?

Average Error: 18.4 → 0.3
Time: 13.7s
Precision: binary64
Cost: 46276

?

\[\left(0 \leq x \land x \leq 1.79 \cdot 10^{+308}\right) \land \left(0 \leq y \land y \leq 1.79 \cdot 10^{+308}\right)\]
\[{x}^{y} - {y}^{x} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{expm1}\left(\log x \cdot y\right) - x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(y \cdot 3\right)} - {y}^{\left(x \cdot 3\right)}}{\mathsf{fma}\left({y}^{x}, {y}^{x} + {x}^{y}, {x}^{\left(y \cdot 2\right)}\right)}\\ \end{array} \]
(FPCore (x y) :precision binary64 (- (pow x y) (pow y x)))
↓
(FPCore (x y)
 :precision binary64
 (if (<= x 1.9e-11)
   (- (expm1 (* (log x) y)) (* x (log y)))
   (/
    (- (pow x (* y 3.0)) (pow y (* x 3.0)))
    (fma (pow y x) (+ (pow y x) (pow x y)) (pow x (* y 2.0))))))
double code(double x, double y) {
	return pow(x, y) - pow(y, x);
}
↓
double code(double x, double y) {
	double tmp;
	if (x <= 1.9e-11) {
		tmp = expm1((log(x) * y)) - (x * log(y));
	} else {
		tmp = (pow(x, (y * 3.0)) - pow(y, (x * 3.0))) / fma(pow(y, x), (pow(y, x) + pow(x, y)), pow(x, (y * 2.0)));
	}
	return tmp;
}

function code(x, y)
	return Float64((x ^ y) - (y ^ x))
end
↓
function code(x, y)
	tmp = 0.0
	if (x <= 1.9e-11)
		tmp = Float64(expm1(Float64(log(x) * y)) - Float64(x * log(y)));
	else
		tmp = Float64(Float64((x ^ Float64(y * 3.0)) - (y ^ Float64(x * 3.0))) / fma((y ^ x), Float64((y ^ x) + (x ^ y)), (x ^ Float64(y * 2.0))));
	end
	return tmp
end

code[x_, y_] := N[(N[Power[x, y], $MachinePrecision] - N[Power[y, x], $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_] := If[LessEqual[x, 1.9e-11], N[(N[(Exp[N[(N[Log[x], $MachinePrecision] * y), $MachinePrecision]] - 1), $MachinePrecision] - N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, N[(y * 3.0), $MachinePrecision]], $MachinePrecision] - N[Power[y, N[(x * 3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Power[y, x], $MachinePrecision] * N[(N[Power[y, x], $MachinePrecision] + N[Power[x, y], $MachinePrecision]), $MachinePrecision] + N[Power[x, N[(y * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

{x}^{y} - {y}^{x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq 1.9 \cdot 10^{-11}:\\
\;\;\;\;\mathsf{expm1}\left(\log x \cdot y\right) - x \cdot \log y\\

\mathbf{else}:\\
\;\;\;\;\frac{{x}^{\left(y \cdot 3\right)} - {y}^{\left(x \cdot 3\right)}}{\mathsf{fma}\left({y}^{x}, {y}^{x} + {x}^{y}, {x}^{\left(y \cdot 2\right)}\right)}\\


\end{array}

Error?

Derivation?

  1. Split input into 2 regimes

  2. if x < 1.8999999999999999e-11

    1. Initial program 28.8

      \[{x}^{y} - {y}^{x} \]

    2. Taylor expanded in x around 0 28.8

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\log y \cdot x\right) + {x}^{y}\right) - 1} \]

    3. Simplified15.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(-\log y, x, {x}^{y} + -1\right)} \]
      Proof

      [Start]28.8

      \[ \left(-1 \cdot \left(\log y \cdot x\right) + {x}^{y}\right) - 1 \]

      associate--l+ [=>]15.5

      \[ \color{blue}{-1 \cdot \left(\log y \cdot x\right) + \left({x}^{y} - 1\right)} \]

      mul-1-neg [=>]15.5

      \[ \color{blue}{\left(-\log y \cdot x\right)} + \left({x}^{y} - 1\right) \]

      distribute-lft-neg-in [=>]15.5

      \[ \color{blue}{\left(-\log y\right) \cdot x} + \left({x}^{y} - 1\right) \]

      log-rec [<=]15.5

      \[ \color{blue}{\log \left(\frac{1}{y}\right)} \cdot x + \left({x}^{y} - 1\right) \]

      fma-def [=>]15.5

      \[ \color{blue}{\mathsf{fma}\left(\log \left(\frac{1}{y}\right), x, {x}^{y} - 1\right)} \]

      log-rec [=>]15.5

      \[ \mathsf{fma}\left(\color{blue}{-\log y}, x, {x}^{y} - 1\right) \]

      sub-neg [=>]15.5

      \[ \mathsf{fma}\left(-\log y, x, \color{blue}{{x}^{y} + \left(-1\right)}\right) \]

      metadata-eval [=>]15.5

      \[ \mathsf{fma}\left(-\log y, x, {x}^{y} + \color{blue}{-1}\right) \]

    4. Taylor expanded in x around inf 28.8

      \[\leadsto \color{blue}{\left(-1 \cdot \left(\log y \cdot x\right) + e^{-1 \cdot \left(y \cdot \log \left(\frac{1}{x}\right)\right)}\right) - 1} \]

    5. Simplified0.3

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\left(-\log x\right) \cdot \left(-y\right)\right) - x \cdot \log y} \]
      Proof

      [Start]28.8

      \[ \left(-1 \cdot \left(\log y \cdot x\right) + e^{-1 \cdot \left(y \cdot \log \left(\frac{1}{x}\right)\right)}\right) - 1 \]

      sub-neg [=>]28.8

      \[ \color{blue}{\left(-1 \cdot \left(\log y \cdot x\right) + e^{-1 \cdot \left(y \cdot \log \left(\frac{1}{x}\right)\right)}\right) + \left(-1\right)} \]

      metadata-eval [=>]28.8

      \[ \left(-1 \cdot \left(\log y \cdot x\right) + e^{-1 \cdot \left(y \cdot \log \left(\frac{1}{x}\right)\right)}\right) + \color{blue}{-1} \]

      +-commutative [=>]28.8

      \[ \color{blue}{-1 + \left(-1 \cdot \left(\log y \cdot x\right) + e^{-1 \cdot \left(y \cdot \log \left(\frac{1}{x}\right)\right)}\right)} \]

      +-commutative [=>]28.8

      \[ -1 + \color{blue}{\left(e^{-1 \cdot \left(y \cdot \log \left(\frac{1}{x}\right)\right)} + -1 \cdot \left(\log y \cdot x\right)\right)} \]

      *-commutative [<=]28.8

      \[ -1 + \left(e^{-1 \cdot \left(y \cdot \log \left(\frac{1}{x}\right)\right)} + -1 \cdot \color{blue}{\left(x \cdot \log y\right)}\right) \]

      mul-1-neg [=>]28.8

      \[ -1 + \left(e^{-1 \cdot \left(y \cdot \log \left(\frac{1}{x}\right)\right)} + \color{blue}{\left(-x \cdot \log y\right)}\right) \]

      unsub-neg [=>]28.8

      \[ -1 + \color{blue}{\left(e^{-1 \cdot \left(y \cdot \log \left(\frac{1}{x}\right)\right)} - x \cdot \log y\right)} \]

      associate-+r- [=>]15.5

      \[ \color{blue}{\left(-1 + e^{-1 \cdot \left(y \cdot \log \left(\frac{1}{x}\right)\right)}\right) - x \cdot \log y} \]

    if 1.8999999999999999e-11 < x

    1. Initial program 0.3

      \[{x}^{y} - {y}^{x} \]

    2. Applied egg-rr0.4

      \[\leadsto \color{blue}{\left({x}^{\left(y \cdot 3\right)} - {y}^{\left(x \cdot 3\right)}\right) \cdot \frac{1}{{x}^{\left(2 \cdot y\right)} + {y}^{x} \cdot \left({x}^{y} + {y}^{x}\right)}} \]

    3. Simplified0.4

      \[\leadsto \color{blue}{\frac{{x}^{\left(y \cdot 3\right)} - {y}^{\left(x \cdot 3\right)}}{\mathsf{fma}\left({y}^{x}, {x}^{y} + {y}^{x}, {x}^{\left(y \cdot 2\right)}\right)}} \]
      Proof

      [Start]0.4

      \[ \left({x}^{\left(y \cdot 3\right)} - {y}^{\left(x \cdot 3\right)}\right) \cdot \frac{1}{{x}^{\left(2 \cdot y\right)} + {y}^{x} \cdot \left({x}^{y} + {y}^{x}\right)} \]

      associate-*r/ [=>]0.4

      \[ \color{blue}{\frac{\left({x}^{\left(y \cdot 3\right)} - {y}^{\left(x \cdot 3\right)}\right) \cdot 1}{{x}^{\left(2 \cdot y\right)} + {y}^{x} \cdot \left({x}^{y} + {y}^{x}\right)}} \]

      *-rgt-identity [=>]0.4

      \[ \frac{\color{blue}{{x}^{\left(y \cdot 3\right)} - {y}^{\left(x \cdot 3\right)}}}{{x}^{\left(2 \cdot y\right)} + {y}^{x} \cdot \left({x}^{y} + {y}^{x}\right)} \]

      +-commutative [=>]0.4

      \[ \frac{{x}^{\left(y \cdot 3\right)} - {y}^{\left(x \cdot 3\right)}}{\color{blue}{{y}^{x} \cdot \left({x}^{y} + {y}^{x}\right) + {x}^{\left(2 \cdot y\right)}}} \]

      fma-def [=>]0.4

      \[ \frac{{x}^{\left(y \cdot 3\right)} - {y}^{\left(x \cdot 3\right)}}{\color{blue}{\mathsf{fma}\left({y}^{x}, {x}^{y} + {y}^{x}, {x}^{\left(2 \cdot y\right)}\right)}} \]

      *-commutative [=>]0.4

      \[ \frac{{x}^{\left(y \cdot 3\right)} - {y}^{\left(x \cdot 3\right)}}{\mathsf{fma}\left({y}^{x}, {x}^{y} + {y}^{x}, {x}^{\color{blue}{\left(y \cdot 2\right)}}\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{expm1}\left(\log x \cdot y\right) - x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(y \cdot 3\right)} - {y}^{\left(x \cdot 3\right)}}{\mathsf{fma}\left({y}^{x}, {y}^{x} + {x}^{y}, {x}^{\left(y \cdot 2\right)}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error0.3
Cost40004
\[\begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{expm1}\left(\log x \cdot y\right) - x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(y \cdot 3\right)} - {y}^{\left(x \cdot 3\right)}}{{x}^{\left(y \cdot 2\right)} + {y}^{x} \cdot \left({y}^{x} + {x}^{y}\right)}\\ \end{array} \]
Alternative 2
Error0.5
Cost39752
\[\begin{array}{l} t_0 := {x}^{y} - {y}^{x}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 10^{-12}:\\ \;\;\;\;\log x \cdot y - x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{t_0}}\\ \end{array} \]
Alternative 3
Error0.5
Cost39625
\[\begin{array}{l} t_0 := {x}^{y} - {y}^{x}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-8} \lor \neg \left(t_0 \leq 10^{-12}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\log x \cdot y - x \cdot \log y\\ \end{array} \]
Alternative 4
Error10.0
Cost39497
\[\begin{array}{l} t_0 := {x}^{y} - {y}^{x}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{-14} \lor \neg \left(t_0 \leq 10^{-12}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-x\right)\\ \end{array} \]
Alternative 5
Error0.3
Cost19780
\[\begin{array}{l} \mathbf{if}\;x \leq 2.35 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{expm1}\left(\log x \cdot y\right) - x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{{x}^{y} - {y}^{x}}}\\ \end{array} \]
Alternative 6
Error19.9
Cost6920
\[\begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-30}:\\ \;\;\;\;{x}^{y} + -1\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{-11}:\\ \;\;\;\;\log y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - {y}^{x}\\ \end{array} \]
Alternative 7
Error31.4
Cost6788
\[\begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-11}:\\ \;\;\;\;\log y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - {y}^{x}\\ \end{array} \]
Alternative 8
Error52.0
Cost6656
\[\log y \cdot \left(-x\right) \]

Error

Reproduce?

herbie shell --seed 1 
(FPCore (x y)
  :name "x^y - y^x"
  :precision binary64
  :pre (and (and (<= 0.0 x) (<= x 1.79e+308)) (and (<= 0.0 y) (<= y 1.79e+308)))
  (- (pow x y) (pow y x)))