 128\right) \cdot 128 \]
2. Taylor expanded in x1 around 0
  \[\leadsto \color{blue}{\frac{x1 \cdot y1}{y1 + y2} + \frac{x2 \cdot y2}{y1 + y2}} \]
3. Step-by-step derivation
  1. lower-+.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{y1 + y2} + \color{blue}{\frac{x2 \cdot y2}{y1 + y2}} \]
  2. lower-/.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{y1 + y2} + \frac{\color{blue}{x2 \cdot y2}}{y1 + y2} \]
  3. lower-*.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{y1 + y2} + \frac{\color{blue}{x2} \cdot y2}{y1 + y2} \]
  4. lower-+.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{y1 + y2} + \frac{x2 \cdot \color{blue}{y2}}{y1 + y2} \]
  5. lower-/.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{y1 + y2} + \frac{x2 \cdot y2}{\color{blue}{y1 + y2}} \]
  6. lower-*.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{y1 + y2} + \frac{x2 \cdot y2}{\color{blue}{y1} + y2} \]
  7. lower-+.f6479.5%
    \[\leadsto \frac{x1 \cdot y1}{y1 + y2} + \frac{x2 \cdot y2}{y1 + \color{blue}{y2}} \]
4. Applied rewrites79.5%
  \[\leadsto \color{blue}{\frac{x1 \cdot y1}{y1 + y2} + \frac{x2 \cdot y2}{y1 + y2}} \]
Recombined 2 regimes into one program.
Add Preprocessing

Alternative 6: 87.7% accurate, 0.7× speedup?

\[\begin{array}{l} t_0 := \left(128 \cdot 128\right) \cdot 128\\ \mathbf{if}\;\left(\left(\frac{\frac{x1}{t\_0} \cdot y1 + \frac{x2}{t\_0} \cdot y2}{y1 + y2} \cdot 128\right) \cdot 128\right) \cdot 128 \leq 2 \cdot 10^{-265}:\\ \;\;\;\;\frac{y1}{y1 + y2} \cdot x1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x1, y1, x2 \cdot y2\right)}{y1 + y2}\\ \end{array} \]

(FPCore (x1 y1 x2 y2)
  :precision binary64
  (let* ((t_0 (* (* 128.0 128.0) 128.0)))
  (if (<=
       (*
        (*
         (*
          (/ (+ (* (/ x1 t_0) y1) (* (/ x2 t_0) y2)) (+ y1 y2))
          128.0)
         128.0)
        128.0)
       2e-265)
    (* (/ y1 (+ y1 y2)) x1)
    (/ (fma x1 y1 (* x2 y2)) (+ y1 y2)))))

double code(double x1, double y1, double x2, double y2) {
	double t_0 = (128.0 * 128.0) * 128.0;
	double tmp;
	if ((((((((x1 / t_0) * y1) + ((x2 / t_0) * y2)) / (y1 + y2)) * 128.0) * 128.0) * 128.0) <= 2e-265) {
		tmp = (y1 / (y1 + y2)) * x1;
	} else {
		tmp = fma(x1, y1, (x2 * y2)) / (y1 + y2);
	}
	return tmp;
}

function code(x1, y1, x2, y2)
	t_0 = Float64(Float64(128.0 * 128.0) * 128.0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(Float64(x1 / t_0) * y1) + Float64(Float64(x2 / t_0) * y2)) / Float64(y1 + y2)) * 128.0) * 128.0) * 128.0) <= 2e-265)
		tmp = Float64(Float64(y1 / Float64(y1 + y2)) * x1);
	else
		tmp = Float64(fma(x1, y1, Float64(x2 * y2)) / Float64(y1 + y2));
	end
	return tmp
end

code[x1_, y1_, x2_, y2_] := Block[{t$95$0 = N[(N[(128.0 * 128.0), $MachinePrecision] * 128.0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(N[(x1 / t$95$0), $MachinePrecision] * y1), $MachinePrecision] + N[(N[(x2 / t$95$0), $MachinePrecision] * y2), $MachinePrecision]), $MachinePrecision] / N[(y1 + y2), $MachinePrecision]), $MachinePrecision] * 128.0), $MachinePrecision] * 128.0), $MachinePrecision] * 128.0), $MachinePrecision], 2e-265], N[(N[(y1 / N[(y1 + y2), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision], N[(N[(x1 * y1 + N[(x2 * y2), $MachinePrecision]), $MachinePrecision] / N[(y1 + y2), $MachinePrecision]), $MachinePrecision]]]

\begin{array}{l}
t_0 := \left(128 \cdot 128\right) \cdot 128\\
\mathbf{if}\;\left(\left(\frac{\frac{x1}{t\_0} \cdot y1 + \frac{x2}{t\_0} \cdot y2}{y1 + y2} \cdot 128\right) \cdot 128\right) \cdot 128 \leq 2 \cdot 10^{-265}:\\
\;\;\;\;\frac{y1}{y1 + y2} \cdot x1\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(x1, y1, x2 \cdot y2\right)}{y1 + y2}\\


\end{array}

Derivation

Split input into 2 regimes
if (*.f64 (*.f64 (*.f64 (/.f64 (+.f64 (*.f64 (/.f64 x1 (*.f64 (*.f64 #s(literal 128 binary64) #s(literal 128 binary64)) #s(literal 128 binary64))) y1) (*.f64 (/.f64 x2 (*.f64 (*.f64 #s(literal 128 binary64) #s(literal 128 binary64)) #s(literal 128 binary64))) y2)) (+.f64 y1 y2)) #s(literal 128 binary64)) #s(literal 128 binary64)) #s(literal 128 binary64)) < 2e-265
1. Initial program 77.8%
  \[\left(\left(\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2} \cdot 128\right) \cdot 128\right) \cdot 128 \]
2. Taylor expanded in x1 around inf
  \[\leadsto \color{blue}{\frac{x1 \cdot y1}{y1 + y2}} \]
3. Step-by-step derivation
  1. lower-/.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{\color{blue}{y1 + y2}} \]
  2. lower-*.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{\color{blue}{y1} + y2} \]
  3. lower-+.f6442.1%
    \[\leadsto \frac{x1 \cdot y1}{y1 + \color{blue}{y2}} \]
4. Applied rewrites42.1%
  \[\leadsto \color{blue}{\frac{x1 \cdot y1}{y1 + y2}} \]
5. Step-by-step derivation
  1. lift-/.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{\color{blue}{y1 + y2}} \]
  2. lift-*.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{\color{blue}{y1} + y2} \]
  3. associate-/l*N/A
    \[\leadsto x1 \cdot \color{blue}{\frac{y1}{y1 + y2}} \]
  4. lift-/.f64N/A
    \[\leadsto x1 \cdot \frac{y1}{\color{blue}{y1 + y2}} \]
  5. *-commutativeN/A
    \[\leadsto \frac{y1}{y1 + y2} \cdot \color{blue}{x1} \]
  6. lower-*.f6451.6%
    \[\leadsto \frac{y1}{y1 + y2} \cdot \color{blue}{x1} \]
6. Applied rewrites51.6%
  \[\leadsto \frac{y1}{y1 + y2} \cdot \color{blue}{x1} \]
if 2e-265 < (*.f64 (*.f64 (*.f64 (/.f64 (+.f64 (*.f64 (/.f64 x1 (*.f64 (*.f64 #s(literal 128 binary64) #s(literal 128 binary64)) #s(literal 128 binary64))) y1) (*.f64 (/.f64 x2 (*.f64 (*.f64 #s(literal 128 binary64) #s(literal 128 binary64)) #s(literal 128 binary64))) y2)) (+.f64 y1 y2)) #s(literal 128 binary64)) #s(literal 128 binary64)) #s(literal 128 binary64))
1. Initial program 77.8%
  \[\left(\left(\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2} \cdot 128\right) \cdot 128\right) \cdot 128 \]
2. Taylor expanded in x1 around 0
  \[\leadsto \color{blue}{\frac{x1 \cdot y1}{y1 + y2} + \frac{x2 \cdot y2}{y1 + y2}} \]
3. Step-by-step derivation
  1. lower-+.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{y1 + y2} + \color{blue}{\frac{x2 \cdot y2}{y1 + y2}} \]
  2. lower-/.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{y1 + y2} + \frac{\color{blue}{x2 \cdot y2}}{y1 + y2} \]
  3. lower-*.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{y1 + y2} + \frac{\color{blue}{x2} \cdot y2}{y1 + y2} \]
  4. lower-+.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{y1 + y2} + \frac{x2 \cdot \color{blue}{y2}}{y1 + y2} \]
  5. lower-/.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{y1 + y2} + \frac{x2 \cdot y2}{\color{blue}{y1 + y2}} \]
  6. lower-*.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{y1 + y2} + \frac{x2 \cdot y2}{\color{blue}{y1} + y2} \]
  7. lower-+.f6479.5%
    \[\leadsto \frac{x1 \cdot y1}{y1 + y2} + \frac{x2 \cdot y2}{y1 + \color{blue}{y2}} \]
4. Applied rewrites79.5%
  \[\leadsto \color{blue}{\frac{x1 \cdot y1}{y1 + y2} + \frac{x2 \cdot y2}{y1 + y2}} \]
5. Step-by-step derivation
  1. lift-+.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{y1 + y2} + \color{blue}{\frac{x2 \cdot y2}{y1 + y2}} \]
  2. lift-/.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{y1 + y2} + \frac{\color{blue}{x2 \cdot y2}}{y1 + y2} \]
  3. lift-/.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{y1 + y2} + \frac{x2 \cdot y2}{\color{blue}{y1 + y2}} \]
  4. div-add-revN/A
    \[\leadsto \frac{x1 \cdot y1 + x2 \cdot y2}{\color{blue}{y1 + y2}} \]
  5. lower-/.f64N/A
    \[\leadsto \frac{x1 \cdot y1 + x2 \cdot y2}{\color{blue}{y1 + y2}} \]
  6. lift-*.f64N/A
    \[\leadsto \frac{x1 \cdot y1 + x2 \cdot y2}{y1 + y2} \]
  7. lower-fma.f6479.5%
    \[\leadsto \frac{\mathsf{fma}\left(x1, y1, x2 \cdot y2\right)}{\color{blue}{y1} + y2} \]
6. Applied rewrites79.5%
  \[\leadsto \frac{\mathsf{fma}\left(x1, y1, x2 \cdot y2\right)}{\color{blue}{y1 + y2}} \]
Recombined 2 regimes into one program.
Add Preprocessing

Alternative 7: 71.6% accurate, 3.1× speedup?

\[\begin{array}{l} \mathbf{if}\;y2 \leq 9.5 \cdot 10^{-140}:\\ \;\;\;\;x1 + y2 \cdot \frac{x2}{y1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y2}{y1 + y2} \cdot x2\\ \end{array} \]

(FPCore (x1 y1 x2 y2)
  :precision binary64
  (if (<= y2 9.5e-140) (+ x1 (* y2 (/ x2 y1))) (* (/ y2 (+ y1 y2)) x2)))

double code(double x1, double y1, double x2, double y2) {
	double tmp;
	if (y2 <= 9.5e-140) {
		tmp = x1 + (y2 * (x2 / y1));
	} else {
		tmp = (y2 / (y1 + y2)) * x2;
	}
	return tmp;
}

real(8) function code(x1, y1, x2, y2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: y1
    real(8), intent (in) :: x2
    real(8), intent (in) :: y2
    real(8) :: tmp
    if (y2 <= 9.5d-140) then
        tmp = x1 + (y2 * (x2 / y1))
    else
        tmp = (y2 / (y1 + y2)) * x2
    end if
    code = tmp
end function

public static double code(double x1, double y1, double x2, double y2) {
	double tmp;
	if (y2 <= 9.5e-140) {
		tmp = x1 + (y2 * (x2 / y1));
	} else {
		tmp = (y2 / (y1 + y2)) * x2;
	}
	return tmp;
}

def code(x1, y1, x2, y2):
	tmp = 0
	if y2 <= 9.5e-140:
		tmp = x1 + (y2 * (x2 / y1))
	else:
		tmp = (y2 / (y1 + y2)) * x2
	return tmp

function code(x1, y1, x2, y2)
	tmp = 0.0
	if (y2 <= 9.5e-140)
		tmp = Float64(x1 + Float64(y2 * Float64(x2 / y1)));
	else
		tmp = Float64(Float64(y2 / Float64(y1 + y2)) * x2);
	end
	return tmp
end

function tmp_2 = code(x1, y1, x2, y2)
	tmp = 0.0;
	if (y2 <= 9.5e-140)
		tmp = x1 + (y2 * (x2 / y1));
	else
		tmp = (y2 / (y1 + y2)) * x2;
	end
	tmp_2 = tmp;
end

code[x1_, y1_, x2_, y2_] := If[LessEqual[y2, 9.5e-140], N[(x1 + N[(y2 * N[(x2 / y1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y2 / N[(y1 + y2), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision]]

\begin{array}{l}
\mathbf{if}\;y2 \leq 9.5 \cdot 10^{-140}:\\
\;\;\;\;x1 + y2 \cdot \frac{x2}{y1}\\

\mathbf{else}:\\
\;\;\;\;\frac{y2}{y1 + y2} \cdot x2\\


\end{array}

Derivation

Split input into 2 regimes
if y2 < 9.5000000000000002e-140
1. Initial program 77.8%
  \[\left(\left(\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2} \cdot 128\right) \cdot 128\right) \cdot 128 \]
2. Taylor expanded in y1 around inf
  \[\leadsto \color{blue}{x1 + 2097152 \cdot \frac{\frac{1}{2097152} \cdot \left(x2 \cdot y2\right) - \frac{1}{2097152} \cdot \left(x1 \cdot y2\right)}{y1}} \]
3. Step-by-step derivation
  1. metadata-evalN/A
    \[\leadsto x1 + \left(16384 \cdot 128\right) \cdot \frac{\color{blue}{\frac{1}{2097152} \cdot \left(x2 \cdot y2\right) - \frac{1}{2097152} \cdot \left(x1 \cdot y2\right)}}{y1} \]
  2. metadata-evalN/A
    \[\leadsto x1 + \left(\left(128 \cdot 128\right) \cdot 128\right) \cdot \frac{\color{blue}{\frac{1}{2097152} \cdot \left(x2 \cdot y2\right)} - \frac{1}{2097152} \cdot \left(x1 \cdot y2\right)}{y1} \]
  3. lower-+.f64N/A
    \[\leadsto x1 + \color{blue}{\left(\left(128 \cdot 128\right) \cdot 128\right) \cdot \frac{\frac{1}{2097152} \cdot \left(x2 \cdot y2\right) - \frac{1}{2097152} \cdot \left(x1 \cdot y2\right)}{y1}} \]
  4. lower-*.f64N/A
    \[\leadsto x1 + \left(\left(128 \cdot 128\right) \cdot 128\right) \cdot \color{blue}{\frac{\frac{1}{2097152} \cdot \left(x2 \cdot y2\right) - \frac{1}{2097152} \cdot \left(x1 \cdot y2\right)}{y1}} \]
  5. metadata-evalN/A
    \[\leadsto x1 + \left(16384 \cdot 128\right) \cdot \frac{\color{blue}{\frac{1}{2097152} \cdot \left(x2 \cdot y2\right)} - \frac{1}{2097152} \cdot \left(x1 \cdot y2\right)}{y1} \]
  6. metadata-evalN/A
    \[\leadsto x1 + 2097152 \cdot \frac{\color{blue}{\frac{1}{2097152} \cdot \left(x2 \cdot y2\right) - \frac{1}{2097152} \cdot \left(x1 \cdot y2\right)}}{y1} \]
  7. lower-/.f64N/A
    \[\leadsto x1 + 2097152 \cdot \frac{\frac{1}{2097152} \cdot \left(x2 \cdot y2\right) - \frac{1}{2097152} \cdot \left(x1 \cdot y2\right)}{\color{blue}{y1}} \]
4. Applied rewrites47.0%
  \[\leadsto \color{blue}{x1 + 2097152 \cdot \frac{4.76837158203125 \cdot 10^{-7} \cdot \left(x2 \cdot y2\right) - 4.76837158203125 \cdot 10^{-7} \cdot \left(x1 \cdot y2\right)}{y1}} \]
5. Taylor expanded in x1 around 0
  \[\leadsto x1 + \frac{x2 \cdot y2}{\color{blue}{y1}} \]
6. Step-by-step derivation
  1. lower-/.f64N/A
    \[\leadsto x1 + \frac{x2 \cdot y2}{y1} \]
  2. lower-*.f6447.8%
    \[\leadsto x1 + \frac{x2 \cdot y2}{y1} \]
7. Applied rewrites47.8%
  \[\leadsto x1 + \frac{x2 \cdot y2}{\color{blue}{y1}} \]
8. Step-by-step derivation
  1. lift-/.f64N/A
    \[\leadsto x1 + \frac{x2 \cdot y2}{y1} \]
  2. lift-*.f64N/A
    \[\leadsto x1 + \frac{x2 \cdot y2}{y1} \]
  3. *-commutativeN/A
    \[\leadsto x1 + \frac{y2 \cdot x2}{y1} \]
  4. associate-/l*N/A
    \[\leadsto x1 + y2 \cdot \frac{x2}{\color{blue}{y1}} \]
  5. lift-/.f64N/A
    \[\leadsto x1 + y2 \cdot \frac{x2}{y1} \]
  6. lower-*.f6451.1%
    \[\leadsto x1 + y2 \cdot \frac{x2}{\color{blue}{y1}} \]
9. Applied rewrites51.1%
  \[\leadsto x1 + y2 \cdot \frac{x2}{\color{blue}{y1}} \]
if 9.5000000000000002e-140 < y2
1. Initial program 77.8%
  \[\left(\left(\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2} \cdot 128\right) \cdot 128\right) \cdot 128 \]
2. Taylor expanded in x1 around 0
  \[\leadsto \color{blue}{\frac{x2 \cdot y2}{y1 + y2}} \]
3. Step-by-step derivation
  1. lower-/.f64N/A
    \[\leadsto \frac{x2 \cdot y2}{\color{blue}{y1 + y2}} \]
  2. lower-*.f64N/A
    \[\leadsto \frac{x2 \cdot y2}{\color{blue}{y1} + y2} \]
  3. lower-+.f6441.4%
    \[\leadsto \frac{x2 \cdot y2}{y1 + \color{blue}{y2}} \]
4. Applied rewrites41.4%
  \[\leadsto \color{blue}{\frac{x2 \cdot y2}{y1 + y2}} \]
5. Step-by-step derivation
  1. lift-/.f64N/A
    \[\leadsto \frac{x2 \cdot y2}{\color{blue}{y1 + y2}} \]
  2. lift-*.f64N/A
    \[\leadsto \frac{x2 \cdot y2}{\color{blue}{y1} + y2} \]
  3. associate-/l*N/A
    \[\leadsto x2 \cdot \color{blue}{\frac{y2}{y1 + y2}} \]
  4. lift-+.f64N/A
    \[\leadsto x2 \cdot \frac{y2}{y1 + \color{blue}{y2}} \]
  5. +-commutativeN/A
    \[\leadsto x2 \cdot \frac{y2}{y2 + \color{blue}{y1}} \]
  6. lift-+.f64N/A
    \[\leadsto x2 \cdot \frac{y2}{y2 + \color{blue}{y1}} \]
  7. lift-/.f64N/A
    \[\leadsto x2 \cdot \frac{y2}{\color{blue}{y2 + y1}} \]
  8. *-commutativeN/A
    \[\leadsto \frac{y2}{y2 + y1} \cdot \color{blue}{x2} \]
  9. lower-*.f6451.5%
    \[\leadsto \frac{y2}{y2 + y1} \cdot \color{blue}{x2} \]
  10. lift-+.f64N/A
    \[\leadsto \frac{y2}{y2 + y1} \cdot x2 \]
  11. +-commutativeN/A
    \[\leadsto \frac{y2}{y1 + y2} \cdot x2 \]
  12. lift-+.f6451.5%
    \[\leadsto \frac{y2}{y1 + y2} \cdot x2 \]
6. Applied rewrites51.5%
  \[\leadsto \frac{y2}{y1 + y2} \cdot \color{blue}{x2} \]
Recombined 2 regimes into one program.
Add Preprocessing

Alternative 8: 68.2% accurate, 3.1× speedup?

\[\begin{array}{l} \mathbf{if}\;y2 \leq 9.5 \cdot 10^{-140}:\\ \;\;\;\;\frac{y1}{y1 + y2} \cdot x1\\ \mathbf{else}:\\ \;\;\;\;\frac{y2}{y1 + y2} \cdot x2\\ \end{array} \]

(FPCore (x1 y1 x2 y2)
  :precision binary64
  (if (<= y2 9.5e-140) (* (/ y1 (+ y1 y2)) x1) (* (/ y2 (+ y1 y2)) x2)))

double code(double x1, double y1, double x2, double y2) {
	double tmp;
	if (y2 <= 9.5e-140) {
		tmp = (y1 / (y1 + y2)) * x1;
	} else {
		tmp = (y2 / (y1 + y2)) * x2;
	}
	return tmp;
}

real(8) function code(x1, y1, x2, y2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: y1
    real(8), intent (in) :: x2
    real(8), intent (in) :: y2
    real(8) :: tmp
    if (y2 <= 9.5d-140) then
        tmp = (y1 / (y1 + y2)) * x1
    else
        tmp = (y2 / (y1 + y2)) * x2
    end if
    code = tmp
end function

public static double code(double x1, double y1, double x2, double y2) {
	double tmp;
	if (y2 <= 9.5e-140) {
		tmp = (y1 / (y1 + y2)) * x1;
	} else {
		tmp = (y2 / (y1 + y2)) * x2;
	}
	return tmp;
}

def code(x1, y1, x2, y2):
	tmp = 0
	if y2 <= 9.5e-140:
		tmp = (y1 / (y1 + y2)) * x1
	else:
		tmp = (y2 / (y1 + y2)) * x2
	return tmp

function code(x1, y1, x2, y2)
	tmp = 0.0
	if (y2 <= 9.5e-140)
		tmp = Float64(Float64(y1 / Float64(y1 + y2)) * x1);
	else
		tmp = Float64(Float64(y2 / Float64(y1 + y2)) * x2);
	end
	return tmp
end

function tmp_2 = code(x1, y1, x2, y2)
	tmp = 0.0;
	if (y2 <= 9.5e-140)
		tmp = (y1 / (y1 + y2)) * x1;
	else
		tmp = (y2 / (y1 + y2)) * x2;
	end
	tmp_2 = tmp;
end

code[x1_, y1_, x2_, y2_] := If[LessEqual[y2, 9.5e-140], N[(N[(y1 / N[(y1 + y2), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision], N[(N[(y2 / N[(y1 + y2), $MachinePrecision]), $MachinePrecision] * x2), $MachinePrecision]]

\begin{array}{l}
\mathbf{if}\;y2 \leq 9.5 \cdot 10^{-140}:\\
\;\;\;\;\frac{y1}{y1 + y2} \cdot x1\\

\mathbf{else}:\\
\;\;\;\;\frac{y2}{y1 + y2} \cdot x2\\


\end{array}

Derivation

Split input into 2 regimes
if y2 < 9.5000000000000002e-140
1. Initial program 77.8%
  \[\left(\left(\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2} \cdot 128\right) \cdot 128\right) \cdot 128 \]
2. Taylor expanded in x1 around inf
  \[\leadsto \color{blue}{\frac{x1 \cdot y1}{y1 + y2}} \]
3. Step-by-step derivation
  1. lower-/.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{\color{blue}{y1 + y2}} \]
  2. lower-*.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{\color{blue}{y1} + y2} \]
  3. lower-+.f6442.1%
    \[\leadsto \frac{x1 \cdot y1}{y1 + \color{blue}{y2}} \]
4. Applied rewrites42.1%
  \[\leadsto \color{blue}{\frac{x1 \cdot y1}{y1 + y2}} \]
5. Step-by-step derivation
  1. lift-/.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{\color{blue}{y1 + y2}} \]
  2. lift-*.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{\color{blue}{y1} + y2} \]
  3. associate-/l*N/A
    \[\leadsto x1 \cdot \color{blue}{\frac{y1}{y1 + y2}} \]
  4. lift-/.f64N/A
    \[\leadsto x1 \cdot \frac{y1}{\color{blue}{y1 + y2}} \]
  5. *-commutativeN/A
    \[\leadsto \frac{y1}{y1 + y2} \cdot \color{blue}{x1} \]
  6. lower-*.f6451.6%
    \[\leadsto \frac{y1}{y1 + y2} \cdot \color{blue}{x1} \]
6. Applied rewrites51.6%
  \[\leadsto \frac{y1}{y1 + y2} \cdot \color{blue}{x1} \]
if 9.5000000000000002e-140 < y2
1. Initial program 77.8%
  \[\left(\left(\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2} \cdot 128\right) \cdot 128\right) \cdot 128 \]
2. Taylor expanded in x1 around 0
  \[\leadsto \color{blue}{\frac{x2 \cdot y2}{y1 + y2}} \]
3. Step-by-step derivation
  1. lower-/.f64N/A
    \[\leadsto \frac{x2 \cdot y2}{\color{blue}{y1 + y2}} \]
  2. lower-*.f64N/A
    \[\leadsto \frac{x2 \cdot y2}{\color{blue}{y1} + y2} \]
  3. lower-+.f6441.4%
    \[\leadsto \frac{x2 \cdot y2}{y1 + \color{blue}{y2}} \]
4. Applied rewrites41.4%
  \[\leadsto \color{blue}{\frac{x2 \cdot y2}{y1 + y2}} \]
5. Step-by-step derivation
  1. lift-/.f64N/A
    \[\leadsto \frac{x2 \cdot y2}{\color{blue}{y1 + y2}} \]
  2. lift-*.f64N/A
    \[\leadsto \frac{x2 \cdot y2}{\color{blue}{y1} + y2} \]
  3. associate-/l*N/A
    \[\leadsto x2 \cdot \color{blue}{\frac{y2}{y1 + y2}} \]
  4. lift-+.f64N/A
    \[\leadsto x2 \cdot \frac{y2}{y1 + \color{blue}{y2}} \]
  5. +-commutativeN/A
    \[\leadsto x2 \cdot \frac{y2}{y2 + \color{blue}{y1}} \]
  6. lift-+.f64N/A
    \[\leadsto x2 \cdot \frac{y2}{y2 + \color{blue}{y1}} \]
  7. lift-/.f64N/A
    \[\leadsto x2 \cdot \frac{y2}{\color{blue}{y2 + y1}} \]
  8. *-commutativeN/A
    \[\leadsto \frac{y2}{y2 + y1} \cdot \color{blue}{x2} \]
  9. lower-*.f6451.5%
    \[\leadsto \frac{y2}{y2 + y1} \cdot \color{blue}{x2} \]
  10. lift-+.f64N/A
    \[\leadsto \frac{y2}{y2 + y1} \cdot x2 \]
  11. +-commutativeN/A
    \[\leadsto \frac{y2}{y1 + y2} \cdot x2 \]
  12. lift-+.f6451.5%
    \[\leadsto \frac{y2}{y1 + y2} \cdot x2 \]
6. Applied rewrites51.5%
  \[\leadsto \frac{y2}{y1 + y2} \cdot \color{blue}{x2} \]
Recombined 2 regimes into one program.
Add Preprocessing

Alternative 9: 68.1% accurate, 3.1× speedup?

\[\begin{array}{l} \mathbf{if}\;y2 \leq 9.5 \cdot 10^{-140}:\\ \;\;\;\;\frac{y1}{y1 + y2} \cdot x1\\ \mathbf{else}:\\ \;\;\;\;y2 \cdot \frac{x2}{y1 + y2}\\ \end{array} \]

(FPCore (x1 y1 x2 y2)
  :precision binary64
  (if (<= y2 9.5e-140) (* (/ y1 (+ y1 y2)) x1) (* y2 (/ x2 (+ y1 y2)))))

double code(double x1, double y1, double x2, double y2) {
	double tmp;
	if (y2 <= 9.5e-140) {
		tmp = (y1 / (y1 + y2)) * x1;
	} else {
		tmp = y2 * (x2 / (y1 + y2));
	}
	return tmp;
}

real(8) function code(x1, y1, x2, y2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: y1
    real(8), intent (in) :: x2
    real(8), intent (in) :: y2
    real(8) :: tmp
    if (y2 <= 9.5d-140) then
        tmp = (y1 / (y1 + y2)) * x1
    else
        tmp = y2 * (x2 / (y1 + y2))
    end if
    code = tmp
end function

public static double code(double x1, double y1, double x2, double y2) {
	double tmp;
	if (y2 <= 9.5e-140) {
		tmp = (y1 / (y1 + y2)) * x1;
	} else {
		tmp = y2 * (x2 / (y1 + y2));
	}
	return tmp;
}

def code(x1, y1, x2, y2):
	tmp = 0
	if y2 <= 9.5e-140:
		tmp = (y1 / (y1 + y2)) * x1
	else:
		tmp = y2 * (x2 / (y1 + y2))
	return tmp

function code(x1, y1, x2, y2)
	tmp = 0.0
	if (y2 <= 9.5e-140)
		tmp = Float64(Float64(y1 / Float64(y1 + y2)) * x1);
	else
		tmp = Float64(y2 * Float64(x2 / Float64(y1 + y2)));
	end
	return tmp
end

function tmp_2 = code(x1, y1, x2, y2)
	tmp = 0.0;
	if (y2 <= 9.5e-140)
		tmp = (y1 / (y1 + y2)) * x1;
	else
		tmp = y2 * (x2 / (y1 + y2));
	end
	tmp_2 = tmp;
end

code[x1_, y1_, x2_, y2_] := If[LessEqual[y2, 9.5e-140], N[(N[(y1 / N[(y1 + y2), $MachinePrecision]), $MachinePrecision] * x1), $MachinePrecision], N[(y2 * N[(x2 / N[(y1 + y2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}
\mathbf{if}\;y2 \leq 9.5 \cdot 10^{-140}:\\
\;\;\;\;\frac{y1}{y1 + y2} \cdot x1\\

\mathbf{else}:\\
\;\;\;\;y2 \cdot \frac{x2}{y1 + y2}\\


\end{array}

Derivation

Split input into 2 regimes
if y2 < 9.5000000000000002e-140
1. Initial program 77.8%
  \[\left(\left(\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2} \cdot 128\right) \cdot 128\right) \cdot 128 \]
2. Taylor expanded in x1 around inf
  \[\leadsto \color{blue}{\frac{x1 \cdot y1}{y1 + y2}} \]
3. Step-by-step derivation
  1. lower-/.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{\color{blue}{y1 + y2}} \]
  2. lower-*.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{\color{blue}{y1} + y2} \]
  3. lower-+.f6442.1%
    \[\leadsto \frac{x1 \cdot y1}{y1 + \color{blue}{y2}} \]
4. Applied rewrites42.1%
  \[\leadsto \color{blue}{\frac{x1 \cdot y1}{y1 + y2}} \]
5. Step-by-step derivation
  1. lift-/.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{\color{blue}{y1 + y2}} \]
  2. lift-*.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{\color{blue}{y1} + y2} \]
  3. associate-/l*N/A
    \[\leadsto x1 \cdot \color{blue}{\frac{y1}{y1 + y2}} \]
  4. lift-/.f64N/A
    \[\leadsto x1 \cdot \frac{y1}{\color{blue}{y1 + y2}} \]
  5. *-commutativeN/A
    \[\leadsto \frac{y1}{y1 + y2} \cdot \color{blue}{x1} \]
  6. lower-*.f6451.6%
    \[\leadsto \frac{y1}{y1 + y2} \cdot \color{blue}{x1} \]
6. Applied rewrites51.6%
  \[\leadsto \frac{y1}{y1 + y2} \cdot \color{blue}{x1} \]
if 9.5000000000000002e-140 < y2
1. Initial program 77.8%
  \[\left(\left(\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2} \cdot 128\right) \cdot 128\right) \cdot 128 \]
2. Taylor expanded in x1 around 0
  \[\leadsto \color{blue}{\frac{x2 \cdot y2}{y1 + y2}} \]
3. Step-by-step derivation
  1. lower-/.f64N/A
    \[\leadsto \frac{x2 \cdot y2}{\color{blue}{y1 + y2}} \]
  2. lower-*.f64N/A
    \[\leadsto \frac{x2 \cdot y2}{\color{blue}{y1} + y2} \]
  3. lower-+.f6441.4%
    \[\leadsto \frac{x2 \cdot y2}{y1 + \color{blue}{y2}} \]
4. Applied rewrites41.4%
  \[\leadsto \color{blue}{\frac{x2 \cdot y2}{y1 + y2}} \]
5. Step-by-step derivation
  1. lift-/.f64N/A
    \[\leadsto \frac{x2 \cdot y2}{\color{blue}{y1 + y2}} \]
  2. lift-*.f64N/A
    \[\leadsto \frac{x2 \cdot y2}{\color{blue}{y1} + y2} \]
  3. *-commutativeN/A
    \[\leadsto \frac{y2 \cdot x2}{\color{blue}{y1} + y2} \]
  4. associate-/l*N/A
    \[\leadsto y2 \cdot \color{blue}{\frac{x2}{y1 + y2}} \]
  5. lower-*.f64N/A
    \[\leadsto y2 \cdot \color{blue}{\frac{x2}{y1 + y2}} \]
  6. lower-/.f6451.4%
    \[\leadsto y2 \cdot \frac{x2}{\color{blue}{y1 + y2}} \]
6. Applied rewrites51.4%
  \[\leadsto y2 \cdot \color{blue}{\frac{x2}{y1 + y2}} \]
Recombined 2 regimes into one program.
Add Preprocessing

Alternative 10: 68.1% accurate, 3.1× speedup?

\[\begin{array}{l} \mathbf{if}\;y2 \leq 9.5 \cdot 10^{-140}:\\ \;\;\;\;y1 \cdot \frac{x1}{y1 + y2}\\ \mathbf{else}:\\ \;\;\;\;y2 \cdot \frac{x2}{y1 + y2}\\ \end{array} \]

(FPCore (x1 y1 x2 y2)
  :precision binary64
  (if (<= y2 9.5e-140) (* y1 (/ x1 (+ y1 y2))) (* y2 (/ x2 (+ y1 y2)))))

double code(double x1, double y1, double x2, double y2) {
	double tmp;
	if (y2 <= 9.5e-140) {
		tmp = y1 * (x1 / (y1 + y2));
	} else {
		tmp = y2 * (x2 / (y1 + y2));
	}
	return tmp;
}

real(8) function code(x1, y1, x2, y2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: y1
    real(8), intent (in) :: x2
    real(8), intent (in) :: y2
    real(8) :: tmp
    if (y2 <= 9.5d-140) then
        tmp = y1 * (x1 / (y1 + y2))
    else
        tmp = y2 * (x2 / (y1 + y2))
    end if
    code = tmp
end function

public static double code(double x1, double y1, double x2, double y2) {
	double tmp;
	if (y2 <= 9.5e-140) {
		tmp = y1 * (x1 / (y1 + y2));
	} else {
		tmp = y2 * (x2 / (y1 + y2));
	}
	return tmp;
}

def code(x1, y1, x2, y2):
	tmp = 0
	if y2 <= 9.5e-140:
		tmp = y1 * (x1 / (y1 + y2))
	else:
		tmp = y2 * (x2 / (y1 + y2))
	return tmp

function code(x1, y1, x2, y2)
	tmp = 0.0
	if (y2 <= 9.5e-140)
		tmp = Float64(y1 * Float64(x1 / Float64(y1 + y2)));
	else
		tmp = Float64(y2 * Float64(x2 / Float64(y1 + y2)));
	end
	return tmp
end

function tmp_2 = code(x1, y1, x2, y2)
	tmp = 0.0;
	if (y2 <= 9.5e-140)
		tmp = y1 * (x1 / (y1 + y2));
	else
		tmp = y2 * (x2 / (y1 + y2));
	end
	tmp_2 = tmp;
end

code[x1_, y1_, x2_, y2_] := If[LessEqual[y2, 9.5e-140], N[(y1 * N[(x1 / N[(y1 + y2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y2 * N[(x2 / N[(y1 + y2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}
\mathbf{if}\;y2 \leq 9.5 \cdot 10^{-140}:\\
\;\;\;\;y1 \cdot \frac{x1}{y1 + y2}\\

\mathbf{else}:\\
\;\;\;\;y2 \cdot \frac{x2}{y1 + y2}\\


\end{array}

Derivation

Split input into 2 regimes
if y2 < 9.5000000000000002e-140
1. Initial program 77.8%
  \[\left(\left(\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2} \cdot 128\right) \cdot 128\right) \cdot 128 \]
2. Taylor expanded in x1 around inf
  \[\leadsto \color{blue}{\frac{x1 \cdot y1}{y1 + y2}} \]
3. Step-by-step derivation
  1. lower-/.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{\color{blue}{y1 + y2}} \]
  2. lower-*.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{\color{blue}{y1} + y2} \]
  3. lower-+.f6442.1%
    \[\leadsto \frac{x1 \cdot y1}{y1 + \color{blue}{y2}} \]
4. Applied rewrites42.1%
  \[\leadsto \color{blue}{\frac{x1 \cdot y1}{y1 + y2}} \]
5. Step-by-step derivation
  1. lift-/.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{\color{blue}{y1 + y2}} \]
  2. lift-*.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{\color{blue}{y1} + y2} \]
  3. *-commutativeN/A
    \[\leadsto \frac{y1 \cdot x1}{\color{blue}{y1} + y2} \]
  4. associate-/l*N/A
    \[\leadsto y1 \cdot \color{blue}{\frac{x1}{y1 + y2}} \]
  5. lower-*.f64N/A
    \[\leadsto y1 \cdot \color{blue}{\frac{x1}{y1 + y2}} \]
  6. lower-/.f6451.5%
    \[\leadsto y1 \cdot \frac{x1}{\color{blue}{y1 + y2}} \]
6. Applied rewrites51.5%
  \[\leadsto y1 \cdot \color{blue}{\frac{x1}{y1 + y2}} \]
if 9.5000000000000002e-140 < y2
1. Initial program 77.8%
  \[\left(\left(\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2} \cdot 128\right) \cdot 128\right) \cdot 128 \]
2. Taylor expanded in x1 around 0
  \[\leadsto \color{blue}{\frac{x2 \cdot y2}{y1 + y2}} \]
3. Step-by-step derivation
  1. lower-/.f64N/A
    \[\leadsto \frac{x2 \cdot y2}{\color{blue}{y1 + y2}} \]
  2. lower-*.f64N/A
    \[\leadsto \frac{x2 \cdot y2}{\color{blue}{y1} + y2} \]
  3. lower-+.f6441.4%
    \[\leadsto \frac{x2 \cdot y2}{y1 + \color{blue}{y2}} \]
4. Applied rewrites41.4%
  \[\leadsto \color{blue}{\frac{x2 \cdot y2}{y1 + y2}} \]
5. Step-by-step derivation
  1. lift-/.f64N/A
    \[\leadsto \frac{x2 \cdot y2}{\color{blue}{y1 + y2}} \]
  2. lift-*.f64N/A
    \[\leadsto \frac{x2 \cdot y2}{\color{blue}{y1} + y2} \]
  3. *-commutativeN/A
    \[\leadsto \frac{y2 \cdot x2}{\color{blue}{y1} + y2} \]
  4. associate-/l*N/A
    \[\leadsto y2 \cdot \color{blue}{\frac{x2}{y1 + y2}} \]
  5. lower-*.f64N/A
    \[\leadsto y2 \cdot \color{blue}{\frac{x2}{y1 + y2}} \]
  6. lower-/.f6451.4%
    \[\leadsto y2 \cdot \frac{x2}{\color{blue}{y1 + y2}} \]
6. Applied rewrites51.4%
  \[\leadsto y2 \cdot \color{blue}{\frac{x2}{y1 + y2}} \]
Recombined 2 regimes into one program.
Add Preprocessing

Alternative 11: 64.4% accurate, 3.1× speedup?

\[\begin{array}{l} \mathbf{if}\;y2 \leq 4 \cdot 10^{-140}:\\ \;\;\;\;y1 \cdot \frac{x1}{y1 + y2}\\ \mathbf{else}:\\ \;\;\;\;x2\\ \end{array} \]

(FPCore (x1 y1 x2 y2)
  :precision binary64
  (if (<= y2 4e-140) (* y1 (/ x1 (+ y1 y2))) x2))

double code(double x1, double y1, double x2, double y2) {
	double tmp;
	if (y2 <= 4e-140) {
		tmp = y1 * (x1 / (y1 + y2));
	} else {
		tmp = x2;
	}
	return tmp;
}

real(8) function code(x1, y1, x2, y2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: y1
    real(8), intent (in) :: x2
    real(8), intent (in) :: y2
    real(8) :: tmp
    if (y2 <= 4d-140) then
        tmp = y1 * (x1 / (y1 + y2))
    else
        tmp = x2
    end if
    code = tmp
end function

public static double code(double x1, double y1, double x2, double y2) {
	double tmp;
	if (y2 <= 4e-140) {
		tmp = y1 * (x1 / (y1 + y2));
	} else {
		tmp = x2;
	}
	return tmp;
}

def code(x1, y1, x2, y2):
	tmp = 0
	if y2 <= 4e-140:
		tmp = y1 * (x1 / (y1 + y2))
	else:
		tmp = x2
	return tmp

function code(x1, y1, x2, y2)
	tmp = 0.0
	if (y2 <= 4e-140)
		tmp = Float64(y1 * Float64(x1 / Float64(y1 + y2)));
	else
		tmp = x2;
	end
	return tmp
end

function tmp_2 = code(x1, y1, x2, y2)
	tmp = 0.0;
	if (y2 <= 4e-140)
		tmp = y1 * (x1 / (y1 + y2));
	else
		tmp = x2;
	end
	tmp_2 = tmp;
end

code[x1_, y1_, x2_, y2_] := If[LessEqual[y2, 4e-140], N[(y1 * N[(x1 / N[(y1 + y2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x2]

\begin{array}{l}
\mathbf{if}\;y2 \leq 4 \cdot 10^{-140}:\\
\;\;\;\;y1 \cdot \frac{x1}{y1 + y2}\\

\mathbf{else}:\\
\;\;\;\;x2\\


\end{array}

Derivation

Split input into 2 regimes
if y2 < 3.9999999999999999e-140
1. Initial program 77.8%
  \[\left(\left(\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2} \cdot 128\right) \cdot 128\right) \cdot 128 \]
2. Taylor expanded in x1 around inf
  \[\leadsto \color{blue}{\frac{x1 \cdot y1}{y1 + y2}} \]
3. Step-by-step derivation
  1. lower-/.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{\color{blue}{y1 + y2}} \]
  2. lower-*.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{\color{blue}{y1} + y2} \]
  3. lower-+.f6442.1%
    \[\leadsto \frac{x1 \cdot y1}{y1 + \color{blue}{y2}} \]
4. Applied rewrites42.1%
  \[\leadsto \color{blue}{\frac{x1 \cdot y1}{y1 + y2}} \]
5. Step-by-step derivation
  1. lift-/.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{\color{blue}{y1 + y2}} \]
  2. lift-*.f64N/A
    \[\leadsto \frac{x1 \cdot y1}{\color{blue}{y1} + y2} \]
  3. *-commutativeN/A
    \[\leadsto \frac{y1 \cdot x1}{\color{blue}{y1} + y2} \]
  4. associate-/l*N/A
    \[\leadsto y1 \cdot \color{blue}{\frac{x1}{y1 + y2}} \]
  5. lower-*.f64N/A
    \[\leadsto y1 \cdot \color{blue}{\frac{x1}{y1 + y2}} \]
  6. lower-/.f6451.5%
    \[\leadsto y1 \cdot \frac{x1}{\color{blue}{y1 + y2}} \]
6. Applied rewrites51.5%
  \[\leadsto y1 \cdot \color{blue}{\frac{x1}{y1 + y2}} \]
if 3.9999999999999999e-140 < y2
1. Initial program 77.8%
  \[\left(\left(\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2} \cdot 128\right) \cdot 128\right) \cdot 128 \]
2. Taylor expanded in y1 around 0
  \[\leadsto \color{blue}{x2} \]
3. Step-by-step derivation
4. Applied rewrites40.4%
  \[\leadsto \color{blue}{x2} \]
Recombined 2 regimes into one program.
Add Preprocessing

Alternative 12: 60.0% accurate, 4.1× speedup?

\[\begin{array}{l} \mathbf{if}\;y2 \leq 2.2 \cdot 10^{-161}:\\ \;\;\;\;\left(0.0078125 \cdot x1\right) \cdot 128\\ \mathbf{else}:\\ \;\;\;\;x2\\ \end{array} \]

(FPCore (x1 y1 x2 y2)
  :precision binary64
  (if (<= y2 2.2e-161) (* (* 0.0078125 x1) 128.0) x2))

double code(double x1, double y1, double x2, double y2) {
	double tmp;
	if (y2 <= 2.2e-161) {
		tmp = (0.0078125 * x1) * 128.0;
	} else {
		tmp = x2;
	}
	return tmp;
}

real(8) function code(x1, y1, x2, y2)
    real(8), intent (in) :: x1
    real(8), intent (in) :: y1
    real(8), intent (in) :: x2
    real(8), intent (in) :: y2
    real(8) :: tmp
    if (y2 <= 2.2d-161) then
        tmp = (0.0078125d0 * x1) * 128.0d0
    else
        tmp = x2
    end if
    code = tmp
end function

public static double code(double x1, double y1, double x2, double y2) {
	double tmp;
	if (y2 <= 2.2e-161) {
		tmp = (0.0078125 * x1) * 128.0;
	} else {
		tmp = x2;
	}
	return tmp;
}

def code(x1, y1, x2, y2):
	tmp = 0
	if y2 <= 2.2e-161:
		tmp = (0.0078125 * x1) * 128.0
	else:
		tmp = x2
	return tmp

function code(x1, y1, x2, y2)
	tmp = 0.0
	if (y2 <= 2.2e-161)
		tmp = Float64(Float64(0.0078125 * x1) * 128.0);
	else
		tmp = x2;
	end
	return tmp
end

function tmp_2 = code(x1, y1, x2, y2)
	tmp = 0.0;
	if (y2 <= 2.2e-161)
		tmp = (0.0078125 * x1) * 128.0;
	else
		tmp = x2;
	end
	tmp_2 = tmp;
end

code[x1_, y1_, x2_, y2_] := If[LessEqual[y2, 2.2e-161], N[(N[(0.0078125 * x1), $MachinePrecision] * 128.0), $MachinePrecision], x2]

\begin{array}{l}
\mathbf{if}\;y2 \leq 2.2 \cdot 10^{-161}:\\
\;\;\;\;\left(0.0078125 \cdot x1\right) \cdot 128\\

\mathbf{else}:\\
\;\;\;\;x2\\


\end{array}

Derivation

Split input into 2 regimes
if y2 < 2.2e-161
1. Initial program 77.8%
  \[\left(\left(\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2} \cdot 128\right) \cdot 128\right) \cdot 128 \]
2. Taylor expanded in y1 around inf
  \[\leadsto \color{blue}{\left(\frac{1}{128} \cdot x1\right)} \cdot 128 \]
3. Step-by-step derivation
  1. lower-*.f6440.4%
    \[\leadsto \left(0.0078125 \cdot \color{blue}{x1}\right) \cdot 128 \]
4. Applied rewrites40.4%
  \[\leadsto \color{blue}{\left(0.0078125 \cdot x1\right)} \cdot 128 \]
if 2.2e-161 < y2
1. Initial program 77.8%
  \[\left(\left(\frac{\frac{x1}{\left(128 \cdot 128\right) \cdot 128} \cdot y1 + \frac{x2}{\left(128 \cdot 128\right) \cdot 128} \cdot y2}{y1 + y2} \cdot 128\right) \cdot 128\right) \cdot 128 \]
2. Taylor expanded in y1 around 0
  \[\leadsto \color{blue}{x2} \]
3. Step-by-step derivation
4. Applied rewrites40.4%
  \[\leadsto \color{blue}{x2} \]
Recombined 2 regimes into one program.
Add Preprocessing

(x1 / (128128128) * y1 + x2 / (128128128) * y2) / (y1 + y2) * 128128128

Specification

Local Percentage Accuracy vs ?

Accuracy vs Speed?

Initial Program: 77.8% accurate, 1.0× speedup?

Alternative 1: 99.7% accurate, 1.2× speedup?

Alternative 2: 99.6% accurate, 1.2× speedup?

Alternative 3: 99.6% accurate, 1.2× speedup?

Alternative 4: 91.8% accurate, 0.6× speedup?

Alternative 5: 87.7% accurate, 0.6× speedup?

Alternative 6: 87.7% accurate, 0.7× speedup?

Alternative 7: 71.6% accurate, 3.1× speedup?

`if y2 < 9.5000000000000002e-140`

`if 9.5000000000000002e-140 < y2`

Alternative 8: 68.2% accurate, 3.1× speedup?

`if y2 < 9.5000000000000002e-140`

`if 9.5000000000000002e-140 < y2`

Alternative 9: 68.1% accurate, 3.1× speedup?

`if y2 < 9.5000000000000002e-140`

`if 9.5000000000000002e-140 < y2`

Alternative 10: 68.1% accurate, 3.1× speedup?

`if y2 < 9.5000000000000002e-140`

`if 9.5000000000000002e-140 < y2`

Alternative 11: 64.4% accurate, 3.1× speedup?

`if y2 < 3.9999999999999999e-140`

`if 3.9999999999999999e-140 < y2`

Alternative 12: 60.0% accurate, 4.1× speedup?

`if y2 < 2.2e-161`

`if 2.2e-161 < y2`

Alternative 13: 40.4% accurate, 44.6× speedup?

Reproduce

Specification

Local Percentage Accuracy vs ?

Accuracy vs Speed?

Initial Program: 77.8% accurate, 1.0× speedupMathFPCoreCFortranJavaPythonJuliaMATLABWolframTeX?

Alternative 1: 99.7% accurate, 1.2× speedupMathFPCoreCJuliaWolframTeX?

Alternative 2: 99.6% accurate, 1.2× speedupMathFPCoreCJuliaWolframTeX?

Alternative 3: 99.6% accurate, 1.2× speedupMathFPCoreCJuliaWolframTeX?

Alternative 4: 91.8% accurate, 0.6× speedupMathFPCoreCJuliaWolframTeX?

Alternative 5: 87.7% accurate, 0.6× speedupMathFPCoreCFortranJavaPythonJuliaMATLABWolframTeX?

Alternative 6: 87.7% accurate, 0.7× speedupMathFPCoreCJuliaWolframTeX?

Alternative 7: 71.6% accurate, 3.1× speedupMathFPCoreCFortranJavaPythonJuliaMATLABWolframTeX?

if y2 < 9.5000000000000002e-140

if 9.5000000000000002e-140 < y2

Alternative 8: 68.2% accurate, 3.1× speedupMathFPCoreCFortranJavaPythonJuliaMATLABWolframTeX?

if y2 < 9.5000000000000002e-140

if 9.5000000000000002e-140 < y2

Alternative 9: 68.1% accurate, 3.1× speedupMathFPCoreCFortranJavaPythonJuliaMATLABWolframTeX?

if y2 < 9.5000000000000002e-140

if 9.5000000000000002e-140 < y2

Alternative 10: 68.1% accurate, 3.1× speedupMathFPCoreCFortranJavaPythonJuliaMATLABWolframTeX?

if y2 < 9.5000000000000002e-140

if 9.5000000000000002e-140 < y2

Alternative 11: 64.4% accurate, 3.1× speedupMathFPCoreCFortranJavaPythonJuliaMATLABWolframTeX?

if y2 < 3.9999999999999999e-140

if 3.9999999999999999e-140 < y2

Alternative 12: 60.0% accurate, 4.1× speedupMathFPCoreCFortranJavaPythonJuliaMATLABWolframTeX?

if y2 < 2.2e-161

if 2.2e-161 < y2

Alternative 13: 40.4% accurate, 44.6× speedupMathFPCoreCFortranJavaPythonJuliaMATLABWolframTeX?

Reproduce

Initial Program: 77.8% accurate, 1.0× speedup?

Alternative 1: 99.7% accurate, 1.2× speedup?

Alternative 2: 99.6% accurate, 1.2× speedup?

Alternative 3: 99.6% accurate, 1.2× speedup?

Alternative 4: 91.8% accurate, 0.6× speedup?

Alternative 5: 87.7% accurate, 0.6× speedup?

Alternative 6: 87.7% accurate, 0.7× speedup?

Alternative 7: 71.6% accurate, 3.1× speedup?

`if y2 < 9.5000000000000002e-140`

`if 9.5000000000000002e-140 < y2`

Alternative 8: 68.2% accurate, 3.1× speedup?

`if y2 < 9.5000000000000002e-140`

`if 9.5000000000000002e-140 < y2`

Alternative 9: 68.1% accurate, 3.1× speedup?

`if y2 < 9.5000000000000002e-140`

`if 9.5000000000000002e-140 < y2`

Alternative 10: 68.1% accurate, 3.1× speedup?

`if y2 < 9.5000000000000002e-140`

`if 9.5000000000000002e-140 < y2`

Alternative 11: 64.4% accurate, 3.1× speedup?

`if y2 < 3.9999999999999999e-140`

`if 3.9999999999999999e-140 < y2`

Alternative 12: 60.0% accurate, 4.1× speedup?

`if y2 < 2.2e-161`

`if 2.2e-161 < y2`

Alternative 13: 40.4% accurate, 44.6× speedup?