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

Average Error: 23.95% → 0.5%

Time: 11.1s

Precision: binary64

Cost: 32836

Math

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

↓

\[\begin{array}{l} t_0 := \log \left(\sqrt[3]{x}\right)\\ \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(2 \cdot t_0 + \left(t_0 - \log y\right)\right) - z\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (cbrt x))))
   (if (<= y -5e-310)
     (- (* x (- (log (- x)) (log (- y)))) z)
     (- (* x (+ (* 2.0 t_0) (- t_0 (log y)))) 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));
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * (log(-x) - log(-y))) - z;
	} else {
		tmp = (x * ((2.0 * t_0) + (t_0 - log(y)))) - z;
	}
	return tmp;
}

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));
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * (Math.log(-x) - Math.log(-y))) - z;
	} else {
		tmp = (x * ((2.0 * t_0) + (t_0 - Math.log(y)))) - z;
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z)
	t_0 = log(cbrt(x))
	tmp = 0.0
	if (y <= -5e-310)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-y)))) - z);
	else
		tmp = Float64(Float64(x * Float64(Float64(2.0 * t_0) + Float64(t_0 - log(y)))) - z);
	end
	return tmp
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[Power[x, 1/3], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, -5e-310], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x * N[(N[(2.0 * t$95$0), $MachinePrecision] + N[(t$95$0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]

Target

Original	23.95%
Target	12.19%
Herbie	0.5%

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

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 23.53

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

   2. Applied egg-rr 0.5

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

   if -4.999999999999985e-310 < y

   1. Initial program 24.37

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

   2. Applied egg-rr 63.93

      \[\leadsto x \cdot \color{blue}{{\left({\log \left(\frac{x}{y}\right)}^{3}\right)}^{0.3333333333333333}} - z \]

   3. Applied egg-rr 0.51

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

   4. Simplified 0.51

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

      [Start] 0.51	\[ x \cdot \left(\log \left({\left(\sqrt[3]{x}\right)}^{2}\right) + \left(\log \left(\sqrt[3]{x}\right) + \log \left({y}^{-1}\right)\right)\right) - z \]
      associate-+r+ [=>] 0.52	\[ x \cdot \color{blue}{\left(\left(\log \left({\left(\sqrt[3]{x}\right)}^{2}\right) + \log \left(\sqrt[3]{x}\right)\right) + \log \left({y}^{-1}\right)\right)} - z \]
      log-pow [=>] 0.52	\[ x \cdot \left(\left(\log \left({\left(\sqrt[3]{x}\right)}^{2}\right) + \log \left(\sqrt[3]{x}\right)\right) + \color{blue}{-1 \cdot \log y}\right) - z \]
      mul-1-neg [=>] 0.52	\[ x \cdot \left(\left(\log \left({\left(\sqrt[3]{x}\right)}^{2}\right) + \log \left(\sqrt[3]{x}\right)\right) + \color{blue}{\left(-\log y\right)}\right) - z \]
      associate-+r+ [<=] 0.51	\[ x \cdot \color{blue}{\left(\log \left({\left(\sqrt[3]{x}\right)}^{2}\right) + \left(\log \left(\sqrt[3]{x}\right) + \left(-\log y\right)\right)\right)} - z \]
      log-pow [=>] 0.51	\[ x \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{x}\right)} + \left(\log \left(\sqrt[3]{x}\right) + \left(-\log y\right)\right)\right) - z \]
      sub-neg [<=] 0.51	\[ x \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right) + \color{blue}{\left(\log \left(\sqrt[3]{x}\right) - \log y\right)}\right) - z \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right) + \left(\log \left(\sqrt[3]{x}\right) - \log y\right)\right) - z\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5%
Cost	32708

\[\begin{array}{l} t_0 := \log \left(\sqrt{y}\right)\\ \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(\log x - t_0\right) - t_0\right) - z\\ \end{array} \]

Alternative 2
Error	11.32%
Cost	20425

\[\begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 10^{+308}\right):\\ \;\;\;\;x \cdot \log \left(y \cdot x\right) - z\\ \mathbf{else}:\\ \;\;\;\;t_0 - z\\ \end{array} \]

Alternative 3
Error	11.98%
Cost	20424

\[\begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t_0 \leq 10^{+308}:\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 4
Error	13.35%
Cost	13648

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+214}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-159}:\\ \;\;\;\;\left(-z\right) - x \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{elif}\;x \leq 10^{-178}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+136}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \]

Alternative 5
Error	6.99%
Cost	13644

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+213}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-159}:\\ \;\;\;\;\left(-z\right) - x \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 6
Error	0.48%
Cost	13508

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

Alternative 7
Error	33.09%
Cost	7577

\[\begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+70}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{+55}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-32}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{-132} \lor \neg \left(z \leq 2.25 \cdot 10^{-86}\right) \land z \leq 4 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \left(-\log \left(\frac{y}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 8
Error	33.41%
Cost	7514

\[\begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{+70}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -6 \cdot 10^{+55} \lor \neg \left(z \leq -1.15 \cdot 10^{-36}\right) \land \left(z \leq 9 \cdot 10^{-132} \lor \neg \left(z \leq 1.02 \cdot 10^{-86}\right) \land z \leq 6.2 \cdot 10^{-38}\right):\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 9
Error	47.98%
Cost	128

\[-z \]

Reproduce

herbie shell --seed 2023090 
(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))