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

Percentage Accurate: 77.4% → 99.7%

Time: 12.4s

Alternatives: 12

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: 77.4% 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: 99.7% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (- (* x (* (log (/ (cbrt x) (cbrt y))) 3.0)) z))

C

double code(double x, double y, double z) {
	return (x * (log((cbrt(x) / cbrt(y))) * 3.0)) - z;
}

Java

public static double code(double x, double y, double z) {
	return (x * (Math.log((Math.cbrt(x) / Math.cbrt(y))) * 3.0)) - z;
}

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(log(Float64(cbrt(x) / cbrt(y))) * 3.0)) - z)
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(N[Log[N[(N[Power[x, 1/3], $MachinePrecision] / N[Power[y, 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

   1. add-cube-cbrt 80.2%

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

   2. pow3 80.2%

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

4. Applied egg-rr 80.2%

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

   1. pow-to-exp 80.2%

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

   2. rem-log-exp 80.2%

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

6. Applied egg-rr 80.2%

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

   1. cbrt-div 99.7%

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

   2. div-inv 99.7%

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

8. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

10. Simplified 99.7%

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

Alternative 2: 86.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right)\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\mathsf{fma}\left(x, t\_0, -z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (/ x y))) (t_1 (* x t_0)))
   (if (<= t_1 (- INFINITY))
     (* x (- (log (- x)) (log (- y))))
     (if (<= t_1 4e+288) (fma x t_0 (- z)) (* x (- (log x) (log y)))))))

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)) {
		tmp = x * (log(-x) - log(-y));
	} else if (t_1 <= 4e+288) {
		tmp = fma(x, t_0, -z);
	} else {
		tmp = x * (log(x) - log(y));
	}
	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))
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (t_1 <= 4e+288)
		tmp = fma(x, t_0, Float64(-z));
	else
		tmp = Float64(x * Float64(log(x) - log(y)));
	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[LessEqual[t$95$1, (-Infinity)], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+288], N[(x * t$95$0 + (-z)), $MachinePrecision], N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 4.5%

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

   3. Taylor expanded in z around 0 4.5%

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

   4. Taylor expanded in y around -inf 47.6%

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

      1. metadata-eval 47.6%

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

      2. distribute-neg-frac 47.6%

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

      3. distribute-frac-neg2 47.6%

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

      4. log-rec 47.6%

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

      5. sub-neg 47.6%

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

      6. neg-mul-1 47.6%

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

   6. Simplified 47.6%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 4e288

   1. Initial program 99.8%

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

      1. fmm-def 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

   if 4e288 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 8.5%

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

      1. add-sqr-sqrt 8.5%

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

      2. pow2 8.5%

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

   4. Applied egg-rr 8.5%

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

   5. Taylor expanded in x around inf 52.6%

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

      1. log-rec 52.6%

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

      2. distribute-rgt-in 52.6%

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

      3. mul-1-neg 52.6%

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

      4. log-rec 52.6%

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

      5. remove-double-neg 52.6%

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

      6. distribute-rgt-in 52.6%

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

      7. +-commutative 52.6%

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

      8. sub-neg 52.6%

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

   7. Simplified 52.6%

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

Alternative 3: 87.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right)\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;x \cdot \log \left(x \cdot y\right) - z\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\mathsf{fma}\left(x, t\_0, -z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (/ x y))) (t_1 (* x t_0)))
   (if (<= t_1 (- INFINITY))
     (- (* x (log (* x y))) z)
     (if (<= t_1 4e+288) (fma x t_0 (- z)) (* x (- (log x) (log y)))))))

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)) {
		tmp = (x * log((x * y))) - z;
	} else if (t_1 <= 4e+288) {
		tmp = fma(x, t_0, -z);
	} else {
		tmp = x * (log(x) - log(y));
	}
	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))
		tmp = Float64(Float64(x * log(Float64(x * y))) - z);
	elseif (t_1 <= 4e+288)
		tmp = fma(x, t_0, Float64(-z));
	else
		tmp = Float64(x * Float64(log(x) - log(y)));
	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[LessEqual[t$95$1, (-Infinity)], N[(N[(x * N[Log[N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[t$95$1, 4e+288], N[(x * t$95$0 + (-z)), $MachinePrecision], N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 4.5%

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

      1. add-cube-cbrt 4.5%

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

      2. associate-/l* 4.5%

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

      3. log-prod 68.9%

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

      4. pow2 68.9%

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

   4. Applied egg-rr 68.9%

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

      1. log-div 52.1%

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

      2. associate-+r- 52.1%

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

      3. log-prod 52.1%

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

      4. unpow2 52.1%

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

      5. add-cube-cbrt 52.1%

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

      6. log-div 4.5%

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

      7. frac-2neg 4.5%

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

      8. diff-log 47.6%

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

      9. sub-neg 47.6%

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

      10. distribute-rgt-in 47.4%

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

      11. add-sqr-sqrt 47.4%

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

      12. sqrt-unprod 7.4%

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

      13. sqr-neg 7.4%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 0.0%

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

   6. Applied egg-rr 52.0%

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

      1. distribute-rgt-out 52.1%

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

      2. sub-neg 52.1%

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

      3. log-div 4.5%

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

      4. *-commutative 4.5%

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

      5. log-div 52.1%

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

      6. sub-neg 52.1%

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

      7. add-log-exp 52.1%

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

      8. sum-log 1.2%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 38.7%

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

      11. sqr-neg 38.7%

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

      12. sqrt-unprod 38.7%

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

      13. add-sqr-sqrt 38.7%

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

      14. add-exp-log 43.1%

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

   8. Applied egg-rr 43.1%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 4e288

   1. Initial program 99.8%

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

      1. fmm-def 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

   if 4e288 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 8.5%

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

      1. add-sqr-sqrt 8.5%

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

      2. pow2 8.5%

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

   4. Applied egg-rr 8.5%

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

   5. Taylor expanded in x around inf 52.6%

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

      1. log-rec 52.6%

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

      2. distribute-rgt-in 52.6%

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

      3. mul-1-neg 52.6%

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

      4. log-rec 52.6%

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

      5. remove-double-neg 52.6%

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

      6. distribute-rgt-in 52.6%

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

      7. +-commutative 52.6%

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

      8. sub-neg 52.6%

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

   7. Simplified 52.6%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;x \cdot \log \left(x \cdot y\right) - z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 4 \cdot 10^{+288}:\\ \;\;\;\;\mathsf{fma}\left(x, \log \left(\frac{x}{y}\right), -z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;x \cdot \log \left(x \cdot y\right) - z\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+288}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY))
     (- (* x (log (* x y))) z)
     (if (<= t_0 4e+288) (- t_0 z) (* x (- (log x) (log y)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (x * log((x * y))) - z;
	} else if (t_0 <= 4e+288) {
		tmp = t_0 - z;
	} else {
		tmp = x * (log(x) - log(y));
	}
	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) {
		tmp = (x * Math.log((x * y))) - z;
	} else if (t_0 <= 4e+288) {
		tmp = t_0 - z;
	} else {
		tmp = x * (Math.log(x) - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (x * math.log((x * y))) - z
	elif t_0 <= 4e+288:
		tmp = t_0 - z
	else:
		tmp = x * (math.log(x) - math.log(y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(x * log(Float64(x * y))) - z);
	elseif (t_0 <= 4e+288)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(x * Float64(log(x) - log(y)));
	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)
		tmp = (x * log((x * y))) - z;
	elseif (t_0 <= 4e+288)
		tmp = t_0 - z;
	else
		tmp = x * (log(x) - log(y));
	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[LessEqual[t$95$0, (-Infinity)], N[(N[(x * N[Log[N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[t$95$0, 4e+288], N[(t$95$0 - z), $MachinePrecision], N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 4.5%

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

      1. add-cube-cbrt 4.5%

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

      2. associate-/l* 4.5%

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

      3. log-prod 68.9%

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

      4. pow2 68.9%

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

   4. Applied egg-rr 68.9%

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

      1. log-div 52.1%

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

      2. associate-+r- 52.1%

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

      3. log-prod 52.1%

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

      4. unpow2 52.1%

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

      5. add-cube-cbrt 52.1%

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

      6. log-div 4.5%

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

      7. frac-2neg 4.5%

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

      8. diff-log 47.6%

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

      9. sub-neg 47.6%

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

      10. distribute-rgt-in 47.4%

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

      11. add-sqr-sqrt 47.4%

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

      12. sqrt-unprod 7.4%

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

      13. sqr-neg 7.4%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 0.0%

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

   6. Applied egg-rr 52.0%

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

      1. distribute-rgt-out 52.1%

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

      2. sub-neg 52.1%

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

      3. log-div 4.5%

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

      4. *-commutative 4.5%

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

      5. log-div 52.1%

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

      6. sub-neg 52.1%

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

      7. add-log-exp 52.1%

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

      8. sum-log 1.2%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 38.7%

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

      11. sqr-neg 38.7%

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

      12. sqrt-unprod 38.7%

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

      13. add-sqr-sqrt 38.7%

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

      14. add-exp-log 43.1%

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

   8. Applied egg-rr 43.1%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 4e288

   1. Initial program 99.8%

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

   if 4e288 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 8.5%

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

      1. add-sqr-sqrt 8.5%

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

      2. pow2 8.5%

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

   4. Applied egg-rr 8.5%

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

   5. Taylor expanded in x around inf 52.6%

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

      1. log-rec 52.6%

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

      2. distribute-rgt-in 52.6%

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

      3. mul-1-neg 52.6%

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

      4. log-rec 52.6%

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

      5. remove-double-neg 52.6%

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

      6. distribute-rgt-in 52.6%

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

      7. +-commutative 52.6%

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

      8. sub-neg 52.6%

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

   7. Simplified 52.6%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;x \cdot \log \left(x \cdot y\right) - z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 4 \cdot 10^{+288}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 10^{+302}\right):\\ \;\;\;\;x \cdot \log \left(x \cdot y\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 1e+302)))
     (- (* x (log (* x y))) 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 <= 1e+302)) {
		tmp = (x * log((x * y))) - 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 <= 1e+302)) {
		tmp = (x * Math.log((x * y))) - 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 <= 1e+302):
		tmp = (x * math.log((x * y))) - 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 <= 1e+302))
		tmp = Float64(Float64(x * log(Float64(x * y))) - 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 <= 1e+302)))
		tmp = (x * log((x * y))) - 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, 1e+302]], $MachinePrecision]], N[(N[(x * N[Log[N[(x * y), $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 1.0000000000000001e302 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 5.1%

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

      1. add-cube-cbrt 5.1%

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

      2. associate-/l* 5.1%

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

      3. log-prod 65.0%

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

      4. pow2 65.0%

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

   4. Applied egg-rr 65.0%

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

      1. log-div 56.4%

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

      2. associate-+r- 56.4%

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

      3. log-prod 56.4%

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

      4. unpow2 56.4%

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

      5. add-cube-cbrt 56.4%

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

      6. log-div 5.1%

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

      7. frac-2neg 5.1%

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

      8. diff-log 43.3%

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

      9. sub-neg 43.3%

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

      10. distribute-rgt-in 43.2%

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

      11. add-sqr-sqrt 43.2%

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

      12. sqrt-unprod 11.2%

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

      13. sqr-neg 11.2%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 0.0%

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

   6. Applied egg-rr 56.4%

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

      1. distribute-rgt-out 56.4%

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

      2. sub-neg 56.4%

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

      3. log-div 5.1%

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

      4. *-commutative 5.1%

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

      5. log-div 56.4%

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

      6. sub-neg 56.4%

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

      7. add-log-exp 56.4%

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

      8. sum-log 2.9%

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

      9. add-sqr-sqrt 2.3%

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

      10. sqrt-unprod 19.1%

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

      11. sqr-neg 19.1%

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

      12. sqrt-unprod 16.8%

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

      13. add-sqr-sqrt 23.1%

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

      14. add-exp-log 40.6%

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

   8. Applied egg-rr 40.6%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1.0000000000000001e302

   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 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 10^{+302}\right):\\ \;\;\;\;x \cdot \log \left(x \cdot y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 10^{+302}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 1e+302))) (- 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 <= 1e+302)) {
		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 <= 1e+302)) {
		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 <= 1e+302):
		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 <= 1e+302))
		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 <= 1e+302)))
		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, 1e+302]], $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 1.0000000000000001e302 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 5.1%

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

   3. Taylor expanded in x around 0 37.8%

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

      1. mul-1-neg 37.8%

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

   5. Simplified 37.8%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1.0000000000000001e302

   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 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 10^{+302}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+159}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-68}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.9e+159)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -4.5e-68)
     (- (* x (log (/ x y))) z)
     (if (<= x -2e-309) (- z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.9e+159) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -4.5e-68) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -2e-309) {
		tmp = -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 (x <= (-1.9d+159)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-4.5d-68)) then
        tmp = (x * log((x / y))) - z
    else if (x <= (-2d-309)) then
        tmp = -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 (x <= -1.9e+159) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -4.5e-68) {
		tmp = (x * Math.log((x / y))) - z;
	} else if (x <= -2e-309) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.9e+159:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -4.5e-68:
		tmp = (x * math.log((x / y))) - z
	elif x <= -2e-309:
		tmp = -z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.9e+159)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -4.5e-68)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -2e-309)
		tmp = Float64(-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 (x <= -1.9e+159)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -4.5e-68)
		tmp = (x * log((x / y))) - z;
	elseif (x <= -2e-309)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.9e+159], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.5e-68], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -2e-309], (-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 < -1.89999999999999983e159

   1. Initial program 65.5%

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

   3. Taylor expanded in z around 0 56.3%

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

   4. Taylor expanded in y around -inf 89.8%

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

      1. metadata-eval 98.9%

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

      2. distribute-neg-frac 98.9%

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

      3. distribute-frac-neg2 98.9%

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

      4. log-rec 98.9%

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

      5. sub-neg 98.9%

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

      6. neg-mul-1 98.9%

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

   6. Simplified 89.8%

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

   if -1.89999999999999983e159 < x < -4.49999999999999999e-68

   1. Initial program 97.4%

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

   if -4.49999999999999999e-68 < x < -1.9999999999999988e-309

   1. Initial program 78.1%

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

   3. Taylor expanded in x around 0 89.1%

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

      1. mul-1-neg 89.1%

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

   5. Simplified 89.1%

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

   if -1.9999999999999988e-309 < x

   1. Initial program 78.8%

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

   3. Taylor expanded in x around 0 99.7%

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

      1. log-rec 99.7%

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

      2. sub-neg 99.7%

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

   5. Simplified 99.7%

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

Alternative 8: 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 81.8%

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

   3. Taylor expanded in y around -inf 99.5%

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

      1. metadata-eval 99.5%

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

      2. distribute-neg-frac 99.5%

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

      3. distribute-frac-neg2 99.5%

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

      4. log-rec 99.5%

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

      5. sub-neg 99.5%

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

      6. neg-mul-1 99.5%

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

   5. Simplified 99.5%

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

   if -1.999999999999994e-310 < y

   1. Initial program 78.8%

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

   3. Taylor expanded in x around 0 99.7%

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

      1. log-rec 99.7%

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

      2. sub-neg 99.7%

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

   5. Simplified 99.7%

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

Alternative 9: 66.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -21000000000000:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-39}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\log \left(\frac{y}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -21000000000000.0)
   (* x (log (/ x y)))
   (if (<= x 3.2e-39) (- z) (* x (- (log (/ y x)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -21000000000000.0) {
		tmp = x * log((x / y));
	} else if (x <= 3.2e-39) {
		tmp = -z;
	} else {
		tmp = x * -log((y / x));
	}
	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 <= (-21000000000000.0d0)) then
        tmp = x * log((x / y))
    else if (x <= 3.2d-39) then
        tmp = -z
    else
        tmp = x * -log((y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -21000000000000.0) {
		tmp = x * Math.log((x / y));
	} else if (x <= 3.2e-39) {
		tmp = -z;
	} else {
		tmp = x * -Math.log((y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -21000000000000.0:
		tmp = x * math.log((x / y))
	elif x <= 3.2e-39:
		tmp = -z
	else:
		tmp = x * -math.log((y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -21000000000000.0)
		tmp = Float64(x * log(Float64(x / y)));
	elseif (x <= 3.2e-39)
		tmp = Float64(-z);
	else
		tmp = Float64(x * Float64(-log(Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -21000000000000.0)
		tmp = x * log((x / y));
	elseif (x <= 3.2e-39)
		tmp = -z;
	else
		tmp = x * -log((y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -21000000000000.0], N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.2e-39], (-z), N[(x * (-N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1e13

   1. Initial program 80.8%

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

   3. Taylor expanded in z around 0 69.1%

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

   if -2.1e13 < x < 3.1999999999999998e-39

   1. Initial program 82.0%

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

   3. Taylor expanded in x around 0 81.8%

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

      1. mul-1-neg 81.8%

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

   5. Simplified 81.8%

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

   if 3.1999999999999998e-39 < x

   1. Initial program 76.5%

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

   3. Taylor expanded in z around 0 53.5%

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

      1. clear-num 53.5%

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

      2. neg-log 56.0%

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

   5. Applied egg-rr 56.0%

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

Alternative 10: 67.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{-54} \lor \neg \left(z \leq 5.5 \cdot 10^{-46}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -9.2e-54) (not (<= z 5.5e-46))) (- z) (* x (log (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -9.2e-54) || !(z <= 5.5e-46)) {
		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 ((z <= (-9.2d-54)) .or. (.not. (z <= 5.5d-46))) 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 ((z <= -9.2e-54) || !(z <= 5.5e-46)) {
		tmp = -z;
	} else {
		tmp = x * Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -9.2e-54) or not (z <= 5.5e-46):
		tmp = -z
	else:
		tmp = x * math.log((x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -9.2e-54) || !(z <= 5.5e-46))
		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 ((z <= -9.2e-54) || ~((z <= 5.5e-46)))
		tmp = -z;
	else
		tmp = x * log((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -9.2e-54], N[Not[LessEqual[z, 5.5e-46]], $MachinePrecision]], (-z), N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.1999999999999996e-54 or 5.49999999999999983e-46 < z

   1. Initial program 80.7%

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

   3. Taylor expanded in x around 0 74.6%

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

      1. mul-1-neg 74.6%

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

   5. Simplified 74.6%

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

   if -9.1999999999999996e-54 < z < 5.49999999999999983e-46

   1. Initial program 79.5%

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

   3. Taylor expanded in z around 0 66.5%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{-54} \lor \neg \left(z \leq 5.5 \cdot 10^{-46}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.8% 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 80.2%

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

3. Taylor expanded in x around 0 51.9%

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

   1. mul-1-neg 51.9%

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

5. Simplified 51.9%

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

Alternative 12: 2.2% 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 80.2%

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

3. Taylor expanded in x around 0 51.9%

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

   1. mul-1-neg 51.9%

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

5. Simplified 51.9%

   \[\leadsto \color{blue}{-z} \]
6. Step-by-step derivation

   1. neg-sub0 51.9%

      \[\leadsto \color{blue}{0 - z} \]

   2. sub-neg 51.9%

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

   3. add-sqr-sqrt 25.4%

      \[\leadsto 0 + \color{blue}{\sqrt{-z} \cdot \sqrt{-z}} \]

   4. sqrt-unprod 15.0%

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

   5. sqr-neg 15.0%

      \[\leadsto 0 + \sqrt{\color{blue}{z \cdot z}} \]

   6. sqrt-unprod 1.1%

      \[\leadsto 0 + \color{blue}{\sqrt{z} \cdot \sqrt{z}} \]

   7. add-sqr-sqrt 2.1%

      \[\leadsto 0 + \color{blue}{z} \]

7. Applied egg-rr 2.1%

   \[\leadsto \color{blue}{0 + z} \]
8. Step-by-step derivation

   1. +-lft-identity 2.1%

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

9. Simplified 2.1%

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

Developer Target 1: 88.6% 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 2024163 
(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))