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

Average Accuracy: 76.2% → 99.7%

Time: 15.1s

Precision: binary64

Cost: 19904

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Target

Original	76.2%
Target	87.7%
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 76.2%

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

2. Applied egg-rr 76.1%

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

   [Start] 76.2	\[ x \cdot \log \left(\frac{x}{y}\right) - z \]
   add-cube-cbrt [=>] 76.2	\[ x \cdot \log \color{blue}{\left(\left(\sqrt[3]{\frac{x}{y}} \cdot \sqrt[3]{\frac{x}{y}}\right) \cdot \sqrt[3]{\frac{x}{y}}\right)} - z \]
   log-prod [=>] 76.1	\[ x \cdot \color{blue}{\left(\log \left(\sqrt[3]{\frac{x}{y}} \cdot \sqrt[3]{\frac{x}{y}}\right) + \log \left(\sqrt[3]{\frac{x}{y}}\right)\right)} - z \]
   distribute-rgt-in [=>] 76.1	\[ \color{blue}{\left(\log \left(\sqrt[3]{\frac{x}{y}} \cdot \sqrt[3]{\frac{x}{y}}\right) \cdot x + \log \left(\sqrt[3]{\frac{x}{y}}\right) \cdot x\right)} - z \]
   pow2 [=>] 76.1	\[ \left(\log \color{blue}{\left({\left(\sqrt[3]{\frac{x}{y}}\right)}^{2}\right)} \cdot x + \log \left(\sqrt[3]{\frac{x}{y}}\right) \cdot x\right) - z \]

3. Applied egg-rr 76.2%

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

   [Start] 76.1	\[ \left(\log \left({\left(\sqrt[3]{\frac{x}{y}}\right)}^{2}\right) \cdot x + \log \left(\sqrt[3]{\frac{x}{y}}\right) \cdot x\right) - z \]
   *-commutative [=>] 76.1	\[ \left(\color{blue}{x \cdot \log \left({\left(\sqrt[3]{\frac{x}{y}}\right)}^{2}\right)} + \log \left(\sqrt[3]{\frac{x}{y}}\right) \cdot x\right) - z \]
   fma-def [=>] 76.2	\[ \color{blue}{\mathsf{fma}\left(x, \log \left({\left(\sqrt[3]{\frac{x}{y}}\right)}^{2}\right), \log \left(\sqrt[3]{\frac{x}{y}}\right) \cdot x\right)} - z \]
   log-pow [=>] 76.2	\[ \mathsf{fma}\left(x, \color{blue}{2 \cdot \log \left(\sqrt[3]{\frac{x}{y}}\right)}, \log \left(\sqrt[3]{\frac{x}{y}}\right) \cdot x\right) - z \]
   *-commutative [=>] 76.2	\[ \mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{\frac{x}{y}}\right), \color{blue}{x \cdot \log \left(\sqrt[3]{\frac{x}{y}}\right)}\right) - z \]

4. Simplified 76.1%

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

   [Start] 76.2	\[ \mathsf{fma}\left(x, 2 \cdot \log \left(\sqrt[3]{\frac{x}{y}}\right), x \cdot \log \left(\sqrt[3]{\frac{x}{y}}\right)\right) - z \]
   fma-udef [=>] 76.1	\[ \color{blue}{\left(x \cdot \left(2 \cdot \log \left(\sqrt[3]{\frac{x}{y}}\right)\right) + x \cdot \log \left(\sqrt[3]{\frac{x}{y}}\right)\right)} - z \]
   associate-*l* [<=] 76.1	\[ \left(\color{blue}{\left(x \cdot 2\right) \cdot \log \left(\sqrt[3]{\frac{x}{y}}\right)} + x \cdot \log \left(\sqrt[3]{\frac{x}{y}}\right)\right) - z \]
   distribute-rgt-out [=>] 76.1	\[ \color{blue}{\log \left(\sqrt[3]{\frac{x}{y}}\right) \cdot \left(x \cdot 2 + x\right)} - z \]

5. Applied egg-rr 99.7%

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

   [Start] 76.1	\[ \log \left(\sqrt[3]{\frac{x}{y}}\right) \cdot \left(x \cdot 2 + x\right) - z \]
   cbrt-div [=>] 99.7	\[ \log \color{blue}{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)} \cdot \left(x \cdot 2 + x\right) - z \]
   div-inv [=>] 99.7	\[ \log \color{blue}{\left(\sqrt[3]{x} \cdot \frac{1}{\sqrt[3]{y}}\right)} \cdot \left(x \cdot 2 + x\right) - z \]

6. Simplified 99.7%

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

   [Start] 99.7	\[ \log \left(\sqrt[3]{x} \cdot \frac{1}{\sqrt[3]{y}}\right) \cdot \left(x \cdot 2 + x\right) - z \]
   associate-*r/ [=>] 99.7	\[ \log \color{blue}{\left(\frac{\sqrt[3]{x} \cdot 1}{\sqrt[3]{y}}\right)} \cdot \left(x \cdot 2 + x\right) - z \]
   *-rgt-identity [=>] 99.7	\[ \log \left(\frac{\color{blue}{\sqrt[3]{x}}}{\sqrt[3]{y}}\right) \cdot \left(x \cdot 2 + x\right) - z \]

7. Final simplification 99.7%

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

Alternatives

Alternative 1
Accuracy	87.3%
Cost	20424

\[\begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+299}:\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 2
Accuracy	86.6%
Cost	13776

\[\begin{array}{l} t_0 := \log \left(\frac{y}{x}\right)\\ \mathbf{if}\;x \leq -1.14 \cdot 10^{+114}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-177}:\\ \;\;\;\;\frac{x}{\frac{1}{-t_0}} - z\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-243}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+199}:\\ \;\;\;\;\left(-z\right) - x \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log x - x \cdot \log y\\ \end{array} \]

Alternative 3
Accuracy	86.8%
Cost	13648

\[\begin{array}{l} t_0 := \log \left(\frac{y}{x}\right)\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{+114}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.22 \cdot 10^{-175}:\\ \;\;\;\;\frac{x}{\frac{1}{-t_0}} - z\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-243}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+165}:\\ \;\;\;\;\left(-z\right) - x \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \]

Alternative 4
Accuracy	93.0%
Cost	13644

\[\begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{+112}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-177}:\\ \;\;\;\;\frac{x}{\frac{1}{-\log \left(\frac{y}{x}\right)}} - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-303}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 5
Accuracy	99.5%
Cost	13636

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{x}{\frac{1}{\log \left(-x\right) - \log \left(-y\right)}} - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 6
Accuracy	99.5%
Cost	13508

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-311}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 7
Accuracy	67.2%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{-39}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 3.45 \cdot 10^{-35}:\\ \;\;\;\;x \cdot \left(-\log \left(\frac{y}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 8
Accuracy	67.1%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{-36}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 9
Accuracy	49.7%
Cost	128

\[-z \]

Reproduce

herbie shell --seed 2023140 
(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))