Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2

Average Accuracy: 75.8% → 99.7%

Time: 12.7s

Precision: binary64

Cost: 39232

Math

\[x \cdot \log \left(\frac{x}{y}\right) - z \]

↓

\[\begin{array}{l} t_0 := \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\\ x \cdot \left(t_0 + 2 \cdot t_0\right) - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (* x (log (/ x y))) z))

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (/ (cbrt x) (cbrt y))))) (- (* x (+ t_0 (* 2.0 t_0))) z)))

C

double code(double x, double y, double z) {
	return (x * log((x / y))) - z;
}

↓

double code(double x, double y, double z) {
	double t_0 = log((cbrt(x) / cbrt(y)));
	return (x * (t_0 + (2.0 * t_0))) - z;
}

Java

public static double code(double x, double y, double z) {
	return (x * Math.log((x / y))) - z;
}

↓

public static double code(double x, double y, double z) {
	double t_0 = Math.log((Math.cbrt(x) / Math.cbrt(y)));
	return (x * (t_0 + (2.0 * t_0))) - z;
}

Julia

function code(x, y, z)
	return Float64(Float64(x * log(Float64(x / y))) - z)
end

↓

function code(x, y, z)
	t_0 = log(Float64(cbrt(x) / cbrt(y)))
	return Float64(Float64(x * Float64(t_0 + Float64(2.0 * t_0))) - z)
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[Log[N[(N[Power[x, 1/3], $MachinePrecision] / N[Power[y, 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(x * N[(t$95$0 + N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Target

Original	75.8%
Target	87.8%
Herbie	99.7%

\[\begin{array}{l} \mathbf{if}\;y < 7.595077799083773 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Derivation

1. Initial program 75.8%

   \[x \cdot \log \left(\frac{x}{y}\right) - z \]

2. Applied egg-rr 48.8%

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

   [Start] 75.8	\[ x \cdot \log \left(\frac{x}{y}\right) - z \]
   frac-2neg [=>] 75.8	\[ x \cdot \log \color{blue}{\left(\frac{-x}{-y}\right)} - z \]
   log-div [=>] 48.8	\[ x \cdot \color{blue}{\left(\log \left(-x\right) - \log \left(-y\right)\right)} - z \]

3. Applied egg-rr 94.5%

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

   [Start] 48.8	\[ x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z \]
   diff-log [=>] 75.8	\[ x \cdot \color{blue}{\log \left(\frac{-x}{-y}\right)} - z \]
   add-cube-cbrt [=>] 75.8	\[ x \cdot \log \left(\frac{\color{blue}{\left(\sqrt[3]{-x} \cdot \sqrt[3]{-x}\right) \cdot \sqrt[3]{-x}}}{-y}\right) - z \]
   add-cube-cbrt [=>] 75.8	\[ x \cdot \log \left(\frac{\left(\sqrt[3]{-x} \cdot \sqrt[3]{-x}\right) \cdot \sqrt[3]{-x}}{\color{blue}{\left(\sqrt[3]{-y} \cdot \sqrt[3]{-y}\right) \cdot \sqrt[3]{-y}}}\right) - z \]
   times-frac [=>] 75.8	\[ x \cdot \log \color{blue}{\left(\frac{\sqrt[3]{-x} \cdot \sqrt[3]{-x}}{\sqrt[3]{-y} \cdot \sqrt[3]{-y}} \cdot \frac{\sqrt[3]{-x}}{\sqrt[3]{-y}}\right)} - z \]
   log-prod [=>] 94.5	\[ x \cdot \color{blue}{\left(\log \left(\frac{\sqrt[3]{-x} \cdot \sqrt[3]{-x}}{\sqrt[3]{-y} \cdot \sqrt[3]{-y}}\right) + \log \left(\frac{\sqrt[3]{-x}}{\sqrt[3]{-y}}\right)\right)} - z \]
   cbrt-unprod [=>] 52.0	\[ x \cdot \left(\log \left(\frac{\color{blue}{\sqrt[3]{\left(-x\right) \cdot \left(-x\right)}}}{\sqrt[3]{-y} \cdot \sqrt[3]{-y}}\right) + \log \left(\frac{\sqrt[3]{-x}}{\sqrt[3]{-y}}\right)\right) - z \]
   sqr-neg [=>] 52.0	\[ x \cdot \left(\log \left(\frac{\sqrt[3]{\color{blue}{x \cdot x}}}{\sqrt[3]{-y} \cdot \sqrt[3]{-y}}\right) + \log \left(\frac{\sqrt[3]{-x}}{\sqrt[3]{-y}}\right)\right) - z \]
   cbrt-unprod [<=] 94.5	\[ x \cdot \left(\log \left(\frac{\color{blue}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{-y} \cdot \sqrt[3]{-y}}\right) + \log \left(\frac{\sqrt[3]{-x}}{\sqrt[3]{-y}}\right)\right) - z \]
   pow2 [=>] 94.5	\[ x \cdot \left(\log \left(\frac{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}}{\sqrt[3]{-y} \cdot \sqrt[3]{-y}}\right) + \log \left(\frac{\sqrt[3]{-x}}{\sqrt[3]{-y}}\right)\right) - z \]
   cbrt-unprod [=>] 51.0	\[ x \cdot \left(\log \left(\frac{{\left(\sqrt[3]{x}\right)}^{2}}{\color{blue}{\sqrt[3]{\left(-y\right) \cdot \left(-y\right)}}}\right) + \log \left(\frac{\sqrt[3]{-x}}{\sqrt[3]{-y}}\right)\right) - z \]
   sqr-neg [=>] 51.0	\[ x \cdot \left(\log \left(\frac{{\left(\sqrt[3]{x}\right)}^{2}}{\sqrt[3]{\color{blue}{y \cdot y}}}\right) + \log \left(\frac{\sqrt[3]{-x}}{\sqrt[3]{-y}}\right)\right) - z \]
   cbrt-unprod [<=] 94.5	\[ x \cdot \left(\log \left(\frac{{\left(\sqrt[3]{x}\right)}^{2}}{\color{blue}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}\right) + \log \left(\frac{\sqrt[3]{-x}}{\sqrt[3]{-y}}\right)\right) - z \]
   pow2 [=>] 94.5	\[ x \cdot \left(\log \left(\frac{{\left(\sqrt[3]{x}\right)}^{2}}{\color{blue}{{\left(\sqrt[3]{y}\right)}^{2}}}\right) + \log \left(\frac{\sqrt[3]{-x}}{\sqrt[3]{-y}}\right)\right) - z \]

4. Simplified 99.7%

   \[\leadsto x \cdot \color{blue}{\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)} - z \]
   Proof

   [Start] 94.5	\[ x \cdot \left(\log \left(\frac{{\left(\sqrt[3]{x}\right)}^{2}}{{\left(\sqrt[3]{y}\right)}^{2}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) - z \]
   log-div [=>] 99.6	\[ x \cdot \left(\color{blue}{\left(\log \left({\left(\sqrt[3]{x}\right)}^{2}\right) - \log \left({\left(\sqrt[3]{y}\right)}^{2}\right)\right)} + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) - z \]
   log-pow [=>] 50.8	\[ x \cdot \left(\left(\color{blue}{2 \cdot \log \left(\sqrt[3]{x}\right)} - \log \left({\left(\sqrt[3]{y}\right)}^{2}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) - z \]
   log-pow [=>] 50.8	\[ x \cdot \left(\left(2 \cdot \log \left(\sqrt[3]{x}\right) - \color{blue}{2 \cdot \log \left(\sqrt[3]{y}\right)}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) - z \]
   distribute-lft-out-- [=>] 50.8	\[ x \cdot \left(\color{blue}{2 \cdot \left(\log \left(\sqrt[3]{x}\right) - \log \left(\sqrt[3]{y}\right)\right)} + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) - z \]
   log-div [<=] 99.7	\[ x \cdot \left(2 \cdot \color{blue}{\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)} + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) - z \]

5. Final simplification 99.7%

   \[\leadsto x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + 2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) - z \]

Alternatives

Alternative 1
Accuracy	86.8%
Cost	26696

\[\begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t_0 \leq 10^{+290}:\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \]

Alternative 2
Accuracy	87.3%
Cost	20424

\[\begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t_0 \leq 10^{+290}:\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 3
Accuracy	99.6%
Cost	19908

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(2 \cdot \log \left(\frac{\sqrt{x}}{\sqrt{y}}\right)\right) - z\\ \end{array} \]

Alternative 4
Accuracy	86.6%
Cost	13648

\[\begin{array}{l} t_0 := \left(-z\right) - x \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{if}\;x \leq -2.8 \cdot 10^{+171}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -2.55 \cdot 10^{-154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-178}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+123}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \]

Alternative 5
Accuracy	93.4%
Cost	13644

\[\begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+171}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-154}:\\ \;\;\;\;\left(-z\right) - x \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 6
Accuracy	99.5%
Cost	13508

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 7
Accuracy	67.2%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+17}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{-84}:\\ \;\;\;\;x \cdot \left(-\log \left(\frac{y}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 8
Accuracy	67.2%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+16}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-84}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 9
Accuracy	50.9%
Cost	128

\[-z \]

Reproduce

herbie shell --seed 2023151 
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< y 7.595077799083773e-308) (- (* x (log (/ x y))) z) (- (* x (- (log x) (log y))) z))

  (- (* x (log (/ x y))) z))