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

Average Error: 15.0 → 0.2

Time: 11.7s

Precision: binary64

Cost: 58560

Math

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

↓

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

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 (+ t_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 + (t_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 + (t_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(t_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[(t$95$0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Target

Original	15.0
Target	7.8
Herbie	0.2

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

Derivation

1. Initial program 15.0

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

2. Applied egg-rr 32.7

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

3. Applied egg-rr 3.7

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

4. Applied egg-rr 0.2

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

5. Final simplification 0.2

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

Alternatives

Alternative 1
Error	7.8
Cost	20424

\[\begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 2
Error	0.4
Cost	19908

\[\begin{array}{l} \mathbf{if}\;x \leq 0:\\ \;\;\;\;\mathsf{fma}\left(x, \log \left(\frac{-1}{y}\right) + \log \left(-x\right), -z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x \cdot \log y\right) - z\\ \end{array} \]

Alternative 3
Error	5.9
Cost	13512

\[\begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-168}:\\ \;\;\;\;\left(-z\right) - x \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 4
Error	0.3
Cost	13508

\[\begin{array}{l} \mathbf{if}\;y \leq 0:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 5
Error	0.3
Cost	13508

\[\begin{array}{l} \mathbf{if}\;y \leq 0:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x \cdot \log y\right) - z\\ \end{array} \]

Alternative 6
Error	21.4
Cost	7312

\[\begin{array}{l} \mathbf{if}\;z \leq -5.548096587988062 \cdot 10^{+34}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -1.937295576961193 \cdot 10^{-23}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{elif}\;z \leq -1.4325113804953457 \cdot 10^{-64}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 3.737836133362085 \cdot 10^{-99}:\\ \;\;\;\;x \cdot \left(-\log \left(\frac{y}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 7
Error	21.4
Cost	7248

\[\begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;z \leq -5.548096587988062 \cdot 10^{+34}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -1.937295576961193 \cdot 10^{-23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.4325113804953457 \cdot 10^{-64}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 3.737836133362085 \cdot 10^{-99}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 8
Error	31.4
Cost	128

\[-z \]

Alternative 9
Error	62.6
Cost	64

\[z \]

Reproduce

herbie shell --seed 2022300 
(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))