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

Percentage Accurate: 76.7% → 99.5%

Time: 12.0s

Alternatives: 10

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * log((x / y))) - z
end function

Java

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

Python

def code(x, y, z):
	return (x * math.log((x / y))) - z

Julia

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

MATLAB

function tmp = code(x, y, z)
	tmp = (x * log((x / y))) - z;
end

Wolfram

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

Initial Program: 76.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * log((x / y))) - z
end function

Java

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

Python

def code(x, y, z):
	return (x * math.log((x / y))) - z

Julia

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

MATLAB

function tmp = code(x, y, z)
	tmp = (x * log((x / y))) - z;
end

Wolfram

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

Alternative 1: 99.5% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-310)
   (- (* x (- (- (log (- x)) (log (- (cbrt y)))) (log (pow (cbrt y) 2.0)))) z)
   (- (* x (- (log x) (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-310) {
		tmp = (x * ((log(-x) - log(-cbrt(y))) - log(pow(cbrt(y), 2.0)))) - z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-310) {
		tmp = (x * ((Math.log(-x) - Math.log(-Math.cbrt(y))) - Math.log(Math.pow(Math.cbrt(y), 2.0)))) - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2e-310)
		tmp = Float64(Float64(x * Float64(Float64(log(Float64(-x)) - log(Float64(-cbrt(y)))) - log((cbrt(y) ^ 2.0)))) - z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2e-310], N[(N[(x * N[(N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-N[Power[y, 1/3], $MachinePrecision])], $MachinePrecision]), $MachinePrecision] - N[Log[N[Power[N[Power[y, 1/3], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.999999999999994e-310

   1. Initial program 76.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 76.4%

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

      2. add-cube-cbrt 76.4%

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

      3. times-frac 76.4%

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

      4. log-prod 88.8%

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

      5. pow2 88.8%

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

   4. Applied egg-rr 88.8%

      \[\leadsto x \cdot \color{blue}{\left(\log \left(\frac{1}{{\left(\sqrt[3]{y}\right)}^{2}}\right) + \log \left(\frac{x}{\sqrt[3]{y}}\right)\right)} - z \]
   5. Step-by-step derivation

      1. +-commutative 88.8%

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

      2. log-rec 88.8%

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

      3. unsub-neg 88.8%

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

   6. Simplified 88.8%

      \[\leadsto x \cdot \color{blue}{\left(\log \left(\frac{x}{\sqrt[3]{y}}\right) - \log \left({\left(\sqrt[3]{y}\right)}^{2}\right)\right)} - z \]
   7. Step-by-step derivation

      1. frac-2neg 88.8%

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

      2. log-div 99.4%

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

   8. Applied egg-rr 99.4%

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

   if -1.999999999999994e-310 < y

   1. Initial program 75.0%

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

   3. Taylor expanded in x around 0 99.6%

      \[\leadsto \color{blue}{x \cdot \left(\log x + \log \left(\frac{1}{y}\right)\right)} - z \]
   4. Step-by-step derivation

      1. log-rec 99.6%

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

      2. sub-neg 99.6%

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

   5. Simplified 99.6%

      \[\leadsto \color{blue}{x \cdot \left(\log x - \log y\right)} - z \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 88.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right)\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 2 \cdot 10^{+303}\right):\\ \;\;\;\;x \cdot \log \left(y \cdot x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, t\_0, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (/ x y))) (t_1 (* x t_0)))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 2e+303)))
     (- (* x (log (* y x))) z)
     (fma x t_0 (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y));
	double t_1 = x * t_0;
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 2e+303)) {
		tmp = (x * log((y * x))) - z;
	} else {
		tmp = fma(x, t_0, -z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = log(Float64(x / y))
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 2e+303))
		tmp = Float64(Float64(x * log(Float64(y * x))) - z);
	else
		tmp = fma(x, t_0, Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 2e+303]], $MachinePrecision]], N[(N[(x * N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x * t$95$0 + (-z)), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 2e303 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 6.3%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 6.3%

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

      2. add-cube-cbrt 6.3%

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

      3. times-frac 6.3%

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

      4. log-prod 67.9%

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

      5. pow2 67.9%

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

   4. Applied egg-rr 67.9%

      \[\leadsto x \cdot \color{blue}{\left(\log \left(\frac{1}{{\left(\sqrt[3]{y}\right)}^{2}}\right) + \log \left(\frac{x}{\sqrt[3]{y}}\right)\right)} - z \]
   5. Step-by-step derivation

      1. +-commutative 67.9%

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

      2. log-rec 67.9%

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

      3. unsub-neg 67.9%

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

   6. Simplified 67.9%

      \[\leadsto x \cdot \color{blue}{\left(\log \left(\frac{x}{\sqrt[3]{y}}\right) - \log \left({\left(\sqrt[3]{y}\right)}^{2}\right)\right)} - z \]
   7. Step-by-step derivation

      1. diff-log 6.3%

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

      2. associate-/l/ 6.3%

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

      3. unpow2 6.3%

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

      4. add-cube-cbrt 6.3%

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

      5. expm1-log1p-u 6.3%

         \[\leadsto x \cdot \log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{y}\right)\right)\right)} - z \]

      6. expm1-log1p-u 6.3%

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

      7. diff-log 52.9%

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

      8. sub-neg 52.9%

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

      9. distribute-rgt-in 51.5%

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

   8. Applied egg-rr 51.5%

      \[\leadsto \color{blue}{\left(\log x \cdot x + \left(-\log y\right) \cdot x\right)} - z \]
   9. Step-by-step derivation

      1. +-commutative 51.5%

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

      2. distribute-rgt-out 52.9%

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

      3. neg-sub0 52.9%

         \[\leadsto x \cdot \left(\color{blue}{\left(0 - \log y\right)} + \log x\right) - z \]

      4. associate--r- 52.9%

         \[\leadsto x \cdot \color{blue}{\left(0 - \left(\log y - \log x\right)\right)} - z \]

      5. log-div 7.5%

         \[\leadsto x \cdot \left(0 - \color{blue}{\log \left(\frac{y}{x}\right)}\right) - z \]

      6. neg-sub0 7.5%

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

      7. *-commutative 7.5%

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

      8. neg-log 6.3%

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

      9. clear-num 6.3%

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

      10. div-inv 6.3%

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

      11. add-exp-log 3.0%

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

      12. neg-log 3.0%

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

      13. add-sqr-sqrt 1.0%

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

      14. sqrt-unprod 27.4%

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

      15. sqr-neg 27.4%

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

      16. sqrt-unprod 26.5%

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

      17. add-sqr-sqrt 33.5%

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

      18. add-exp-log 59.4%

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

   10. Applied egg-rr 59.4%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 2e303

   1. Initial program 99.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Step-by-step derivation

      1. fma-neg 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log \left(\frac{x}{y}\right), -z\right)} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 2 \cdot 10^{+303}\right):\\ \;\;\;\;x \cdot \log \left(y \cdot x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log \left(\frac{x}{y}\right), -z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 2 \cdot 10^{+303}\right):\\ \;\;\;\;x \cdot \log \left(y \cdot x\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 2e+303)))
     (- (* x (log (* y x))) z)
     (- t_0 z))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 2e+303)) {
		tmp = (x * log((y * x))) - z;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 2e+303)) {
		tmp = (x * Math.log((y * x))) - z;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 2e+303):
		tmp = (x * math.log((y * x))) - z
	else:
		tmp = t_0 - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 2e+303))
		tmp = Float64(Float64(x * log(Float64(y * x))) - z);
	else
		tmp = Float64(t_0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 2e+303)))
		tmp = (x * log((y * x))) - z;
	else
		tmp = t_0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 2e+303]], $MachinePrecision]], N[(N[(x * N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(t$95$0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 2e303 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 6.3%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 6.3%

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

      2. add-cube-cbrt 6.3%

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

      3. times-frac 6.3%

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

      4. log-prod 67.9%

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

      5. pow2 67.9%

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

   4. Applied egg-rr 67.9%

      \[\leadsto x \cdot \color{blue}{\left(\log \left(\frac{1}{{\left(\sqrt[3]{y}\right)}^{2}}\right) + \log \left(\frac{x}{\sqrt[3]{y}}\right)\right)} - z \]
   5. Step-by-step derivation

      1. +-commutative 67.9%

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

      2. log-rec 67.9%

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

      3. unsub-neg 67.9%

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

   6. Simplified 67.9%

      \[\leadsto x \cdot \color{blue}{\left(\log \left(\frac{x}{\sqrt[3]{y}}\right) - \log \left({\left(\sqrt[3]{y}\right)}^{2}\right)\right)} - z \]
   7. Step-by-step derivation

      1. diff-log 6.3%

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

      2. associate-/l/ 6.3%

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

      3. unpow2 6.3%

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

      4. add-cube-cbrt 6.3%

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

      5. expm1-log1p-u 6.3%

         \[\leadsto x \cdot \log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{y}\right)\right)\right)} - z \]

      6. expm1-log1p-u 6.3%

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

      7. diff-log 52.9%

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

      8. sub-neg 52.9%

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

      9. distribute-rgt-in 51.5%

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

   8. Applied egg-rr 51.5%

      \[\leadsto \color{blue}{\left(\log x \cdot x + \left(-\log y\right) \cdot x\right)} - z \]
   9. Step-by-step derivation

      1. +-commutative 51.5%

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

      2. distribute-rgt-out 52.9%

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

      3. neg-sub0 52.9%

         \[\leadsto x \cdot \left(\color{blue}{\left(0 - \log y\right)} + \log x\right) - z \]

      4. associate--r- 52.9%

         \[\leadsto x \cdot \color{blue}{\left(0 - \left(\log y - \log x\right)\right)} - z \]

      5. log-div 7.5%

         \[\leadsto x \cdot \left(0 - \color{blue}{\log \left(\frac{y}{x}\right)}\right) - z \]

      6. neg-sub0 7.5%

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

      7. *-commutative 7.5%

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

      8. neg-log 6.3%

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

      9. clear-num 6.3%

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

      10. div-inv 6.3%

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

      11. add-exp-log 3.0%

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

      12. neg-log 3.0%

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

      13. add-sqr-sqrt 1.0%

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

      14. sqrt-unprod 27.4%

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

      15. sqr-neg 27.4%

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

      16. sqrt-unprod 26.5%

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

      17. add-sqr-sqrt 33.5%

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

      18. add-exp-log 59.4%

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

   10. Applied egg-rr 59.4%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 2e303

   1. Initial program 99.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 2 \cdot 10^{+303}\right):\\ \;\;\;\;x \cdot \log \left(y \cdot x\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 2 \cdot 10^{+303}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 2e+303))) (- z) (- t_0 z))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 2e+303)) {
		tmp = -z;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 2e+303)) {
		tmp = -z;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 2e+303):
		tmp = -z
	else:
		tmp = t_0 - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 2e+303))
		tmp = Float64(-z);
	else
		tmp = Float64(t_0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 2e+303)))
		tmp = -z;
	else
		tmp = t_0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 2e+303]], $MachinePrecision]], (-z), N[(t$95$0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 2e303 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 6.3%

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

   3. Taylor expanded in x around 0 54.6%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   4. Step-by-step derivation

      1. mul-1-neg 54.6%

         \[\leadsto \color{blue}{-z} \]

   5. Simplified 54.6%

      \[\leadsto \color{blue}{-z} \]

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 2e303

   1. Initial program 99.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 2 \cdot 10^{+303}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+114}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -2.85 \cdot 10^{-154}:\\ \;\;\;\;\mathsf{fma}\left(x, \log \left(\frac{x}{y}\right), -z\right)\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.5e+114)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -2.85e-154)
     (fma x (log (/ x y)) (- z))
     (if (<= x -4e-308) (- z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.5e+114) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -2.85e-154) {
		tmp = fma(x, log((x / y)), -z);
	} else if (x <= -4e-308) {
		tmp = -z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.5e+114)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -2.85e-154)
		tmp = fma(x, log(Float64(x / y)), Float64(-z));
	elseif (x <= -4e-308)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.5e+114], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.85e-154], N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] + (-z)), $MachinePrecision], If[LessEqual[x, -4e-308], (-z), N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.5e114

   1. Initial program 70.4%

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

   3. Taylor expanded in z around 0 64.9%

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

   4. Taylor expanded in y around -inf 93.7%

      \[\leadsto \color{blue}{x \cdot \left(\log \left(-1 \cdot x\right) + \log \left(\frac{-1}{y}\right)\right)} \]
   5. Step-by-step derivation

      1. metadata-eval 99.2%

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

      2. distribute-neg-frac 99.2%

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

      3. distribute-frac-neg2 99.2%

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

      4. log-rec 99.2%

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

      5. sub-neg 99.2%

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

      6. neg-mul-1 99.2%

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

   6. Simplified 93.7%

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

   if -2.5e114 < x < -2.8499999999999999e-154

   1. Initial program 96.1%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Step-by-step derivation

      1. fma-neg 96.2%

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

   3. Simplified 96.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log \left(\frac{x}{y}\right), -z\right)} \]
   4. Add Preprocessing

   if -2.8499999999999999e-154 < x < -4.00000000000000013e-308

   1. Initial program 49.4%

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

   3. Taylor expanded in x around 0 94.1%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   4. Step-by-step derivation

      1. mul-1-neg 94.1%

         \[\leadsto \color{blue}{-z} \]

   5. Simplified 94.1%

      \[\leadsto \color{blue}{-z} \]

   if -4.00000000000000013e-308 < x

   1. Initial program 75.0%

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

   3. Taylor expanded in x around 0 99.6%

      \[\leadsto \color{blue}{x \cdot \left(\log x + \log \left(\frac{1}{y}\right)\right)} - z \]
   4. Step-by-step derivation

      1. log-rec 99.6%

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

      2. sub-neg 99.6%

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

   5. Simplified 99.6%

      \[\leadsto \color{blue}{x \cdot \left(\log x - \log y\right)} - z \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-310)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (- (* x (- (log x) (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-310) {
		tmp = (x * (log(-x) - log(-y))) - z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2d-310)) then
        tmp = (x * (log(-x) - log(-y))) - z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-310) {
		tmp = (x * (Math.log(-x) - Math.log(-y))) - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2e-310:
		tmp = (x * (math.log(-x) - math.log(-y))) - z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2e-310)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-y)))) - z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2e-310)
		tmp = (x * (log(-x) - log(-y))) - z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2e-310], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.999999999999994e-310

   1. Initial program 76.4%

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

   3. Taylor expanded in y around -inf 99.4%

      \[\leadsto \color{blue}{x \cdot \left(\log \left(-1 \cdot x\right) + \log \left(\frac{-1}{y}\right)\right)} - z \]
   4. Step-by-step derivation

      1. metadata-eval 99.4%

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

      2. distribute-neg-frac 99.4%

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

      3. distribute-frac-neg2 99.4%

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

      4. log-rec 99.4%

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

      5. sub-neg 99.4%

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

      6. neg-mul-1 99.4%

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

   5. Simplified 99.4%

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

   if -1.999999999999994e-310 < y

   1. Initial program 75.0%

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

   3. Taylor expanded in x around 0 99.6%

      \[\leadsto \color{blue}{x \cdot \left(\log x + \log \left(\frac{1}{y}\right)\right)} - z \]
   4. Step-by-step derivation

      1. log-rec 99.6%

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

      2. sub-neg 99.6%

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

   5. Simplified 99.6%

      \[\leadsto \color{blue}{x \cdot \left(\log x - \log y\right)} - z \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 61.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-24} \lor \neg \left(x \leq 2.6 \cdot 10^{-116}\right) \land \left(x \leq 9.2 \cdot 10^{+53} \lor \neg \left(x \leq 4.7 \cdot 10^{+157}\right)\right):\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.02e-24)
         (and (not (<= x 2.6e-116)) (or (<= x 9.2e+53) (not (<= x 4.7e+157)))))
   (* x (log (/ x y)))
   (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.02e-24) || (!(x <= 2.6e-116) && ((x <= 9.2e+53) || !(x <= 4.7e+157)))) {
		tmp = x * log((x / y));
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.02d-24)) .or. (.not. (x <= 2.6d-116)) .and. (x <= 9.2d+53) .or. (.not. (x <= 4.7d+157))) then
        tmp = x * log((x / y))
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.02e-24) || (!(x <= 2.6e-116) && ((x <= 9.2e+53) || !(x <= 4.7e+157)))) {
		tmp = x * Math.log((x / y));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.02e-24) or (not (x <= 2.6e-116) and ((x <= 9.2e+53) or not (x <= 4.7e+157))):
		tmp = x * math.log((x / y))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.02e-24) || (!(x <= 2.6e-116) && ((x <= 9.2e+53) || !(x <= 4.7e+157))))
		tmp = Float64(x * log(Float64(x / y)));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.02e-24) || (~((x <= 2.6e-116)) && ((x <= 9.2e+53) || ~((x <= 4.7e+157)))))
		tmp = x * log((x / y));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.02e-24], And[N[Not[LessEqual[x, 2.6e-116]], $MachinePrecision], Or[LessEqual[x, 9.2e+53], N[Not[LessEqual[x, 4.7e+157]], $MachinePrecision]]]], N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -1.0200000000000001e-24 or 2.6e-116 < x < 9.20000000000000079e53 or 4.7000000000000003e157 < x

   1. Initial program 85.2%

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

   3. Taylor expanded in z around 0 69.1%

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

   if -1.0200000000000001e-24 < x < 2.6e-116 or 9.20000000000000079e53 < x < 4.7000000000000003e157

   1. Initial program 68.0%

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

   3. Taylor expanded in x around 0 76.2%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   4. Step-by-step derivation

      1. mul-1-neg 76.2%

         \[\leadsto \color{blue}{-z} \]

   5. Simplified 76.2%

      \[\leadsto \color{blue}{-z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-24} \lor \neg \left(x \leq 2.6 \cdot 10^{-116}\right) \land \left(x \leq 9.2 \cdot 10^{+53} \lor \neg \left(x \leq 4.7 \cdot 10^{+157}\right)\right):\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-24}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-116} \lor \neg \left(x \leq 7.5 \cdot 10^{+54}\right) \land x \leq 4.7 \cdot 10^{+157}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.3e-24)
   (* (- x) (log (/ y x)))
   (if (or (<= x 2.9e-116) (and (not (<= x 7.5e+54)) (<= x 4.7e+157)))
     (- z)
     (* x (log (/ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.3e-24) {
		tmp = -x * log((y / x));
	} else if ((x <= 2.9e-116) || (!(x <= 7.5e+54) && (x <= 4.7e+157))) {
		tmp = -z;
	} else {
		tmp = x * log((x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.3d-24)) then
        tmp = -x * log((y / x))
    else if ((x <= 2.9d-116) .or. (.not. (x <= 7.5d+54)) .and. (x <= 4.7d+157)) then
        tmp = -z
    else
        tmp = x * log((x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.3e-24) {
		tmp = -x * Math.log((y / x));
	} else if ((x <= 2.9e-116) || (!(x <= 7.5e+54) && (x <= 4.7e+157))) {
		tmp = -z;
	} else {
		tmp = x * Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.3e-24:
		tmp = -x * math.log((y / x))
	elif (x <= 2.9e-116) or (not (x <= 7.5e+54) and (x <= 4.7e+157)):
		tmp = -z
	else:
		tmp = x * math.log((x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.3e-24)
		tmp = Float64(Float64(-x) * log(Float64(y / x)));
	elseif ((x <= 2.9e-116) || (!(x <= 7.5e+54) && (x <= 4.7e+157)))
		tmp = Float64(-z);
	else
		tmp = Float64(x * log(Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.3e-24)
		tmp = -x * log((y / x));
	elseif ((x <= 2.9e-116) || (~((x <= 7.5e+54)) && (x <= 4.7e+157)))
		tmp = -z;
	else
		tmp = x * log((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.3e-24], N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 2.9e-116], And[N[Not[LessEqual[x, 7.5e+54]], $MachinePrecision], LessEqual[x, 4.7e+157]]], (-z), N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3e-24

   1. Initial program 81.7%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 81.7%

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

      2. neg-log 83.0%

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

   4. Applied egg-rr 83.0%

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

   5. Taylor expanded in z around 0 71.2%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \log \left(\frac{y}{x}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 71.2%

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

      2. neg-mul-1 71.2%

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

   7. Simplified 71.2%

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

   if -1.3e-24 < x < 2.8999999999999998e-116 or 7.50000000000000042e54 < x < 4.7000000000000003e157

   1. Initial program 68.0%

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

   3. Taylor expanded in x around 0 76.2%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   4. Step-by-step derivation

      1. mul-1-neg 76.2%

         \[\leadsto \color{blue}{-z} \]

   5. Simplified 76.2%

      \[\leadsto \color{blue}{-z} \]

   if 2.8999999999999998e-116 < x < 7.50000000000000042e54 or 4.7000000000000003e157 < x

   1. Initial program 89.2%

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

   3. Taylor expanded in z around 0 68.1%

      \[\leadsto \color{blue}{x \cdot \log \left(\frac{x}{y}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-24}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-116} \lor \neg \left(x \leq 7.5 \cdot 10^{+54}\right) \land x \leq 4.7 \cdot 10^{+157}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.1% accurate, 53.5× speedup

Math

\[\begin{array}{l} \\ -z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- z))

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = -z
end function

Java

public static double code(double x, double y, double z) {
	return -z;
}

Python

def code(x, y, z):
	return -z

Julia

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

MATLAB

function tmp = code(x, y, z)
	tmp = -z;
end

Wolfram

code[x_, y_, z_] := (-z)

Derivation

1. Initial program 75.7%

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

3. Taylor expanded in x around 0 49.7%

   \[\leadsto \color{blue}{-1 \cdot z} \]
4. Step-by-step derivation

   1. mul-1-neg 49.7%

      \[\leadsto \color{blue}{-z} \]

5. Simplified 49.7%

   \[\leadsto \color{blue}{-z} \]
6. Add Preprocessing

Alternative 10: 2.3% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 z)

C

double code(double x, double y, double z) {
	return z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = z
end function

Java

public static double code(double x, double y, double z) {
	return z;
}

Python

def code(x, y, z):
	return z

Julia

function code(x, y, z)
	return z
end

MATLAB

function tmp = code(x, y, z)
	tmp = z;
end

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 75.7%

   \[x \cdot \log \left(\frac{x}{y}\right) - z \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cube-cbrt 75.0%

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

   2. associate-*l* 75.0%

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

   3. fma-neg 75.0%

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

   4. pow2 75.0%

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

   5. add-sqr-sqrt 37.1%

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

   6. sqrt-unprod 49.1%

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

   7. sqr-neg 49.1%

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

   8. sqrt-unprod 20.4%

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

   9. add-sqr-sqrt 39.3%

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

4. Applied egg-rr 39.3%

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

5. Taylor expanded in x around 0 2.3%

   \[\leadsto \color{blue}{z} \]
6. Add Preprocessing

Developer Target 1: 88.2% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< y 7.595077799083773e-308)
   (- (* x (log (/ x y))) z)
   (- (* x (- (log x) (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y < 7.595077799083773e-308) {
		tmp = (x * log((x / y))) - z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y < 7.595077799083773d-308) then
        tmp = (x * log((x / y))) - z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y < 7.595077799083773e-308) {
		tmp = (x * Math.log((x / y))) - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y < 7.595077799083773e-308:
		tmp = (x * math.log((x / y))) - z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y < 7.595077799083773e-308)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y < 7.595077799083773e-308)
		tmp = (x * log((x / y))) - z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[y, 7.595077799083773e-308], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Reproduce

herbie shell --seed 2024146 
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y 7595077799083773/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* x (log (/ x y))) z) (- (* x (- (log x) (log y))) z)))

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