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

Percentage Accurate: 77.2% → 99.7%

Time: 11.0s

Alternatives: 11

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: 77.2% 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

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

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(Float64(x * log(Float64(cbrt(x) / cbrt(y)))) * 3.0) - z)
end

Wolfram

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

Derivation

1. Initial program 78.3%

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

   1. add-cbrt-cube 52.7%

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

   2. pow3 52.7%

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

4. Applied egg-rr 52.7%

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

   1. rem-cbrt-cube 78.3%

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

   2. *-un-lft-identity 78.3%

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

   3. metadata-eval 78.3%

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

   4. associate-*r* 78.2%

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

   5. associate-*r* 78.2%

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

   6. *-commutative 78.2%

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

   7. add-log-exp 78.2%

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

   8. exp-to-pow 78.2%

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

   9. pow1/3 78.2%

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

   10. cbrt-undiv 99.6%

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

   11. associate-*r* 99.6%

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

   12. *-commutative 99.6%

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

   13. associate-*r* 99.6%

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

   14. cbrt-undiv 78.2%

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

6. Applied egg-rr 78.2%

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

   1. cbrt-div 99.6%

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

   2. div-inv 99.6%

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

8. Applied egg-rr 99.6%

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

   1. associate-*r/ 99.6%

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

   2. *-rgt-identity 99.6%

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

10. Simplified 99.6%

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

11. Final simplification 99.6%

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

Alternative 2: 87.0% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 2e+299))) (- 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+299)) {
		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+299)) {
		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+299):
		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+299))
		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+299)))
		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+299]], $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 2.0000000000000001e299 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 6.6%

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

      1. remove-double-neg 6.6%

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

      2. sub-neg 6.6%

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

      3. distribute-neg-in 6.6%

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

      4. distribute-rgt-neg-in 6.6%

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

      5. remove-double-neg 6.6%

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

      6. fma-udef 6.6%

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

      7. log-div 47.2%

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

      8. sub-neg 47.2%

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

      9. distribute-neg-in 47.2%

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

      10. remove-double-neg 47.2%

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

      11. +-commutative 47.2%

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

      12. sub-neg 47.2%

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

      13. log-div 9.3%

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

   3. Simplified 9.3%

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

   5. Taylor expanded in x around 0 42.4%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 2.0000000000000001e299

   1. Initial program 99.7%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 2 \cdot 10^{+299}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-310)
   (- (fma x (- (log (- y)) (log (- x))) z))
   (- (- (* x (log x)) (* x (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = -fma(x, (log(-y) - log(-x)), z);
	} else {
		tmp = ((x * log(x)) - (x * log(y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-310)
		tmp = Float64(-fma(x, Float64(log(Float64(-y)) - log(Float64(-x))), z));
	else
		tmp = Float64(Float64(Float64(x * log(x)) - Float64(x * log(y))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5e-310], (-N[(x * N[(N[Log[(-y)], $MachinePrecision] - N[Log[(-x)], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), N[(N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 77.9%

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

      1. remove-double-neg 77.9%

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

      2. sub-neg 77.9%

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

      3. distribute-neg-in 77.9%

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

      4. distribute-rgt-neg-in 77.9%

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

      5. remove-double-neg 77.9%

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

      6. fma-udef 77.9%

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

      7. log-div 0.0%

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

      8. sub-neg 0.0%

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

      9. distribute-neg-in 0.0%

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

      10. remove-double-neg 0.0%

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

      11. +-commutative 0.0%

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

      12. sub-neg 0.0%

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

      13. log-div 77.3%

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

   3. Simplified 77.3%

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

      1. frac-2neg 77.3%

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

      2. log-div 99.5%

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

   6. Applied egg-rr 99.5%

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

   if -4.999999999999985e-310 < y

   1. Initial program 78.7%

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

      1. add-cube-cbrt 78.7%

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

      2. log-prod 78.6%

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

      3. pow2 78.6%

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

   4. Applied egg-rr 78.6%

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

      1. log-pow 78.7%

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

      2. distribute-lft1-in 78.7%

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

      3. metadata-eval 78.7%

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

   6. Simplified 78.7%

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

      1. pow1/3 78.5%

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

      2. log-pow 78.6%

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

   8. Applied egg-rr 78.6%

      \[\leadsto x \cdot \left(3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(\frac{x}{y}\right)\right)}\right) - z \]
   9. Step-by-step derivation

      1. clear-num 78.6%

         \[\leadsto x \cdot \left(3 \cdot \left(0.3333333333333333 \cdot \log \color{blue}{\left(\frac{1}{\frac{y}{x}}\right)}\right)\right) - z \]

      2. log-rec 79.6%

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

   10. Applied egg-rr 79.6%

       \[\leadsto x \cdot \left(3 \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(-\log \left(\frac{y}{x}\right)\right)}\right)\right) - z \]
   11. Step-by-step derivation

       1. associate-*r* 79.7%

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

       2. metadata-eval 79.7%

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

       3. *-un-lft-identity 79.7%

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

       4. neg-log 78.7%

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

       5. clear-num 78.7%

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

       6. diff-log 99.4%

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

       7. sub-neg 99.4%

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

       8. distribute-rgt-in 99.4%

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

   12. Applied egg-rr 99.4%

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

4. Final simplification 99.5%

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

Alternative 4: 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 78.3%

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

   1. add-cube-cbrt 78.3%

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

   2. log-prod 78.2%

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

   3. pow2 78.2%

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

4. Applied egg-rr 78.2%

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

   1. log-pow 78.2%

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

   2. distribute-lft1-in 78.2%

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

   3. metadata-eval 78.2%

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

6. Simplified 78.2%

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

   1. cbrt-div 99.6%

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

   2. div-inv 99.6%

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

8. Applied egg-rr 99.6%

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

   1. associate-*r/ 99.6%

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

   2. *-rgt-identity 99.6%

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

10. Simplified 99.6%

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

11. Final simplification 99.6%

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

Alternative 5: 93.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+201}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-147}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.05e+201)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -2e-147)
     (- (* x (log (/ x y))) z)
     (if (<= x -5e-308) (- z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.05e+201) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -2e-147) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -5e-308) {
		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 <= (-2.05d+201)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-2d-147)) then
        tmp = (x * log((x / y))) - z
    else if (x <= (-5d-308)) 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 <= -2.05e+201) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -2e-147) {
		tmp = (x * Math.log((x / y))) - z;
	} else if (x <= -5e-308) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.05e+201:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -2e-147:
		tmp = (x * math.log((x / y))) - z
	elif x <= -5e-308:
		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 <= -2.05e+201)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -2e-147)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -5e-308)
		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 <= -2.05e+201)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -2e-147)
		tmp = (x * log((x / y))) - z;
	elseif (x <= -5e-308)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.05e+201], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2e-147], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -5e-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 < -2.0500000000000001e201

   1. Initial program 51.8%

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

      1. remove-double-neg 51.8%

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

      2. sub-neg 51.8%

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

      3. distribute-neg-in 51.8%

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

      4. distribute-rgt-neg-in 51.8%

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

      5. remove-double-neg 51.8%

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

      6. fma-udef 51.8%

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

      7. log-div 0.0%

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

      8. sub-neg 0.0%

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

      9. distribute-neg-in 0.0%

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

      10. remove-double-neg 0.0%

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

      11. +-commutative 0.0%

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

      12. sub-neg 0.0%

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

      13. log-div 53.0%

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

   3. Simplified 53.0%

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

      1. frac-2neg 53.0%

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

      2. log-div 98.8%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. sub-neg 0.0%

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

      2. distribute-rgt-in 0.0%

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

      3. log-rec 0.0%

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

      4. mul-1-neg 0.0%

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

      5. remove-double-neg 0.0%

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

      6. log-prod 96.2%

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

      7. neg-mul-1 96.2%

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

      8. distribute-rgt-in 96.6%

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

      9. sub-neg 96.6%

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

   9. Simplified 96.6%

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

   if -2.0500000000000001e201 < x < -1.9999999999999999e-147

   1. Initial program 90.4%

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

   if -1.9999999999999999e-147 < x < -4.99999999999999955e-308

   1. Initial program 70.2%

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

      1. remove-double-neg 70.2%

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

      2. sub-neg 70.2%

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

      3. distribute-neg-in 70.2%

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

      4. distribute-rgt-neg-in 70.2%

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

      5. remove-double-neg 70.2%

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

      6. fma-udef 70.2%

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

      7. log-div 0.0%

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

      8. sub-neg 0.0%

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

      9. distribute-neg-in 0.0%

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

      10. remove-double-neg 0.0%

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

      11. +-commutative 0.0%

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

      12. sub-neg 0.0%

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

      13. log-div 70.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in x around 0 90.9%

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

   if -4.99999999999999955e-308 < x

   1. Initial program 78.7%

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

      1. log-div 99.4%

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

   4. Applied egg-rr 99.4%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+201}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-147}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-148}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -6.5e-148)
   (- (* x (log (/ x y))) z)
   (if (<= x -1e-308) (- z) (- (* x (- (log x) (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.5e-148) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -1e-308) {
		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 <= (-6.5d-148)) then
        tmp = (x * log((x / y))) - z
    else if (x <= (-1d-308)) 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 <= -6.5e-148) {
		tmp = (x * Math.log((x / y))) - z;
	} else if (x <= -1e-308) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -6.5e-148:
		tmp = (x * math.log((x / y))) - z
	elif x <= -1e-308:
		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 <= -6.5e-148)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -1e-308)
		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 <= -6.5e-148)
		tmp = (x * log((x / y))) - z;
	elseif (x <= -1e-308)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -6.5e-148], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -1e-308], (-z), N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.4999999999999997e-148

   1. Initial program 80.9%

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

   if -6.4999999999999997e-148 < x < -9.9999999999999991e-309

   1. Initial program 70.2%

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

      1. remove-double-neg 70.2%

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

      2. sub-neg 70.2%

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

      3. distribute-neg-in 70.2%

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

      4. distribute-rgt-neg-in 70.2%

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

      5. remove-double-neg 70.2%

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

      6. fma-udef 70.2%

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

      7. log-div 0.0%

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

      8. sub-neg 0.0%

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

      9. distribute-neg-in 0.0%

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

      10. remove-double-neg 0.0%

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

      11. +-commutative 0.0%

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

      12. sub-neg 0.0%

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

      13. log-div 70.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in x around 0 90.9%

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

   if -9.9999999999999991e-309 < x

   1. Initial program 78.7%

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

      1. log-div 99.4%

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

   4. Applied egg-rr 99.4%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-148}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-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 <= -5e-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 <= (-5d-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 <= -5e-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 <= -5e-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 <= -5e-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 <= -5e-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, -5e-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 < -4.999999999999985e-310

   1. Initial program 77.9%

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

      1. frac-2neg 77.9%

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

      2. log-div 99.5%

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

   4. Applied egg-rr 99.5%

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

   if -4.999999999999985e-310 < y

   1. Initial program 78.7%

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

      1. log-div 99.4%

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

   4. Applied egg-rr 99.4%

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

4. Final simplification 99.5%

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

Alternative 8: 99.3% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-310)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (- (- (* x (log x)) (* x (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * (log(-x) - log(-y))) - z;
	} else {
		tmp = ((x * log(x)) - (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 <= (-5d-310)) then
        tmp = (x * (log(-x) - log(-y))) - z
    else
        tmp = ((x * log(x)) - (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 <= -5e-310) {
		tmp = (x * (Math.log(-x) - Math.log(-y))) - z;
	} else {
		tmp = ((x * Math.log(x)) - (x * Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5e-310:
		tmp = (x * (math.log(-x) - math.log(-y))) - z
	else:
		tmp = ((x * math.log(x)) - (x * math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-310)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-y)))) - z);
	else
		tmp = Float64(Float64(Float64(x * log(x)) - Float64(x * log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5e-310)
		tmp = (x * (log(-x) - log(-y))) - z;
	else
		tmp = ((x * log(x)) - (x * log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5e-310], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 77.9%

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

      1. frac-2neg 77.9%

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

      2. log-div 99.5%

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

   4. Applied egg-rr 99.5%

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

   if -4.999999999999985e-310 < y

   1. Initial program 78.7%

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

      1. add-cube-cbrt 78.7%

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

      2. log-prod 78.6%

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

      3. pow2 78.6%

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

   4. Applied egg-rr 78.6%

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

      1. log-pow 78.7%

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

      2. distribute-lft1-in 78.7%

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

      3. metadata-eval 78.7%

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

   6. Simplified 78.7%

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

      1. pow1/3 78.5%

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

      2. log-pow 78.6%

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

   8. Applied egg-rr 78.6%

      \[\leadsto x \cdot \left(3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(\frac{x}{y}\right)\right)}\right) - z \]
   9. Step-by-step derivation

      1. clear-num 78.6%

         \[\leadsto x \cdot \left(3 \cdot \left(0.3333333333333333 \cdot \log \color{blue}{\left(\frac{1}{\frac{y}{x}}\right)}\right)\right) - z \]

      2. log-rec 79.6%

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

   10. Applied egg-rr 79.6%

       \[\leadsto x \cdot \left(3 \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(-\log \left(\frac{y}{x}\right)\right)}\right)\right) - z \]
   11. Step-by-step derivation

       1. associate-*r* 79.7%

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

       2. metadata-eval 79.7%

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

       3. *-un-lft-identity 79.7%

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

       4. neg-log 78.7%

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

       5. clear-num 78.7%

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

       6. diff-log 99.4%

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

       7. sub-neg 99.4%

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

       8. distribute-rgt-in 99.4%

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

   12. Applied egg-rr 99.4%

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

4. Final simplification 99.5%

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

Alternative 9: 64.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-44} \lor \neg \left(x \leq 10^{-31}\right):\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1e-44) (not (<= x 1e-31))) (* (- x) (log (/ y x))) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1e-44) || !(x <= 1e-31)) {
		tmp = -x * log((y / x));
	} 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 <= (-1d-44)) .or. (.not. (x <= 1d-31))) then
        tmp = -x * log((y / x))
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1e-44) || !(x <= 1e-31)) {
		tmp = -x * Math.log((y / x));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1e-44) or not (x <= 1e-31):
		tmp = -x * math.log((y / x))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1e-44) || !(x <= 1e-31))
		tmp = Float64(Float64(-x) * log(Float64(y / x)));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1e-44) || ~((x <= 1e-31)))
		tmp = -x * log((y / x));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1e-44], N[Not[LessEqual[x, 1e-31]], $MachinePrecision]], N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -9.99999999999999953e-45 or 1e-31 < x

   1. Initial program 78.4%

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

      1. remove-double-neg 78.4%

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

      2. sub-neg 78.4%

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

      3. distribute-neg-in 78.4%

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

      4. distribute-rgt-neg-in 78.4%

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

      5. remove-double-neg 78.4%

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

      6. fma-udef 78.4%

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

      7. log-div 52.3%

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

      8. sub-neg 52.3%

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

      9. distribute-neg-in 52.3%

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

      10. remove-double-neg 52.3%

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

      11. +-commutative 52.3%

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

      12. sub-neg 52.3%

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

      13. log-div 79.5%

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

   3. Simplified 79.5%

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

   5. Taylor expanded in x around inf 40.8%

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

      1. log-rec 40.8%

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

      2. neg-mul-1 40.8%

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

      3. neg-mul-1 40.8%

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

      4. sub-neg 40.8%

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

      5. log-div 61.0%

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

   7. Simplified 61.0%

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

   if -9.99999999999999953e-45 < x < 1e-31

   1. Initial program 78.2%

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

      1. remove-double-neg 78.2%

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

      2. sub-neg 78.2%

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

      3. distribute-neg-in 78.2%

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

      4. distribute-rgt-neg-in 78.2%

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

      5. remove-double-neg 78.2%

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

      6. fma-udef 78.2%

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

      7. log-div 45.4%

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

      8. sub-neg 45.4%

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

      9. distribute-neg-in 45.4%

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

      10. remove-double-neg 45.4%

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

      11. +-commutative 45.4%

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

      12. sub-neg 45.4%

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

      13. log-div 77.3%

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

   3. Simplified 77.3%

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

   5. Taylor expanded in x around 0 80.7%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-44} \lor \neg \left(x \leq 10^{-31}\right):\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-45} \lor \neg \left(x \leq 1.45 \cdot 10^{-31}\right):\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -8.5e-45) (not (<= x 1.45e-31))) (* x (log (/ x y))) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.5e-45) || !(x <= 1.45e-31)) {
		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 <= (-8.5d-45)) .or. (.not. (x <= 1.45d-31))) 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 <= -8.5e-45) || !(x <= 1.45e-31)) {
		tmp = x * Math.log((x / y));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -8.5e-45) or not (x <= 1.45e-31):
		tmp = x * math.log((x / y))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -8.5e-45) || !(x <= 1.45e-31))
		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 <= -8.5e-45) || ~((x <= 1.45e-31)))
		tmp = x * log((x / y));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -8.5e-45], N[Not[LessEqual[x, 1.45e-31]], $MachinePrecision]], N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -8.50000000000000041e-45 or 1.45e-31 < x

   1. Initial program 78.4%

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

   3. Taylor expanded in z around 0 60.5%

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

   if -8.50000000000000041e-45 < x < 1.45e-31

   1. Initial program 78.2%

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

      1. remove-double-neg 78.2%

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

      2. sub-neg 78.2%

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

      3. distribute-neg-in 78.2%

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

      4. distribute-rgt-neg-in 78.2%

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

      5. remove-double-neg 78.2%

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

      6. fma-udef 78.2%

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

      7. log-div 45.4%

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

      8. sub-neg 45.4%

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

      9. distribute-neg-in 45.4%

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

      10. remove-double-neg 45.4%

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

      11. +-commutative 45.4%

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

      12. sub-neg 45.4%

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

      13. log-div 77.3%

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

   3. Simplified 77.3%

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

   5. Taylor expanded in x around 0 80.7%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-45} \lor \neg \left(x \leq 1.45 \cdot 10^{-31}\right):\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.5% 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 78.3%

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

   1. remove-double-neg 78.3%

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

   2. sub-neg 78.3%

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

   3. distribute-neg-in 78.3%

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

   4. distribute-rgt-neg-in 78.3%

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

   5. remove-double-neg 78.3%

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

   6. fma-udef 78.3%

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

   7. log-div 49.3%

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

   8. sub-neg 49.3%

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

   9. distribute-neg-in 49.3%

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

   10. remove-double-neg 49.3%

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

   11. +-commutative 49.3%

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

   12. sub-neg 49.3%

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

   13. log-div 78.5%

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

3. Simplified 78.5%

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

5. Taylor expanded in x around 0 47.8%

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

6. Final simplification 47.8%

   \[\leadsto -z \]
7. Add Preprocessing

Developer target: 88.9% 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 2024010 
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< y 7.595077799083773e-308) (- (* x (log (/ x y))) z) (- (* x (- (log x) (log y))) z))

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