Result for pow(log(y),sin(x))+(x/y)*cbrt(y*x)

Alternative 1: 99.4% accurate, 0.8× speedup?

\[\mathsf{fma}\left(\sqrt[3]{y} \cdot \frac{x}{y}, \sqrt[3]{x}, {\log y}^{\sin x}\right) \]

(FPCore (y x)
  :precision binary64
  (fma (* (cbrt y) (/ x y)) (cbrt x) (pow (log y) (sin x))))

double code(double y, double x) {
	return fma((cbrt(y) * (x / y)), cbrt(x), pow(log(y), sin(x)));
}

function code(y, x)
	return fma(Float64(cbrt(y) * Float64(x / y)), cbrt(x), (log(y) ^ sin(x)))
end

code[y_, x_] := N[(N[(N[Power[y, 1/3], $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision] + N[Power[N[Log[y], $MachinePrecision], N[Sin[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

\mathsf{fma}\left(\sqrt[3]{y} \cdot \frac{x}{y}, \sqrt[3]{x}, {\log y}^{\sin x}\right)

Derivation

Initial program 78.5%
\[{\log y}^{\sin x} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
Step-by-step derivation
1. lift-+.f64N/A
  \[\leadsto \color{blue}{{\log y}^{\sin x} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x}} \]
2. +-commutativeN/A
  \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x} + {\log y}^{\sin x}} \]
3. add-flipN/A
  \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x} - \left(\mathsf{neg}\left({\log y}^{\sin x}\right)\right)} \]
4. sub-flipN/A
  \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\log y}^{\sin x}\right)\right)\right)\right)} \]
5. lift-*.f64N/A
  \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\log y}^{\sin x}\right)\right)\right)\right) \]
6. lift-/.f64N/A
  \[\leadsto \color{blue}{\frac{x}{y}} \cdot \sqrt[3]{y \cdot x} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\log y}^{\sin x}\right)\right)\right)\right) \]
7. mult-flipN/A
  \[\leadsto \color{blue}{\left(x \cdot \frac{1}{y}\right)} \cdot \sqrt[3]{y \cdot x} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\log y}^{\sin x}\right)\right)\right)\right) \]
8. associate-*l*N/A
  \[\leadsto \color{blue}{x \cdot \left(\frac{1}{y} \cdot \sqrt[3]{y \cdot x}\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\log y}^{\sin x}\right)\right)\right)\right) \]
9. *-commutativeN/A
  \[\leadsto \color{blue}{\left(\frac{1}{y} \cdot \sqrt[3]{y \cdot x}\right) \cdot x} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\log y}^{\sin x}\right)\right)\right)\right) \]
10. remove-double-negN/A
  \[\leadsto \left(\frac{1}{y} \cdot \sqrt[3]{y \cdot x}\right) \cdot x + \color{blue}{{\log y}^{\sin x}} \]
11. lower-fma.f64N/A
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{y} \cdot \sqrt[3]{y \cdot x}, x, {\log y}^{\sin x}\right)} \]
12. *-commutativeN/A
  \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt[3]{y \cdot x} \cdot \frac{1}{y}}, x, {\log y}^{\sin x}\right) \]
13. mult-flipN/A
  \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sqrt[3]{y \cdot x}}{y}}, x, {\log y}^{\sin x}\right) \]
14. lower-/.f6478.5%
  \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sqrt[3]{y \cdot x}}{y}}, x, {\log y}^{\sin x}\right) \]
15. lift-*.f64N/A
  \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{\color{blue}{y \cdot x}}}{y}, x, {\log y}^{\sin x}\right) \]
16. *-commutativeN/A
  \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{\color{blue}{x \cdot y}}}{y}, x, {\log y}^{\sin x}\right) \]
17. lower-*.f6478.5%
  \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{\color{blue}{x \cdot y}}}{y}, x, {\log y}^{\sin x}\right) \]
Applied rewrites78.5%
\[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{x \cdot y}}{y}, x, {\log y}^{\sin x}\right)} \]
Step-by-step derivation
1. lift-/.f64N/A
  \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sqrt[3]{x \cdot y}}{y}}, x, {\log y}^{\sin x}\right) \]
2. div-flip-revN/A
  \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{y}{\sqrt[3]{x \cdot y}}}}, x, {\log y}^{\sin x}\right) \]
3. lift-cbrt.f64N/A
  \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{y}{\color{blue}{\sqrt[3]{x \cdot y}}}}, x, {\log y}^{\sin x}\right) \]
4. lift-*.f64N/A
  \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{y}{\sqrt[3]{\color{blue}{x \cdot y}}}}, x, {\log y}^{\sin x}\right) \]
5. cbrt-prodN/A
  \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{y}{\color{blue}{\sqrt[3]{x} \cdot \sqrt[3]{y}}}}, x, {\log y}^{\sin x}\right) \]
6. lift-cbrt.f64N/A
  \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{y}{\color{blue}{\sqrt[3]{x}} \cdot \sqrt[3]{y}}}, x, {\log y}^{\sin x}\right) \]
7. lift-cbrt.f64N/A
  \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{y}{\sqrt[3]{x} \cdot \color{blue}{\sqrt[3]{y}}}}, x, {\log y}^{\sin x}\right) \]
8. associate-/r*N/A
  \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{\frac{\frac{y}{\sqrt[3]{x}}}{\sqrt[3]{y}}}}, x, {\log y}^{\sin x}\right) \]
9. div-flip-revN/A
  \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sqrt[3]{y}}{\frac{y}{\sqrt[3]{x}}}}, x, {\log y}^{\sin x}\right) \]
10. lower-/.f64N/A
  \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sqrt[3]{y}}{\frac{y}{\sqrt[3]{x}}}}, x, {\log y}^{\sin x}\right) \]
11. lower-/.f6499.4%
  \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{y}}{\color{blue}{\frac{y}{\sqrt[3]{x}}}}, x, {\log y}^{\sin x}\right) \]
Applied rewrites99.4%
\[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sqrt[3]{y}}{\frac{y}{\sqrt[3]{x}}}}, x, {\log y}^{\sin x}\right) \]
Step-by-step derivation
1. lift-/.f64N/A
  \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sqrt[3]{y}}{\frac{y}{\sqrt[3]{x}}}}, x, {\log y}^{\sin x}\right) \]
2. mult-flipN/A
  \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt[3]{y} \cdot \frac{1}{\frac{y}{\sqrt[3]{x}}}}, x, {\log y}^{\sin x}\right) \]
3. *-commutativeN/A
  \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{y}{\sqrt[3]{x}}} \cdot \sqrt[3]{y}}, x, {\log y}^{\sin x}\right) \]
4. lift-/.f64N/A
  \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{\frac{y}{\sqrt[3]{x}}}} \cdot \sqrt[3]{y}, x, {\log y}^{\sin x}\right) \]
5. associate-/r/N/A
  \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{y} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{y}, x, {\log y}^{\sin x}\right) \]
6. lift-/.f64N/A
  \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\frac{1}{y}} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{y}, x, {\log y}^{\sin x}\right) \]
7. associate-*r*N/A
  \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{y} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{y}\right)}, x, {\log y}^{\sin x}\right) \]
8. lift-cbrt.f64N/A
  \[\leadsto \mathsf{fma}\left(\frac{1}{y} \cdot \left(\color{blue}{\sqrt[3]{x}} \cdot \sqrt[3]{y}\right), x, {\log y}^{\sin x}\right) \]
9. lift-cbrt.f64N/A
  \[\leadsto \mathsf{fma}\left(\frac{1}{y} \cdot \left(\sqrt[3]{x} \cdot \color{blue}{\sqrt[3]{y}}\right), x, {\log y}^{\sin x}\right) \]
10. cbrt-prodN/A
  \[\leadsto \mathsf{fma}\left(\frac{1}{y} \cdot \color{blue}{\sqrt[3]{x \cdot y}}, x, {\log y}^{\sin x}\right) \]
11. lift-*.f64N/A
  \[\leadsto \mathsf{fma}\left(\frac{1}{y} \cdot \sqrt[3]{\color{blue}{x \cdot y}}, x, {\log y}^{\sin x}\right) \]
12. lift-cbrt.f64N/A
  \[\leadsto \mathsf{fma}\left(\frac{1}{y} \cdot \color{blue}{\sqrt[3]{x \cdot y}}, x, {\log y}^{\sin x}\right) \]
13. lift-*.f6478.5%
  \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{y} \cdot \sqrt[3]{x \cdot y}}, x, {\log y}^{\sin x}\right) \]
14. lift-fma.f64N/A
  \[\leadsto \color{blue}{\left(\frac{1}{y} \cdot \sqrt[3]{x \cdot y}\right) \cdot x + {\log y}^{\sin x}} \]
15. *-commutativeN/A
  \[\leadsto \color{blue}{x \cdot \left(\frac{1}{y} \cdot \sqrt[3]{x \cdot y}\right)} + {\log y}^{\sin x} \]
Applied rewrites99.4%
\[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{y} \cdot \frac{x}{y}, \sqrt[3]{x}, {\log y}^{\sin x}\right)} \]
Add Preprocessing

Alternative 2: 96.7% accurate, 0.5× speedup?

\[\begin{array}{l} t_0 := {\log y}^{\sin x}\\ \mathbf{if}\;t\_0 + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \leq 2 \cdot 10^{+250}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{x \cdot y}}{y}, x, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{y} \cdot \frac{x}{y}, \sqrt[3]{x}, 1\right)\\ \end{array} \]

(FPCore (y x)
  :precision binary64
  (let* ((t_0 (pow (log y) (sin x))))
  (if (<= (+ t_0 (* (/ x y) (cbrt (* y x)))) 2e+250)
    (fma (/ (cbrt (* x y)) y) x t_0)
    (fma (* (cbrt y) (/ x y)) (cbrt x) 1.0))))

double code(double y, double x) {
	double t_0 = pow(log(y), sin(x));
	double tmp;
	if ((t_0 + ((x / y) * cbrt((y * x)))) <= 2e+250) {
		tmp = fma((cbrt((x * y)) / y), x, t_0);
	} else {
		tmp = fma((cbrt(y) * (x / y)), cbrt(x), 1.0);
	}
	return tmp;
}

function code(y, x)
	t_0 = log(y) ^ sin(x)
	tmp = 0.0
	if (Float64(t_0 + Float64(Float64(x / y) * cbrt(Float64(y * x)))) <= 2e+250)
		tmp = fma(Float64(cbrt(Float64(x * y)) / y), x, t_0);
	else
		tmp = fma(Float64(cbrt(y) * Float64(x / y)), cbrt(x), 1.0);
	end
	return tmp
end

code[y_, x_] := Block[{t$95$0 = N[Power[N[Log[y], $MachinePrecision], N[Sin[x], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$0 + N[(N[(x / y), $MachinePrecision] * N[Power[N[(y * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+250], N[(N[(N[Power[N[(x * y), $MachinePrecision], 1/3], $MachinePrecision] / y), $MachinePrecision] * x + t$95$0), $MachinePrecision], N[(N[(N[Power[y, 1/3], $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision] + 1.0), $MachinePrecision]]]

\begin{array}{l}
t_0 := {\log y}^{\sin x}\\
\mathbf{if}\;t\_0 + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \leq 2 \cdot 10^{+250}:\\
\;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{x \cdot y}}{y}, x, t\_0\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\sqrt[3]{y} \cdot \frac{x}{y}, \sqrt[3]{x}, 1\right)\\


\end{array}

Derivation

Split input into 2 regimes
if (+.f64 (pow.f64 (log.f64 y) (sin.f64 x)) (*.f64 (/.f64 x y) (cbrt.f64 (*.f64 y x)))) < 1.9999999999999998e250
1. Initial program 78.5%
  \[{\log y}^{\sin x} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
2. Step-by-step derivation
  1. lift-+.f64N/A
    \[\leadsto \color{blue}{{\log y}^{\sin x} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x}} \]
  2. +-commutativeN/A
    \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x} + {\log y}^{\sin x}} \]
  3. add-flipN/A
    \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x} - \left(\mathsf{neg}\left({\log y}^{\sin x}\right)\right)} \]
  4. sub-flipN/A
    \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\log y}^{\sin x}\right)\right)\right)\right)} \]
  5. lift-*.f64N/A
    \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\log y}^{\sin x}\right)\right)\right)\right) \]
  6. lift-/.f64N/A
    \[\leadsto \color{blue}{\frac{x}{y}} \cdot \sqrt[3]{y \cdot x} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\log y}^{\sin x}\right)\right)\right)\right) \]
  7. mult-flipN/A
    \[\leadsto \color{blue}{\left(x \cdot \frac{1}{y}\right)} \cdot \sqrt[3]{y \cdot x} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\log y}^{\sin x}\right)\right)\right)\right) \]
  8. associate-*l*N/A
    \[\leadsto \color{blue}{x \cdot \left(\frac{1}{y} \cdot \sqrt[3]{y \cdot x}\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\log y}^{\sin x}\right)\right)\right)\right) \]
  9. *-commutativeN/A
    \[\leadsto \color{blue}{\left(\frac{1}{y} \cdot \sqrt[3]{y \cdot x}\right) \cdot x} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\log y}^{\sin x}\right)\right)\right)\right) \]
  10. remove-double-negN/A
    \[\leadsto \left(\frac{1}{y} \cdot \sqrt[3]{y \cdot x}\right) \cdot x + \color{blue}{{\log y}^{\sin x}} \]
  11. lower-fma.f64N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{y} \cdot \sqrt[3]{y \cdot x}, x, {\log y}^{\sin x}\right)} \]
  12. *-commutativeN/A
    \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt[3]{y \cdot x} \cdot \frac{1}{y}}, x, {\log y}^{\sin x}\right) \]
  13. mult-flipN/A
    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sqrt[3]{y \cdot x}}{y}}, x, {\log y}^{\sin x}\right) \]
  14. lower-/.f6478.5%
    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sqrt[3]{y \cdot x}}{y}}, x, {\log y}^{\sin x}\right) \]
  15. lift-*.f64N/A
    \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{\color{blue}{y \cdot x}}}{y}, x, {\log y}^{\sin x}\right) \]
  16. *-commutativeN/A
    \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{\color{blue}{x \cdot y}}}{y}, x, {\log y}^{\sin x}\right) \]
  17. lower-*.f6478.5%
    \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{\color{blue}{x \cdot y}}}{y}, x, {\log y}^{\sin x}\right) \]
3. Applied rewrites78.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{x \cdot y}}{y}, x, {\log y}^{\sin x}\right)} \]
if 1.9999999999999998e250 < (+.f64 (pow.f64 (log.f64 y) (sin.f64 x)) (*.f64 (/.f64 x y) (cbrt.f64 (*.f64 y x))))
1. Initial program 78.5%
  \[{\log y}^{\sin x} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
2. Taylor expanded in x around 0
  \[\leadsto \color{blue}{1} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
3. Step-by-step derivation
4. Applied rewrites69.5%
  \[\leadsto \color{blue}{1} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
5. Step-by-step derivation
  1. lift-+.f64N/A
    \[\leadsto \color{blue}{1 + \frac{x}{y} \cdot \sqrt[3]{y \cdot x}} \]
  2. +-commutativeN/A
    \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x} + 1} \]
  3. lift-*.f64N/A
    \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x}} + 1 \]
  4. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{y \cdot x}} + 1 \]
  5. lift-*.f64N/A
    \[\leadsto \frac{x}{y} \cdot \sqrt[3]{\color{blue}{y \cdot x}} + 1 \]
  6. cbrt-unprodN/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{x}\right)} + 1 \]
  7. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \left(\color{blue}{\sqrt[3]{y}} \cdot \sqrt[3]{x}\right) + 1 \]
  8. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \left(\sqrt[3]{y} \cdot \color{blue}{\sqrt[3]{x}}\right) + 1 \]
  9. *-commutativeN/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{y}\right)} + 1 \]
  10. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \left(\color{blue}{\sqrt[3]{x}} \cdot \sqrt[3]{y}\right) + 1 \]
  11. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \left(\sqrt[3]{x} \cdot \color{blue}{\sqrt[3]{y}}\right) + 1 \]
  12. cbrt-prodN/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{x \cdot y}} + 1 \]
  13. lift-*.f64N/A
    \[\leadsto \frac{x}{y} \cdot \sqrt[3]{\color{blue}{x \cdot y}} + 1 \]
  14. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{x \cdot y}} + 1 \]
  15. lower-fma.f6469.5%
    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \sqrt[3]{x \cdot y}, 1\right)} \]
6. Applied rewrites69.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \sqrt[3]{x \cdot y}, 1\right)} \]
7. Step-by-step derivation
  1. lift-fma.f64N/A
    \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{x \cdot y} + 1} \]
  2. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{x \cdot y}} + 1 \]
  3. lift-*.f64N/A
    \[\leadsto \frac{x}{y} \cdot \sqrt[3]{\color{blue}{x \cdot y}} + 1 \]
  4. cbrt-prodN/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{y}\right)} + 1 \]
  5. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \left(\color{blue}{\sqrt[3]{x}} \cdot \sqrt[3]{y}\right) + 1 \]
  6. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \left(\sqrt[3]{x} \cdot \color{blue}{\sqrt[3]{y}}\right) + 1 \]
  7. associate-*l*N/A
    \[\leadsto \color{blue}{\left(\frac{x}{y} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{y}} + 1 \]
  8. lift-*.f64N/A
    \[\leadsto \color{blue}{\left(\frac{x}{y} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{y} + 1 \]
  9. *-commutativeN/A
    \[\leadsto \color{blue}{\sqrt[3]{y} \cdot \left(\frac{x}{y} \cdot \sqrt[3]{x}\right)} + 1 \]
  10. lift-*.f64N/A
    \[\leadsto \sqrt[3]{y} \cdot \color{blue}{\left(\frac{x}{y} \cdot \sqrt[3]{x}\right)} + 1 \]
  11. associate-*r*N/A
    \[\leadsto \color{blue}{\left(\sqrt[3]{y} \cdot \frac{x}{y}\right) \cdot \sqrt[3]{x}} + 1 \]
  12. lower-fma.f64N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{y} \cdot \frac{x}{y}, \sqrt[3]{x}, 1\right)} \]
  13. lower-*.f6486.5%
    \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt[3]{y} \cdot \frac{x}{y}}, \sqrt[3]{x}, 1\right) \]
8. Applied rewrites86.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{y} \cdot \frac{x}{y}, \sqrt[3]{x}, 1\right)} \]
Recombined 2 regimes into one program.
Add Preprocessing

Alternative 3: 95.6% accurate, 0.9× speedup?

\[\mathsf{fma}\left({y}^{-0.6666666666666666} \cdot x, \sqrt[3]{x}, {\log y}^{\sin x}\right) \]

(FPCore (y x)
  :precision binary64
  (fma (* (pow y -0.6666666666666666) x) (cbrt x) (pow (log y) (sin x))))

double code(double y, double x) {
	return fma((pow(y, -0.6666666666666666) * x), cbrt(x), pow(log(y), sin(x)));
}

function code(y, x)
	return fma(Float64((y ^ -0.6666666666666666) * x), cbrt(x), (log(y) ^ sin(x)))
end

code[y_, x_] := N[(N[(N[Power[y, -0.6666666666666666], $MachinePrecision] * x), $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision] + N[Power[N[Log[y], $MachinePrecision], N[Sin[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

\mathsf{fma}\left({y}^{-0.6666666666666666} \cdot x, \sqrt[3]{x}, {\log y}^{\sin x}\right)

Derivation

Initial program 78.5%
\[{\log y}^{\sin x} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
Step-by-step derivation
1. lift-+.f64N/A
  \[\leadsto \color{blue}{{\log y}^{\sin x} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x}} \]
2. +-commutativeN/A
  \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x} + {\log y}^{\sin x}} \]
3. add-flipN/A
  \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x} - \left(\mathsf{neg}\left({\log y}^{\sin x}\right)\right)} \]
4. sub-flipN/A
  \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\log y}^{\sin x}\right)\right)\right)\right)} \]
5. lift-*.f64N/A
  \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\log y}^{\sin x}\right)\right)\right)\right) \]
6. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{y \cdot x}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\log y}^{\sin x}\right)\right)\right)\right) \]
7. lift-*.f64N/A
  \[\leadsto \frac{x}{y} \cdot \sqrt[3]{\color{blue}{y \cdot x}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\log y}^{\sin x}\right)\right)\right)\right) \]
8. cbrt-prodN/A
  \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{x}\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\log y}^{\sin x}\right)\right)\right)\right) \]
9. associate-*r*N/A
  \[\leadsto \color{blue}{\left(\frac{x}{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{x}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({\log y}^{\sin x}\right)\right)\right)\right) \]
10. remove-double-negN/A
  \[\leadsto \left(\frac{x}{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{x} + \color{blue}{{\log y}^{\sin x}} \]
11. lower-fma.f64N/A
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y} \cdot \sqrt[3]{y}, \sqrt[3]{x}, {\log y}^{\sin x}\right)} \]
Applied rewrites96.7%
\[\leadsto \color{blue}{\mathsf{fma}\left({y}^{-0.6666666666666666} \cdot x, \sqrt[3]{x}, {\log y}^{\sin x}\right)} \]
Add Preprocessing

Alternative 4: 94.3% accurate, 0.5× speedup?

\[\begin{array}{l} \mathbf{if}\;{\log y}^{\sin x} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \leq 500:\\ \;\;\;\;\frac{1}{\frac{1}{e^{\log \left(-1 \cdot \log \left(\frac{1}{y}\right)\right) \cdot \sin x}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{y} \cdot \frac{x}{y}, \sqrt[3]{x}, 1\right)\\ \end{array} \]

(FPCore (y x)
  :precision binary64
  (if (<= (+ (pow (log y) (sin x)) (* (/ x y) (cbrt (* y x)))) 500.0)
  (/ 1.0 (/ 1.0 (exp (* (log (* -1.0 (log (/ 1.0 y)))) (sin x)))))
  (fma (* (cbrt y) (/ x y)) (cbrt x) 1.0)))

double code(double y, double x) {
	double tmp;
	if ((pow(log(y), sin(x)) + ((x / y) * cbrt((y * x)))) <= 500.0) {
		tmp = 1.0 / (1.0 / exp((log((-1.0 * log((1.0 / y)))) * sin(x))));
	} else {
		tmp = fma((cbrt(y) * (x / y)), cbrt(x), 1.0);
	}
	return tmp;
}

function code(y, x)
	tmp = 0.0
	if (Float64((log(y) ^ sin(x)) + Float64(Float64(x / y) * cbrt(Float64(y * x)))) <= 500.0)
		tmp = Float64(1.0 / Float64(1.0 / exp(Float64(log(Float64(-1.0 * log(Float64(1.0 / y)))) * sin(x)))));
	else
		tmp = fma(Float64(cbrt(y) * Float64(x / y)), cbrt(x), 1.0);
	end
	return tmp
end

code[y_, x_] := If[LessEqual[N[(N[Power[N[Log[y], $MachinePrecision], N[Sin[x], $MachinePrecision]], $MachinePrecision] + N[(N[(x / y), $MachinePrecision] * N[Power[N[(y * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 500.0], N[(1.0 / N[(1.0 / N[Exp[N[(N[Log[N[(-1.0 * N[Log[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[y, 1/3], $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision] + 1.0), $MachinePrecision]]

\begin{array}{l}
\mathbf{if}\;{\log y}^{\sin x} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \leq 500:\\
\;\;\;\;\frac{1}{\frac{1}{e^{\log \left(-1 \cdot \log \left(\frac{1}{y}\right)\right) \cdot \sin x}}}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\sqrt[3]{y} \cdot \frac{x}{y}, \sqrt[3]{x}, 1\right)\\


\end{array}

Derivation

Split input into 2 regimes
if (+.f64 (pow.f64 (log.f64 y) (sin.f64 x)) (*.f64 (/.f64 x y) (cbrt.f64 (*.f64 y x)))) < 500
1. Initial program 78.5%
  \[{\log y}^{\sin x} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
2. Step-by-step derivation
  1. lift-+.f64N/A
    \[\leadsto \color{blue}{{\log y}^{\sin x} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x}} \]
  2. lift-*.f64N/A
    \[\leadsto {\log y}^{\sin x} + \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x}} \]
  3. lift-/.f64N/A
    \[\leadsto {\log y}^{\sin x} + \color{blue}{\frac{x}{y}} \cdot \sqrt[3]{y \cdot x} \]
  4. associate-*l/N/A
    \[\leadsto {\log y}^{\sin x} + \color{blue}{\frac{x \cdot \sqrt[3]{y \cdot x}}{y}} \]
  5. add-to-fractionN/A
    \[\leadsto \color{blue}{\frac{{\log y}^{\sin x} \cdot y + x \cdot \sqrt[3]{y \cdot x}}{y}} \]
  6. div-addN/A
    \[\leadsto \color{blue}{\frac{{\log y}^{\sin x} \cdot y}{y} + \frac{x \cdot \sqrt[3]{y \cdot x}}{y}} \]
  7. common-denominatorN/A
    \[\leadsto \color{blue}{\frac{\left({\log y}^{\sin x} \cdot y\right) \cdot y + \left(x \cdot \sqrt[3]{y \cdot x}\right) \cdot y}{y \cdot y}} \]
  8. div-flipN/A
    \[\leadsto \color{blue}{\frac{1}{\frac{y \cdot y}{\left({\log y}^{\sin x} \cdot y\right) \cdot y + \left(x \cdot \sqrt[3]{y \cdot x}\right) \cdot y}}} \]
  9. lower-unsound-/.f64N/A
    \[\leadsto \color{blue}{\frac{1}{\frac{y \cdot y}{\left({\log y}^{\sin x} \cdot y\right) \cdot y + \left(x \cdot \sqrt[3]{y \cdot x}\right) \cdot y}}} \]
  10. lower-unsound-/.f64N/A
    \[\leadsto \frac{1}{\color{blue}{\frac{y \cdot y}{\left({\log y}^{\sin x} \cdot y\right) \cdot y + \left(x \cdot \sqrt[3]{y \cdot x}\right) \cdot y}}} \]
  11. lower-*.f64N/A
    \[\leadsto \frac{1}{\frac{\color{blue}{y \cdot y}}{\left({\log y}^{\sin x} \cdot y\right) \cdot y + \left(x \cdot \sqrt[3]{y \cdot x}\right) \cdot y}} \]
  12. distribute-rgt-outN/A
    \[\leadsto \frac{1}{\frac{y \cdot y}{\color{blue}{y \cdot \left({\log y}^{\sin x} \cdot y + x \cdot \sqrt[3]{y \cdot x}\right)}}} \]
  13. lower-*.f64N/A
    \[\leadsto \frac{1}{\frac{y \cdot y}{\color{blue}{y \cdot \left({\log y}^{\sin x} \cdot y + x \cdot \sqrt[3]{y \cdot x}\right)}}} \]
  14. lower-fma.f64N/A
    \[\leadsto \frac{1}{\frac{y \cdot y}{y \cdot \color{blue}{\mathsf{fma}\left({\log y}^{\sin x}, y, x \cdot \sqrt[3]{y \cdot x}\right)}}} \]
  15. *-commutativeN/A
    \[\leadsto \frac{1}{\frac{y \cdot y}{y \cdot \mathsf{fma}\left({\log y}^{\sin x}, y, \color{blue}{\sqrt[3]{y \cdot x} \cdot x}\right)}} \]
3. Applied rewrites41.2%
  \[\leadsto \color{blue}{\frac{1}{\frac{y \cdot y}{y \cdot \mathsf{fma}\left({\log y}^{\sin x}, y, \sqrt[3]{x \cdot y} \cdot x\right)}}} \]
4. Taylor expanded in y around inf
  \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{\log \left(-1 \cdot \log \left(\frac{1}{y}\right)\right) \cdot \sin x}}}} \]
5. Step-by-step derivation
  1. lower-/.f64N/A
    \[\leadsto \frac{1}{\frac{1}{\color{blue}{e^{\log \left(-1 \cdot \log \left(\frac{1}{y}\right)\right) \cdot \sin x}}}} \]
  2. lower-exp.f64N/A
    \[\leadsto \frac{1}{\frac{1}{e^{\log \left(-1 \cdot \log \left(\frac{1}{y}\right)\right) \cdot \sin x}}} \]
  3. lower-*.f64N/A
    \[\leadsto \frac{1}{\frac{1}{e^{\log \left(-1 \cdot \log \left(\frac{1}{y}\right)\right) \cdot \sin x}}} \]
  4. lower-log.f64N/A
    \[\leadsto \frac{1}{\frac{1}{e^{\log \left(-1 \cdot \log \left(\frac{1}{y}\right)\right) \cdot \sin x}}} \]
  5. lower-*.f64N/A
    \[\leadsto \frac{1}{\frac{1}{e^{\log \left(-1 \cdot \log \left(\frac{1}{y}\right)\right) \cdot \sin x}}} \]
  6. lower-log.f64N/A
    \[\leadsto \frac{1}{\frac{1}{e^{\log \left(-1 \cdot \log \left(\frac{1}{y}\right)\right) \cdot \sin x}}} \]
  7. lower-/.f64N/A
    \[\leadsto \frac{1}{\frac{1}{e^{\log \left(-1 \cdot \log \left(\frac{1}{y}\right)\right) \cdot \sin x}}} \]
  8. lower-sin.f6464.1%
    \[\leadsto \frac{1}{\frac{1}{e^{\log \left(-1 \cdot \log \left(\frac{1}{y}\right)\right) \cdot \sin x}}} \]
6. Applied rewrites64.1%
  \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{\log \left(-1 \cdot \log \left(\frac{1}{y}\right)\right) \cdot \sin x}}}} \]
if 500 < (+.f64 (pow.f64 (log.f64 y) (sin.f64 x)) (*.f64 (/.f64 x y) (cbrt.f64 (*.f64 y x))))
1. Initial program 78.5%
  \[{\log y}^{\sin x} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
2. Taylor expanded in x around 0
  \[\leadsto \color{blue}{1} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
3. Step-by-step derivation
4. Applied rewrites69.5%
  \[\leadsto \color{blue}{1} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
5. Step-by-step derivation
  1. lift-+.f64N/A
    \[\leadsto \color{blue}{1 + \frac{x}{y} \cdot \sqrt[3]{y \cdot x}} \]
  2. +-commutativeN/A
    \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x} + 1} \]
  3. lift-*.f64N/A
    \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x}} + 1 \]
  4. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{y \cdot x}} + 1 \]
  5. lift-*.f64N/A
    \[\leadsto \frac{x}{y} \cdot \sqrt[3]{\color{blue}{y \cdot x}} + 1 \]
  6. cbrt-unprodN/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{x}\right)} + 1 \]
  7. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \left(\color{blue}{\sqrt[3]{y}} \cdot \sqrt[3]{x}\right) + 1 \]
  8. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \left(\sqrt[3]{y} \cdot \color{blue}{\sqrt[3]{x}}\right) + 1 \]
  9. *-commutativeN/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{y}\right)} + 1 \]
  10. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \left(\color{blue}{\sqrt[3]{x}} \cdot \sqrt[3]{y}\right) + 1 \]
  11. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \left(\sqrt[3]{x} \cdot \color{blue}{\sqrt[3]{y}}\right) + 1 \]
  12. cbrt-prodN/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{x \cdot y}} + 1 \]
  13. lift-*.f64N/A
    \[\leadsto \frac{x}{y} \cdot \sqrt[3]{\color{blue}{x \cdot y}} + 1 \]
  14. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{x \cdot y}} + 1 \]
  15. lower-fma.f6469.5%
    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \sqrt[3]{x \cdot y}, 1\right)} \]
6. Applied rewrites69.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \sqrt[3]{x \cdot y}, 1\right)} \]
7. Step-by-step derivation
  1. lift-fma.f64N/A
    \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{x \cdot y} + 1} \]
  2. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{x \cdot y}} + 1 \]
  3. lift-*.f64N/A
    \[\leadsto \frac{x}{y} \cdot \sqrt[3]{\color{blue}{x \cdot y}} + 1 \]
  4. cbrt-prodN/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{y}\right)} + 1 \]
  5. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \left(\color{blue}{\sqrt[3]{x}} \cdot \sqrt[3]{y}\right) + 1 \]
  6. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \left(\sqrt[3]{x} \cdot \color{blue}{\sqrt[3]{y}}\right) + 1 \]
  7. associate-*l*N/A
    \[\leadsto \color{blue}{\left(\frac{x}{y} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{y}} + 1 \]
  8. lift-*.f64N/A
    \[\leadsto \color{blue}{\left(\frac{x}{y} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{y} + 1 \]
  9. *-commutativeN/A
    \[\leadsto \color{blue}{\sqrt[3]{y} \cdot \left(\frac{x}{y} \cdot \sqrt[3]{x}\right)} + 1 \]
  10. lift-*.f64N/A
    \[\leadsto \sqrt[3]{y} \cdot \color{blue}{\left(\frac{x}{y} \cdot \sqrt[3]{x}\right)} + 1 \]
  11. associate-*r*N/A
    \[\leadsto \color{blue}{\left(\sqrt[3]{y} \cdot \frac{x}{y}\right) \cdot \sqrt[3]{x}} + 1 \]
  12. lower-fma.f64N/A
    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{y} \cdot \frac{x}{y}, \sqrt[3]{x}, 1\right)} \]
  13. lower-*.f6486.5%
    \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt[3]{y} \cdot \frac{x}{y}}, \sqrt[3]{x}, 1\right) \]
8. Applied rewrites86.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{y} \cdot \frac{x}{y}, \sqrt[3]{x}, 1\right)} \]
Recombined 2 regimes into one program.
Add Preprocessing

Alternative 5: 86.5% accurate, 1.9× speedup?

\[\mathsf{fma}\left(\sqrt[3]{y} \cdot \frac{x}{y}, \sqrt[3]{x}, 1\right) \]

(FPCore (y x)
  :precision binary64
  (fma (* (cbrt y) (/ x y)) (cbrt x) 1.0))

double code(double y, double x) {
	return fma((cbrt(y) * (x / y)), cbrt(x), 1.0);
}

function code(y, x)
	return fma(Float64(cbrt(y) * Float64(x / y)), cbrt(x), 1.0)
end

code[y_, x_] := N[(N[(N[Power[y, 1/3], $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision] + 1.0), $MachinePrecision]

\mathsf{fma}\left(\sqrt[3]{y} \cdot \frac{x}{y}, \sqrt[3]{x}, 1\right)

Derivation

Initial program 78.5%
\[{\log y}^{\sin x} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
Taylor expanded in x around 0
\[\leadsto \color{blue}{1} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
Step-by-step derivation
Applied rewrites69.5%
\[\leadsto \color{blue}{1} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
Step-by-step derivation
1. lift-+.f64N/A
  \[\leadsto \color{blue}{1 + \frac{x}{y} \cdot \sqrt[3]{y \cdot x}} \]
2. +-commutativeN/A
  \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x} + 1} \]
3. lift-*.f64N/A
  \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x}} + 1 \]
4. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{y \cdot x}} + 1 \]
5. lift-*.f64N/A
  \[\leadsto \frac{x}{y} \cdot \sqrt[3]{\color{blue}{y \cdot x}} + 1 \]
6. cbrt-unprodN/A
  \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{x}\right)} + 1 \]
7. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \left(\color{blue}{\sqrt[3]{y}} \cdot \sqrt[3]{x}\right) + 1 \]
8. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \left(\sqrt[3]{y} \cdot \color{blue}{\sqrt[3]{x}}\right) + 1 \]
9. *-commutativeN/A
  \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{y}\right)} + 1 \]
10. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \left(\color{blue}{\sqrt[3]{x}} \cdot \sqrt[3]{y}\right) + 1 \]
11. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \left(\sqrt[3]{x} \cdot \color{blue}{\sqrt[3]{y}}\right) + 1 \]
12. cbrt-prodN/A
  \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{x \cdot y}} + 1 \]
13. lift-*.f64N/A
  \[\leadsto \frac{x}{y} \cdot \sqrt[3]{\color{blue}{x \cdot y}} + 1 \]
14. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{x \cdot y}} + 1 \]
15. lower-fma.f6469.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \sqrt[3]{x \cdot y}, 1\right)} \]
Applied rewrites69.5%
\[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \sqrt[3]{x \cdot y}, 1\right)} \]
Step-by-step derivation
1. lift-fma.f64N/A
  \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{x \cdot y} + 1} \]
2. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{x \cdot y}} + 1 \]
3. lift-*.f64N/A
  \[\leadsto \frac{x}{y} \cdot \sqrt[3]{\color{blue}{x \cdot y}} + 1 \]
4. cbrt-prodN/A
  \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{y}\right)} + 1 \]
5. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \left(\color{blue}{\sqrt[3]{x}} \cdot \sqrt[3]{y}\right) + 1 \]
6. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \left(\sqrt[3]{x} \cdot \color{blue}{\sqrt[3]{y}}\right) + 1 \]
7. associate-*l*N/A
  \[\leadsto \color{blue}{\left(\frac{x}{y} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{y}} + 1 \]
8. lift-*.f64N/A
  \[\leadsto \color{blue}{\left(\frac{x}{y} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{y} + 1 \]
9. *-commutativeN/A
  \[\leadsto \color{blue}{\sqrt[3]{y} \cdot \left(\frac{x}{y} \cdot \sqrt[3]{x}\right)} + 1 \]
10. lift-*.f64N/A
  \[\leadsto \sqrt[3]{y} \cdot \color{blue}{\left(\frac{x}{y} \cdot \sqrt[3]{x}\right)} + 1 \]
11. associate-*r*N/A
  \[\leadsto \color{blue}{\left(\sqrt[3]{y} \cdot \frac{x}{y}\right) \cdot \sqrt[3]{x}} + 1 \]
12. lower-fma.f64N/A
  \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{y} \cdot \frac{x}{y}, \sqrt[3]{x}, 1\right)} \]
13. lower-*.f6486.5%
  \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt[3]{y} \cdot \frac{x}{y}}, \sqrt[3]{x}, 1\right) \]
Applied rewrites86.5%
\[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{y} \cdot \frac{x}{y}, \sqrt[3]{x}, 1\right)} \]
Add Preprocessing

Alternative 6: 86.5% accurate, 1.9× speedup?

\[\mathsf{fma}\left(x \cdot \frac{\sqrt[3]{x}}{y}, \sqrt[3]{y}, 1\right) \]

(FPCore (y x)
  :precision binary64
  (fma (* x (/ (cbrt x) y)) (cbrt y) 1.0))

double code(double y, double x) {
	return fma((x * (cbrt(x) / y)), cbrt(y), 1.0);
}

function code(y, x)
	return fma(Float64(x * Float64(cbrt(x) / y)), cbrt(y), 1.0)
end

code[y_, x_] := N[(N[(x * N[(N[Power[x, 1/3], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * N[Power[y, 1/3], $MachinePrecision] + 1.0), $MachinePrecision]

\mathsf{fma}\left(x \cdot \frac{\sqrt[3]{x}}{y}, \sqrt[3]{y}, 1\right)

Derivation

Initial program 78.5%
\[{\log y}^{\sin x} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
Taylor expanded in x around 0
\[\leadsto \color{blue}{1} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
Step-by-step derivation
Applied rewrites69.5%
\[\leadsto \color{blue}{1} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
Step-by-step derivation
1. lift-+.f64N/A
  \[\leadsto \color{blue}{1 + \frac{x}{y} \cdot \sqrt[3]{y \cdot x}} \]
2. +-commutativeN/A
  \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x} + 1} \]
3. lift-*.f64N/A
  \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x}} + 1 \]
4. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{y \cdot x}} + 1 \]
5. lift-*.f64N/A
  \[\leadsto \frac{x}{y} \cdot \sqrt[3]{\color{blue}{y \cdot x}} + 1 \]
6. cbrt-unprodN/A
  \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{x}\right)} + 1 \]
7. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \left(\color{blue}{\sqrt[3]{y}} \cdot \sqrt[3]{x}\right) + 1 \]
8. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \left(\sqrt[3]{y} \cdot \color{blue}{\sqrt[3]{x}}\right) + 1 \]
9. *-commutativeN/A
  \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{y}\right)} + 1 \]
10. associate-*r*N/A
  \[\leadsto \color{blue}{\left(\frac{x}{y} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{y}} + 1 \]
11. lower-fma.f64N/A
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y} \cdot \sqrt[3]{x}, \sqrt[3]{y}, 1\right)} \]
12. lower-*.f6486.5%
  \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{y} \cdot \sqrt[3]{x}}, \sqrt[3]{y}, 1\right) \]
Applied rewrites86.5%
\[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y} \cdot \sqrt[3]{x}, \sqrt[3]{y}, 1\right)} \]
Step-by-step derivation
1. lift-*.f64N/A
  \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{y} \cdot \sqrt[3]{x}}, \sqrt[3]{y}, 1\right) \]
2. lift-/.f64N/A
  \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{y}} \cdot \sqrt[3]{x}, \sqrt[3]{y}, 1\right) \]
3. associate-*l/N/A
  \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x \cdot \sqrt[3]{x}}{y}}, \sqrt[3]{y}, 1\right) \]
4. associate-/l*N/A
  \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \frac{\sqrt[3]{x}}{y}}, \sqrt[3]{y}, 1\right) \]
5. div-flip-revN/A
  \[\leadsto \mathsf{fma}\left(x \cdot \color{blue}{\frac{1}{\frac{y}{\sqrt[3]{x}}}}, \sqrt[3]{y}, 1\right) \]
6. lift-/.f64N/A
  \[\leadsto \mathsf{fma}\left(x \cdot \frac{1}{\color{blue}{\frac{y}{\sqrt[3]{x}}}}, \sqrt[3]{y}, 1\right) \]
7. lower-*.f64N/A
  \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \frac{1}{\frac{y}{\sqrt[3]{x}}}}, \sqrt[3]{y}, 1\right) \]
8. lift-/.f64N/A
  \[\leadsto \mathsf{fma}\left(x \cdot \frac{1}{\color{blue}{\frac{y}{\sqrt[3]{x}}}}, \sqrt[3]{y}, 1\right) \]
9. div-flip-revN/A
  \[\leadsto \mathsf{fma}\left(x \cdot \color{blue}{\frac{\sqrt[3]{x}}{y}}, \sqrt[3]{y}, 1\right) \]
10. lower-/.f6486.5%
  \[\leadsto \mathsf{fma}\left(x \cdot \color{blue}{\frac{\sqrt[3]{x}}{y}}, \sqrt[3]{y}, 1\right) \]
Applied rewrites86.5%
\[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \frac{\sqrt[3]{x}}{y}}, \sqrt[3]{y}, 1\right) \]
Add Preprocessing

Alternative 7: 84.0% accurate, 2.0× speedup?

\[\mathsf{fma}\left(x \cdot {y}^{-0.6666666666666666}, \sqrt[3]{x}, 1\right) \]

(FPCore (y x)
  :precision binary64
  (fma (* x (pow y -0.6666666666666666)) (cbrt x) 1.0))

double code(double y, double x) {
	return fma((x * pow(y, -0.6666666666666666)), cbrt(x), 1.0);
}

function code(y, x)
	return fma(Float64(x * (y ^ -0.6666666666666666)), cbrt(x), 1.0)
end

code[y_, x_] := N[(N[(x * N[Power[y, -0.6666666666666666], $MachinePrecision]), $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision] + 1.0), $MachinePrecision]

\mathsf{fma}\left(x \cdot {y}^{-0.6666666666666666}, \sqrt[3]{x}, 1\right)

Derivation

Initial program 78.5%
\[{\log y}^{\sin x} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
Taylor expanded in x around 0
\[\leadsto \color{blue}{1} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
Step-by-step derivation
Applied rewrites69.5%
\[\leadsto \color{blue}{1} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
Step-by-step derivation
1. lift-+.f64N/A
  \[\leadsto \color{blue}{1 + \frac{x}{y} \cdot \sqrt[3]{y \cdot x}} \]
2. +-commutativeN/A
  \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x} + 1} \]
3. lift-*.f64N/A
  \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x}} + 1 \]
4. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{y \cdot x}} + 1 \]
5. lift-*.f64N/A
  \[\leadsto \frac{x}{y} \cdot \sqrt[3]{\color{blue}{y \cdot x}} + 1 \]
6. cbrt-unprodN/A
  \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{x}\right)} + 1 \]
7. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \left(\color{blue}{\sqrt[3]{y}} \cdot \sqrt[3]{x}\right) + 1 \]
8. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \left(\sqrt[3]{y} \cdot \color{blue}{\sqrt[3]{x}}\right) + 1 \]
9. *-commutativeN/A
  \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{y}\right)} + 1 \]
10. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \left(\color{blue}{\sqrt[3]{x}} \cdot \sqrt[3]{y}\right) + 1 \]
11. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \left(\sqrt[3]{x} \cdot \color{blue}{\sqrt[3]{y}}\right) + 1 \]
12. cbrt-prodN/A
  \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{x \cdot y}} + 1 \]
13. lift-*.f64N/A
  \[\leadsto \frac{x}{y} \cdot \sqrt[3]{\color{blue}{x \cdot y}} + 1 \]
14. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{x \cdot y}} + 1 \]
15. lower-fma.f6469.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \sqrt[3]{x \cdot y}, 1\right)} \]
Applied rewrites69.5%
\[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \sqrt[3]{x \cdot y}, 1\right)} \]
Step-by-step derivation
1. lift-fma.f64N/A
  \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{x \cdot y} + 1} \]
2. lift-/.f64N/A
  \[\leadsto \color{blue}{\frac{x}{y}} \cdot \sqrt[3]{x \cdot y} + 1 \]
3. mult-flipN/A
  \[\leadsto \color{blue}{\left(x \cdot \frac{1}{y}\right)} \cdot \sqrt[3]{x \cdot y} + 1 \]
4. lift-/.f64N/A
  \[\leadsto \left(x \cdot \color{blue}{\frac{1}{y}}\right) \cdot \sqrt[3]{x \cdot y} + 1 \]
5. associate-*l*N/A
  \[\leadsto \color{blue}{x \cdot \left(\frac{1}{y} \cdot \sqrt[3]{x \cdot y}\right)} + 1 \]
6. lift-*.f64N/A
  \[\leadsto x \cdot \color{blue}{\left(\frac{1}{y} \cdot \sqrt[3]{x \cdot y}\right)} + 1 \]
7. *-commutativeN/A
  \[\leadsto \color{blue}{\left(\frac{1}{y} \cdot \sqrt[3]{x \cdot y}\right) \cdot x} + 1 \]
8. lower-fma.f6469.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{y} \cdot \sqrt[3]{x \cdot y}, x, 1\right)} \]
9. lift-*.f64N/A
  \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{y} \cdot \sqrt[3]{x \cdot y}}, x, 1\right) \]
10. *-commutativeN/A
  \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt[3]{x \cdot y} \cdot \frac{1}{y}}, x, 1\right) \]
11. lift-/.f64N/A
  \[\leadsto \mathsf{fma}\left(\sqrt[3]{x \cdot y} \cdot \color{blue}{\frac{1}{y}}, x, 1\right) \]
12. mult-flip-revN/A
  \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sqrt[3]{x \cdot y}}{y}}, x, 1\right) \]
13. lower-/.f6469.5%
  \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sqrt[3]{x \cdot y}}{y}}, x, 1\right) \]
Applied rewrites69.5%
\[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{x \cdot y}}{y}, x, 1\right)} \]
Step-by-step derivation
1. lift-fma.f64N/A
  \[\leadsto \color{blue}{\frac{\sqrt[3]{x \cdot y}}{y} \cdot x + 1} \]
2. +-commutativeN/A
  \[\leadsto \color{blue}{1 + \frac{\sqrt[3]{x \cdot y}}{y} \cdot x} \]
3. lift-/.f64N/A
  \[\leadsto 1 + \color{blue}{\frac{\sqrt[3]{x \cdot y}}{y}} \cdot x \]
4. associate-*l/N/A
  \[\leadsto 1 + \color{blue}{\frac{\sqrt[3]{x \cdot y} \cdot x}{y}} \]
5. *-commutativeN/A
  \[\leadsto 1 + \frac{\color{blue}{x \cdot \sqrt[3]{x \cdot y}}}{y} \]
6. lift-cbrt.f64N/A
  \[\leadsto 1 + \frac{x \cdot \color{blue}{\sqrt[3]{x \cdot y}}}{y} \]
7. lift-*.f64N/A
  \[\leadsto 1 + \frac{x \cdot \sqrt[3]{\color{blue}{x \cdot y}}}{y} \]
8. cbrt-prodN/A
  \[\leadsto 1 + \frac{x \cdot \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{y}\right)}}{y} \]
9. lift-cbrt.f64N/A
  \[\leadsto 1 + \frac{x \cdot \left(\color{blue}{\sqrt[3]{x}} \cdot \sqrt[3]{y}\right)}{y} \]
10. lift-cbrt.f64N/A
  \[\leadsto 1 + \frac{x \cdot \left(\sqrt[3]{x} \cdot \color{blue}{\sqrt[3]{y}}\right)}{y} \]
11. *-commutativeN/A
  \[\leadsto 1 + \frac{x \cdot \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{x}\right)}}{y} \]
12. lift-*.f64N/A
  \[\leadsto 1 + \frac{x \cdot \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{x}\right)}}{y} \]
13. associate-*l/N/A
  \[\leadsto 1 + \color{blue}{\frac{x}{y} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{x}\right)} \]
14. lift-/.f64N/A
  \[\leadsto 1 + \color{blue}{\frac{x}{y}} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{x}\right) \]
15. +-commutativeN/A
  \[\leadsto \color{blue}{\frac{x}{y} \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{x}\right) + 1} \]
Applied rewrites84.0%
\[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot {y}^{-0.6666666666666666}, \sqrt[3]{x}, 1\right)} \]
Add Preprocessing

Alternative 8: 72.8% accurate, 2.0× speedup?

\[\begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \sqrt[3]{x \cdot y}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({y}^{-0.6666666666666666}, {x}^{1.3333333333333333}, 1\right)\\ \end{array} \]

(FPCore (y x)
  :precision binary64
  (if (<= x 1.25e+117)
  (fma (/ x y) (cbrt (* x y)) 1.0)
  (fma (pow y -0.6666666666666666) (pow x 1.3333333333333333) 1.0)))

double code(double y, double x) {
	double tmp;
	if (x <= 1.25e+117) {
		tmp = fma((x / y), cbrt((x * y)), 1.0);
	} else {
		tmp = fma(pow(y, -0.6666666666666666), pow(x, 1.3333333333333333), 1.0);
	}
	return tmp;
}

function code(y, x)
	tmp = 0.0
	if (x <= 1.25e+117)
		tmp = fma(Float64(x / y), cbrt(Float64(x * y)), 1.0);
	else
		tmp = fma((y ^ -0.6666666666666666), (x ^ 1.3333333333333333), 1.0);
	end
	return tmp
end

code[y_, x_] := If[LessEqual[x, 1.25e+117], N[(N[(x / y), $MachinePrecision] * N[Power[N[(x * y), $MachinePrecision], 1/3], $MachinePrecision] + 1.0), $MachinePrecision], N[(N[Power[y, -0.6666666666666666], $MachinePrecision] * N[Power[x, 1.3333333333333333], $MachinePrecision] + 1.0), $MachinePrecision]]

\begin{array}{l}
\mathbf{if}\;x \leq 1.25 \cdot 10^{+117}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \sqrt[3]{x \cdot y}, 1\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left({y}^{-0.6666666666666666}, {x}^{1.3333333333333333}, 1\right)\\


\end{array}

Derivation

Split input into 2 regimes
if x < 1.25e117
1. Initial program 78.5%
  \[{\log y}^{\sin x} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
2. Taylor expanded in x around 0
  \[\leadsto \color{blue}{1} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
3. Step-by-step derivation
4. Applied rewrites69.5%
  \[\leadsto \color{blue}{1} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
5. Step-by-step derivation
  1. lift-+.f64N/A
    \[\leadsto \color{blue}{1 + \frac{x}{y} \cdot \sqrt[3]{y \cdot x}} \]
  2. +-commutativeN/A
    \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x} + 1} \]
  3. lift-*.f64N/A
    \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x}} + 1 \]
  4. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{y \cdot x}} + 1 \]
  5. lift-*.f64N/A
    \[\leadsto \frac{x}{y} \cdot \sqrt[3]{\color{blue}{y \cdot x}} + 1 \]
  6. cbrt-unprodN/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{x}\right)} + 1 \]
  7. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \left(\color{blue}{\sqrt[3]{y}} \cdot \sqrt[3]{x}\right) + 1 \]
  8. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \left(\sqrt[3]{y} \cdot \color{blue}{\sqrt[3]{x}}\right) + 1 \]
  9. *-commutativeN/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{y}\right)} + 1 \]
  10. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \left(\color{blue}{\sqrt[3]{x}} \cdot \sqrt[3]{y}\right) + 1 \]
  11. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \left(\sqrt[3]{x} \cdot \color{blue}{\sqrt[3]{y}}\right) + 1 \]
  12. cbrt-prodN/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{x \cdot y}} + 1 \]
  13. lift-*.f64N/A
    \[\leadsto \frac{x}{y} \cdot \sqrt[3]{\color{blue}{x \cdot y}} + 1 \]
  14. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{x \cdot y}} + 1 \]
  15. lower-fma.f6469.5%
    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \sqrt[3]{x \cdot y}, 1\right)} \]
6. Applied rewrites69.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \sqrt[3]{x \cdot y}, 1\right)} \]
if 1.25e117 < x
1. Initial program 78.5%
  \[{\log y}^{\sin x} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
2. Taylor expanded in x around 0
  \[\leadsto \color{blue}{1} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
3. Step-by-step derivation
4. Applied rewrites69.5%
  \[\leadsto \color{blue}{1} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
5. Step-by-step derivation
  1. lift-+.f64N/A
    \[\leadsto \color{blue}{1 + \frac{x}{y} \cdot \sqrt[3]{y \cdot x}} \]
  2. +-commutativeN/A
    \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x} + 1} \]
  3. lift-*.f64N/A
    \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x}} + 1 \]
  4. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{y \cdot x}} + 1 \]
  5. lift-*.f64N/A
    \[\leadsto \frac{x}{y} \cdot \sqrt[3]{\color{blue}{y \cdot x}} + 1 \]
  6. cbrt-unprodN/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{x}\right)} + 1 \]
  7. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \left(\color{blue}{\sqrt[3]{y}} \cdot \sqrt[3]{x}\right) + 1 \]
  8. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \left(\sqrt[3]{y} \cdot \color{blue}{\sqrt[3]{x}}\right) + 1 \]
  9. *-commutativeN/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{y}\right)} + 1 \]
  10. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \left(\color{blue}{\sqrt[3]{x}} \cdot \sqrt[3]{y}\right) + 1 \]
  11. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \left(\sqrt[3]{x} \cdot \color{blue}{\sqrt[3]{y}}\right) + 1 \]
  12. cbrt-prodN/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{x \cdot y}} + 1 \]
  13. lift-*.f64N/A
    \[\leadsto \frac{x}{y} \cdot \sqrt[3]{\color{blue}{x \cdot y}} + 1 \]
  14. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{x \cdot y}} + 1 \]
  15. lower-fma.f6469.5%
    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \sqrt[3]{x \cdot y}, 1\right)} \]
6. Applied rewrites69.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \sqrt[3]{x \cdot y}, 1\right)} \]
7. Step-by-step derivation
  1. lift-fma.f64N/A
    \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{x \cdot y} + 1} \]
  2. lift-/.f64N/A
    \[\leadsto \color{blue}{\frac{x}{y}} \cdot \sqrt[3]{x \cdot y} + 1 \]
  3. mult-flipN/A
    \[\leadsto \color{blue}{\left(x \cdot \frac{1}{y}\right)} \cdot \sqrt[3]{x \cdot y} + 1 \]
  4. lift-/.f64N/A
    \[\leadsto \left(x \cdot \color{blue}{\frac{1}{y}}\right) \cdot \sqrt[3]{x \cdot y} + 1 \]
  5. associate-*l*N/A
    \[\leadsto \color{blue}{x \cdot \left(\frac{1}{y} \cdot \sqrt[3]{x \cdot y}\right)} + 1 \]
  6. lift-*.f64N/A
    \[\leadsto x \cdot \color{blue}{\left(\frac{1}{y} \cdot \sqrt[3]{x \cdot y}\right)} + 1 \]
  7. *-commutativeN/A
    \[\leadsto \color{blue}{\left(\frac{1}{y} \cdot \sqrt[3]{x \cdot y}\right) \cdot x} + 1 \]
  8. lift-*.f64N/A
    \[\leadsto \color{blue}{\left(\frac{1}{y} \cdot \sqrt[3]{x \cdot y}\right)} \cdot x + 1 \]
  9. *-commutativeN/A
    \[\leadsto \color{blue}{\left(\sqrt[3]{x \cdot y} \cdot \frac{1}{y}\right)} \cdot x + 1 \]
  10. lift-/.f64N/A
    \[\leadsto \left(\sqrt[3]{x \cdot y} \cdot \color{blue}{\frac{1}{y}}\right) \cdot x + 1 \]
  11. mult-flip-revN/A
    \[\leadsto \color{blue}{\frac{\sqrt[3]{x \cdot y}}{y}} \cdot x + 1 \]
  12. lift-cbrt.f64N/A
    \[\leadsto \frac{\color{blue}{\sqrt[3]{x \cdot y}}}{y} \cdot x + 1 \]
  13. lift-*.f64N/A
    \[\leadsto \frac{\sqrt[3]{\color{blue}{x \cdot y}}}{y} \cdot x + 1 \]
  14. *-commutativeN/A
    \[\leadsto \frac{\sqrt[3]{\color{blue}{y \cdot x}}}{y} \cdot x + 1 \]
  15. cbrt-prodN/A
    \[\leadsto \frac{\color{blue}{\sqrt[3]{y} \cdot \sqrt[3]{x}}}{y} \cdot x + 1 \]
  16. lift-cbrt.f64N/A
    \[\leadsto \frac{\color{blue}{\sqrt[3]{y}} \cdot \sqrt[3]{x}}{y} \cdot x + 1 \]
  17. lift-cbrt.f64N/A
    \[\leadsto \frac{\sqrt[3]{y} \cdot \color{blue}{\sqrt[3]{x}}}{y} \cdot x + 1 \]
  18. associate-*l/N/A
    \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{y}}{y} \cdot \sqrt[3]{x}\right)} \cdot x + 1 \]
  19. associate-*l*N/A
    \[\leadsto \color{blue}{\frac{\sqrt[3]{y}}{y} \cdot \left(\sqrt[3]{x} \cdot x\right)} + 1 \]
  20. *-commutativeN/A
    \[\leadsto \frac{\sqrt[3]{y}}{y} \cdot \color{blue}{\left(x \cdot \sqrt[3]{x}\right)} + 1 \]
8. Applied rewrites38.9%
  \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{-0.6666666666666666}, {x}^{1.3333333333333333}, 1\right)} \]
Recombined 2 regimes into one program.
Add Preprocessing

Alternative 9: 72.8% accurate, 2.0× speedup?

\[\begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \sqrt[3]{x \cdot y}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{1.3333333333333333}}{{y}^{0.6666666666666666}}\\ \end{array} \]

(FPCore (y x)
  :precision binary64
  (if (<= x 1.25e+117)
  (fma (/ x y) (cbrt (* x y)) 1.0)
  (/ (pow x 1.3333333333333333) (pow y 0.6666666666666666))))

double code(double y, double x) {
	double tmp;
	if (x <= 1.25e+117) {
		tmp = fma((x / y), cbrt((x * y)), 1.0);
	} else {
		tmp = pow(x, 1.3333333333333333) / pow(y, 0.6666666666666666);
	}
	return tmp;
}

function code(y, x)
	tmp = 0.0
	if (x <= 1.25e+117)
		tmp = fma(Float64(x / y), cbrt(Float64(x * y)), 1.0);
	else
		tmp = Float64((x ^ 1.3333333333333333) / (y ^ 0.6666666666666666));
	end
	return tmp
end

code[y_, x_] := If[LessEqual[x, 1.25e+117], N[(N[(x / y), $MachinePrecision] * N[Power[N[(x * y), $MachinePrecision], 1/3], $MachinePrecision] + 1.0), $MachinePrecision], N[(N[Power[x, 1.3333333333333333], $MachinePrecision] / N[Power[y, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}
\mathbf{if}\;x \leq 1.25 \cdot 10^{+117}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \sqrt[3]{x \cdot y}, 1\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{{x}^{1.3333333333333333}}{{y}^{0.6666666666666666}}\\


\end{array}

Derivation

Split input into 2 regimes
if x < 1.25e117
1. Initial program 78.5%
  \[{\log y}^{\sin x} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
2. Taylor expanded in x around 0
  \[\leadsto \color{blue}{1} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
3. Step-by-step derivation
4. Applied rewrites69.5%
  \[\leadsto \color{blue}{1} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
5. Step-by-step derivation
  1. lift-+.f64N/A
    \[\leadsto \color{blue}{1 + \frac{x}{y} \cdot \sqrt[3]{y \cdot x}} \]
  2. +-commutativeN/A
    \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x} + 1} \]
  3. lift-*.f64N/A
    \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x}} + 1 \]
  4. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{y \cdot x}} + 1 \]
  5. lift-*.f64N/A
    \[\leadsto \frac{x}{y} \cdot \sqrt[3]{\color{blue}{y \cdot x}} + 1 \]
  6. cbrt-unprodN/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{x}\right)} + 1 \]
  7. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \left(\color{blue}{\sqrt[3]{y}} \cdot \sqrt[3]{x}\right) + 1 \]
  8. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \left(\sqrt[3]{y} \cdot \color{blue}{\sqrt[3]{x}}\right) + 1 \]
  9. *-commutativeN/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{y}\right)} + 1 \]
  10. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \left(\color{blue}{\sqrt[3]{x}} \cdot \sqrt[3]{y}\right) + 1 \]
  11. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \left(\sqrt[3]{x} \cdot \color{blue}{\sqrt[3]{y}}\right) + 1 \]
  12. cbrt-prodN/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{x \cdot y}} + 1 \]
  13. lift-*.f64N/A
    \[\leadsto \frac{x}{y} \cdot \sqrt[3]{\color{blue}{x \cdot y}} + 1 \]
  14. lift-cbrt.f64N/A
    \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{x \cdot y}} + 1 \]
  15. lower-fma.f6469.5%
    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \sqrt[3]{x \cdot y}, 1\right)} \]
6. Applied rewrites69.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \sqrt[3]{x \cdot y}, 1\right)} \]
if 1.25e117 < x
1. Initial program 78.5%
  \[{\log y}^{\sin x} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
2. Step-by-step derivation
  1. lift-+.f64N/A
    \[\leadsto \color{blue}{{\log y}^{\sin x} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x}} \]
  2. lift-*.f64N/A
    \[\leadsto {\log y}^{\sin x} + \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x}} \]
  3. lift-/.f64N/A
    \[\leadsto {\log y}^{\sin x} + \color{blue}{\frac{x}{y}} \cdot \sqrt[3]{y \cdot x} \]
  4. associate-*l/N/A
    \[\leadsto {\log y}^{\sin x} + \color{blue}{\frac{x \cdot \sqrt[3]{y \cdot x}}{y}} \]
  5. add-to-fractionN/A
    \[\leadsto \color{blue}{\frac{{\log y}^{\sin x} \cdot y + x \cdot \sqrt[3]{y \cdot x}}{y}} \]
  6. div-addN/A
    \[\leadsto \color{blue}{\frac{{\log y}^{\sin x} \cdot y}{y} + \frac{x \cdot \sqrt[3]{y \cdot x}}{y}} \]
  7. common-denominatorN/A
    \[\leadsto \color{blue}{\frac{\left({\log y}^{\sin x} \cdot y\right) \cdot y + \left(x \cdot \sqrt[3]{y \cdot x}\right) \cdot y}{y \cdot y}} \]
  8. div-flipN/A
    \[\leadsto \color{blue}{\frac{1}{\frac{y \cdot y}{\left({\log y}^{\sin x} \cdot y\right) \cdot y + \left(x \cdot \sqrt[3]{y \cdot x}\right) \cdot y}}} \]
  9. lower-unsound-/.f64N/A
    \[\leadsto \color{blue}{\frac{1}{\frac{y \cdot y}{\left({\log y}^{\sin x} \cdot y\right) \cdot y + \left(x \cdot \sqrt[3]{y \cdot x}\right) \cdot y}}} \]
  10. lower-unsound-/.f64N/A
    \[\leadsto \frac{1}{\color{blue}{\frac{y \cdot y}{\left({\log y}^{\sin x} \cdot y\right) \cdot y + \left(x \cdot \sqrt[3]{y \cdot x}\right) \cdot y}}} \]
  11. lower-*.f64N/A
    \[\leadsto \frac{1}{\frac{\color{blue}{y \cdot y}}{\left({\log y}^{\sin x} \cdot y\right) \cdot y + \left(x \cdot \sqrt[3]{y \cdot x}\right) \cdot y}} \]
  12. distribute-rgt-outN/A
    \[\leadsto \frac{1}{\frac{y \cdot y}{\color{blue}{y \cdot \left({\log y}^{\sin x} \cdot y + x \cdot \sqrt[3]{y \cdot x}\right)}}} \]
  13. lower-*.f64N/A
    \[\leadsto \frac{1}{\frac{y \cdot y}{\color{blue}{y \cdot \left({\log y}^{\sin x} \cdot y + x \cdot \sqrt[3]{y \cdot x}\right)}}} \]
  14. lower-fma.f64N/A
    \[\leadsto \frac{1}{\frac{y \cdot y}{y \cdot \color{blue}{\mathsf{fma}\left({\log y}^{\sin x}, y, x \cdot \sqrt[3]{y \cdot x}\right)}}} \]
  15. *-commutativeN/A
    \[\leadsto \frac{1}{\frac{y \cdot y}{y \cdot \mathsf{fma}\left({\log y}^{\sin x}, y, \color{blue}{\sqrt[3]{y \cdot x} \cdot x}\right)}} \]
3. Applied rewrites41.2%
  \[\leadsto \color{blue}{\frac{1}{\frac{y \cdot y}{y \cdot \mathsf{fma}\left({\log y}^{\sin x}, y, \sqrt[3]{x \cdot y} \cdot x\right)}}} \]
4. Step-by-step derivation
  1. lift-pow.f64N/A
    \[\leadsto \frac{1}{\frac{y \cdot y}{y \cdot \mathsf{fma}\left(\color{blue}{{\log y}^{\sin x}}, y, \sqrt[3]{x \cdot y} \cdot x\right)}} \]
  2. pow-to-expN/A
    \[\leadsto \frac{1}{\frac{y \cdot y}{y \cdot \mathsf{fma}\left(\color{blue}{e^{\log \log y \cdot \sin x}}, y, \sqrt[3]{x \cdot y} \cdot x\right)}} \]
  3. lower-unsound-exp.f64N/A
    \[\leadsto \frac{1}{\frac{y \cdot y}{y \cdot \mathsf{fma}\left(\color{blue}{e^{\log \log y \cdot \sin x}}, y, \sqrt[3]{x \cdot y} \cdot x\right)}} \]
  4. lower-unsound-*.f64N/A
    \[\leadsto \frac{1}{\frac{y \cdot y}{y \cdot \mathsf{fma}\left(e^{\color{blue}{\log \log y \cdot \sin x}}, y, \sqrt[3]{x \cdot y} \cdot x\right)}} \]
  5. lower-unsound-log.f64N/A
    \[\leadsto \frac{1}{\frac{y \cdot y}{y \cdot \mathsf{fma}\left(e^{\log \color{blue}{\log y} \cdot \sin x}, y, \sqrt[3]{x \cdot y} \cdot x\right)}} \]
  6. lower-unsound-log.f64N/A
    \[\leadsto \frac{1}{\frac{y \cdot y}{y \cdot \mathsf{fma}\left(e^{\color{blue}{\log \log y} \cdot \sin x}, y, \sqrt[3]{x \cdot y} \cdot x\right)}} \]
  7. lower-unsound-log.f6441.2%
    \[\leadsto \frac{1}{\frac{y \cdot y}{y \cdot \mathsf{fma}\left(e^{\log \color{blue}{\log y} \cdot \sin x}, y, \sqrt[3]{x \cdot y} \cdot x\right)}} \]
5. Applied rewrites41.2%
  \[\leadsto \frac{1}{\frac{y \cdot y}{y \cdot \mathsf{fma}\left(\color{blue}{e^{\log \log y \cdot \sin x}}, y, \sqrt[3]{x \cdot y} \cdot x\right)}} \]
6. Taylor expanded in y around 0
  \[\leadsto \color{blue}{\frac{{x}^{\frac{4}{3}}}{{y}^{\frac{2}{3}}}} \]
7. Step-by-step derivation
  1. lower-/.f64N/A
    \[\leadsto \frac{{x}^{\frac{4}{3}}}{\color{blue}{{y}^{\frac{2}{3}}}} \]
  2. lower-pow.f64N/A
    \[\leadsto \frac{{x}^{\frac{4}{3}}}{{\color{blue}{y}}^{\frac{2}{3}}} \]
  3. lower-pow.f6414.3%
    \[\leadsto \frac{{x}^{1.3333333333333333}}{{y}^{\color{blue}{0.6666666666666666}}} \]
8. Applied rewrites14.3%
  \[\leadsto \color{blue}{\frac{{x}^{1.3333333333333333}}{{y}^{0.6666666666666666}}} \]
Recombined 2 regimes into one program.
Add Preprocessing

Alternative 10: 69.5% accurate, 3.0× speedup?

\[\mathsf{fma}\left(\frac{\sqrt[3]{x \cdot y}}{y}, x, 1\right) \]

(FPCore (y x)
  :precision binary64
  (fma (/ (cbrt (* x y)) y) x 1.0))

double code(double y, double x) {
	return fma((cbrt((x * y)) / y), x, 1.0);
}

function code(y, x)
	return fma(Float64(cbrt(Float64(x * y)) / y), x, 1.0)
end

code[y_, x_] := N[(N[(N[Power[N[(x * y), $MachinePrecision], 1/3], $MachinePrecision] / y), $MachinePrecision] * x + 1.0), $MachinePrecision]

\mathsf{fma}\left(\frac{\sqrt[3]{x \cdot y}}{y}, x, 1\right)

Derivation

Initial program 78.5%
\[{\log y}^{\sin x} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
Taylor expanded in x around 0
\[\leadsto \color{blue}{1} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
Step-by-step derivation
Applied rewrites69.5%
\[\leadsto \color{blue}{1} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
Step-by-step derivation
1. lift-+.f64N/A
  \[\leadsto \color{blue}{1 + \frac{x}{y} \cdot \sqrt[3]{y \cdot x}} \]
2. +-commutativeN/A
  \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x} + 1} \]
3. lift-*.f64N/A
  \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x}} + 1 \]
4. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{y \cdot x}} + 1 \]
5. lift-*.f64N/A
  \[\leadsto \frac{x}{y} \cdot \sqrt[3]{\color{blue}{y \cdot x}} + 1 \]
6. cbrt-unprodN/A
  \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{x}\right)} + 1 \]
7. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \left(\color{blue}{\sqrt[3]{y}} \cdot \sqrt[3]{x}\right) + 1 \]
8. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \left(\sqrt[3]{y} \cdot \color{blue}{\sqrt[3]{x}}\right) + 1 \]
9. *-commutativeN/A
  \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{y}\right)} + 1 \]
10. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \left(\color{blue}{\sqrt[3]{x}} \cdot \sqrt[3]{y}\right) + 1 \]
11. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \left(\sqrt[3]{x} \cdot \color{blue}{\sqrt[3]{y}}\right) + 1 \]
12. cbrt-prodN/A
  \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{x \cdot y}} + 1 \]
13. lift-*.f64N/A
  \[\leadsto \frac{x}{y} \cdot \sqrt[3]{\color{blue}{x \cdot y}} + 1 \]
14. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{x \cdot y}} + 1 \]
15. lower-fma.f6469.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \sqrt[3]{x \cdot y}, 1\right)} \]
Applied rewrites69.5%
\[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \sqrt[3]{x \cdot y}, 1\right)} \]
Step-by-step derivation
1. lift-fma.f64N/A
  \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{x \cdot y} + 1} \]
2. lift-/.f64N/A
  \[\leadsto \color{blue}{\frac{x}{y}} \cdot \sqrt[3]{x \cdot y} + 1 \]
3. mult-flipN/A
  \[\leadsto \color{blue}{\left(x \cdot \frac{1}{y}\right)} \cdot \sqrt[3]{x \cdot y} + 1 \]
4. lift-/.f64N/A
  \[\leadsto \left(x \cdot \color{blue}{\frac{1}{y}}\right) \cdot \sqrt[3]{x \cdot y} + 1 \]
5. associate-*l*N/A
  \[\leadsto \color{blue}{x \cdot \left(\frac{1}{y} \cdot \sqrt[3]{x \cdot y}\right)} + 1 \]
6. lift-*.f64N/A
  \[\leadsto x \cdot \color{blue}{\left(\frac{1}{y} \cdot \sqrt[3]{x \cdot y}\right)} + 1 \]
7. *-commutativeN/A
  \[\leadsto \color{blue}{\left(\frac{1}{y} \cdot \sqrt[3]{x \cdot y}\right) \cdot x} + 1 \]
8. lower-fma.f6469.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{y} \cdot \sqrt[3]{x \cdot y}, x, 1\right)} \]
9. lift-*.f64N/A
  \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{y} \cdot \sqrt[3]{x \cdot y}}, x, 1\right) \]
10. *-commutativeN/A
  \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt[3]{x \cdot y} \cdot \frac{1}{y}}, x, 1\right) \]
11. lift-/.f64N/A
  \[\leadsto \mathsf{fma}\left(\sqrt[3]{x \cdot y} \cdot \color{blue}{\frac{1}{y}}, x, 1\right) \]
12. mult-flip-revN/A
  \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sqrt[3]{x \cdot y}}{y}}, x, 1\right) \]
13. lower-/.f6469.5%
  \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sqrt[3]{x \cdot y}}{y}}, x, 1\right) \]
Applied rewrites69.5%
\[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{x \cdot y}}{y}, x, 1\right)} \]
Add Preprocessing

Alternative 11: 69.5% accurate, 3.0× speedup?

\[\mathsf{fma}\left(\frac{x}{y}, \sqrt[3]{x \cdot y}, 1\right) \]

(FPCore (y x)
  :precision binary64
  (fma (/ x y) (cbrt (* x y)) 1.0))

double code(double y, double x) {
	return fma((x / y), cbrt((x * y)), 1.0);
}

function code(y, x)
	return fma(Float64(x / y), cbrt(Float64(x * y)), 1.0)
end

code[y_, x_] := N[(N[(x / y), $MachinePrecision] * N[Power[N[(x * y), $MachinePrecision], 1/3], $MachinePrecision] + 1.0), $MachinePrecision]

\mathsf{fma}\left(\frac{x}{y}, \sqrt[3]{x \cdot y}, 1\right)

Derivation

Initial program 78.5%
\[{\log y}^{\sin x} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
Taylor expanded in x around 0
\[\leadsto \color{blue}{1} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
Step-by-step derivation
Applied rewrites69.5%
\[\leadsto \color{blue}{1} + \frac{x}{y} \cdot \sqrt[3]{y \cdot x} \]
Step-by-step derivation
1. lift-+.f64N/A
  \[\leadsto \color{blue}{1 + \frac{x}{y} \cdot \sqrt[3]{y \cdot x}} \]
2. +-commutativeN/A
  \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x} + 1} \]
3. lift-*.f64N/A
  \[\leadsto \color{blue}{\frac{x}{y} \cdot \sqrt[3]{y \cdot x}} + 1 \]
4. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{y \cdot x}} + 1 \]
5. lift-*.f64N/A
  \[\leadsto \frac{x}{y} \cdot \sqrt[3]{\color{blue}{y \cdot x}} + 1 \]
6. cbrt-unprodN/A
  \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{x}\right)} + 1 \]
7. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \left(\color{blue}{\sqrt[3]{y}} \cdot \sqrt[3]{x}\right) + 1 \]
8. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \left(\sqrt[3]{y} \cdot \color{blue}{\sqrt[3]{x}}\right) + 1 \]
9. *-commutativeN/A
  \[\leadsto \frac{x}{y} \cdot \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{y}\right)} + 1 \]
10. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \left(\color{blue}{\sqrt[3]{x}} \cdot \sqrt[3]{y}\right) + 1 \]
11. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \left(\sqrt[3]{x} \cdot \color{blue}{\sqrt[3]{y}}\right) + 1 \]
12. cbrt-prodN/A
  \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{x \cdot y}} + 1 \]
13. lift-*.f64N/A
  \[\leadsto \frac{x}{y} \cdot \sqrt[3]{\color{blue}{x \cdot y}} + 1 \]
14. lift-cbrt.f64N/A
  \[\leadsto \frac{x}{y} \cdot \color{blue}{\sqrt[3]{x \cdot y}} + 1 \]
15. lower-fma.f6469.5%
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \sqrt[3]{x \cdot y}, 1\right)} \]
Applied rewrites69.5%
\[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \sqrt[3]{x \cdot y}, 1\right)} \]
Add Preprocessing

pow(log(y),sin(x))+(x/y)cbrt(yx)

could not determine a ground truth (more)

Specification

Local Percentage Accuracy vs ?

Accuracy vs Speed?

Initial Program: 78.5% accurate, 1.0× speedup?

Alternative 1: 99.4% accurate, 0.8× speedup?

Alternative 2: 96.7% accurate, 0.5× speedup?

`if (+.f64 (pow.f64 (log.f64 y) (sin.f64 x)) (.f64 (/.f64 x y) (cbrt.f64 (.f64 y x)))) < 1.9999999999999998e250`

`if 1.9999999999999998e250 < (+.f64 (pow.f64 (log.f64 y) (sin.f64 x)) (.f64 (/.f64 x y) (cbrt.f64 (.f64 y x))))`

Alternative 3: 95.6% accurate, 0.9× speedup?

Alternative 4: 94.3% accurate, 0.5× speedup?

`if (+.f64 (pow.f64 (log.f64 y) (sin.f64 x)) (.f64 (/.f64 x y) (cbrt.f64 (.f64 y x)))) < 500`

`if 500 < (+.f64 (pow.f64 (log.f64 y) (sin.f64 x)) (.f64 (/.f64 x y) (cbrt.f64 (.f64 y x))))`

Alternative 5: 86.5% accurate, 1.9× speedup?

Alternative 6: 86.5% accurate, 1.9× speedup?

Alternative 7: 84.0% accurate, 2.0× speedup?

Alternative 8: 72.8% accurate, 2.0× speedup?

`if x < 1.25e117`

`if 1.25e117 < x`

Alternative 9: 72.8% accurate, 2.0× speedup?

`if x < 1.25e117`

`if 1.25e117 < x`

Alternative 10: 69.5% accurate, 3.0× speedup?

Alternative 11: 69.5% accurate, 3.0× speedup?

Reproduce

could not determine a ground truth (more)

Specification

Local Percentage Accuracy vs ?

Accuracy vs Speed?

Initial Program: 78.5% accurate, 1.0× speedupMathFPCoreCJavaJuliaWolframTeX?

Alternative 1: 99.4% accurate, 0.8× speedupMathFPCoreCJuliaWolframTeX?

Alternative 2: 96.7% accurate, 0.5× speedupMathFPCoreCJuliaWolframTeX?

if (+.f64 (pow.f64 (log.f64 y) (sin.f64 x)) (*.f64 (/.f64 x y) (cbrt.f64 (*.f64 y x)))) < 1.9999999999999998e250

if 1.9999999999999998e250 < (+.f64 (pow.f64 (log.f64 y) (sin.f64 x)) (*.f64 (/.f64 x y) (cbrt.f64 (*.f64 y x))))

Alternative 3: 95.6% accurate, 0.9× speedupMathFPCoreCJuliaWolframTeX?

Alternative 4: 94.3% accurate, 0.5× speedupMathFPCoreCJuliaWolframTeX?

if (+.f64 (pow.f64 (log.f64 y) (sin.f64 x)) (*.f64 (/.f64 x y) (cbrt.f64 (*.f64 y x)))) < 500

if 500 < (+.f64 (pow.f64 (log.f64 y) (sin.f64 x)) (*.f64 (/.f64 x y) (cbrt.f64 (*.f64 y x))))

Alternative 5: 86.5% accurate, 1.9× speedupMathFPCoreCJuliaWolframTeX?

Alternative 6: 86.5% accurate, 1.9× speedupMathFPCoreCJuliaWolframTeX?

Alternative 7: 84.0% accurate, 2.0× speedupMathFPCoreCJuliaWolframTeX?

Alternative 8: 72.8% accurate, 2.0× speedupMathFPCoreCJuliaWolframTeX?

if x < 1.25e117

if 1.25e117 < x

Alternative 9: 72.8% accurate, 2.0× speedupMathFPCoreCJuliaWolframTeX?

if x < 1.25e117

if 1.25e117 < x

Alternative 10: 69.5% accurate, 3.0× speedupMathFPCoreCJuliaWolframTeX?

Alternative 11: 69.5% accurate, 3.0× speedupMathFPCoreCJuliaWolframTeX?

Reproduce

Initial Program: 78.5% accurate, 1.0× speedup?

Alternative 1: 99.4% accurate, 0.8× speedup?

Alternative 2: 96.7% accurate, 0.5× speedup?

`if (+.f64 (pow.f64 (log.f64 y) (sin.f64 x)) (.f64 (/.f64 x y) (cbrt.f64 (.f64 y x)))) < 1.9999999999999998e250`

`if 1.9999999999999998e250 < (+.f64 (pow.f64 (log.f64 y) (sin.f64 x)) (.f64 (/.f64 x y) (cbrt.f64 (.f64 y x))))`

Alternative 3: 95.6% accurate, 0.9× speedup?

Alternative 4: 94.3% accurate, 0.5× speedup?

`if (+.f64 (pow.f64 (log.f64 y) (sin.f64 x)) (.f64 (/.f64 x y) (cbrt.f64 (.f64 y x)))) < 500`

`if 500 < (+.f64 (pow.f64 (log.f64 y) (sin.f64 x)) (.f64 (/.f64 x y) (cbrt.f64 (.f64 y x))))`

Alternative 5: 86.5% accurate, 1.9× speedup?

Alternative 6: 86.5% accurate, 1.9× speedup?

Alternative 7: 84.0% accurate, 2.0× speedup?

Alternative 8: 72.8% accurate, 2.0× speedup?

`if x < 1.25e117`

`if 1.25e117 < x`

Alternative 9: 72.8% accurate, 2.0× speedup?

`if x < 1.25e117`

`if 1.25e117 < x`

Alternative 10: 69.5% accurate, 3.0× speedup?

Alternative 11: 69.5% accurate, 3.0× speedup?