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

Percentage Accurate: 76.8% → 99.4%

Time: 13.0s

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: 76.8% 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: 99.4% 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 82.1%

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

   3. Taylor expanded in y around -inf 99.6%

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

      1. metadata-eval 99.6%

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

      2. distribute-neg-frac 99.6%

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

      3. distribute-frac-neg2 99.6%

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

      4. neg-mul-1 99.6%

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

      5. log-rec 99.6%

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

      6. sub-neg 99.6%

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

   5. Simplified 99.6%

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

   if -4.999999999999985e-310 < y

   1. Initial program 74.8%

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

   3. Taylor expanded in y around -inf 0.0%

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

      1. metadata-eval 0.0%

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

      2. distribute-neg-frac 0.0%

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

      3. distribute-frac-neg2 0.0%

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

      4. neg-mul-1 0.0%

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

      5. log-rec 0.0%

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

      6. sub-neg 0.0%

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

   5. Simplified 0.0%

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

      1. sub-neg 0.0%

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

      2. distribute-rgt-in 0.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 0.0%

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

      5. sqr-neg 0.0%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 53.7%

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

      10. sqr-neg 53.7%

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

      11. sqrt-unprod 99.6%

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

      12. add-sqr-sqrt 99.6%

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

   7. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 88.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ t_1 := \log \left(y \cdot x\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;x \cdot \left|t\_1\right| - z\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+277}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot t\_1\right| - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))) (t_1 (log (* y x))))
   (if (<= t_0 (- INFINITY))
     (- (* x (fabs t_1)) z)
     (if (<= t_0 5e+277) (- t_0 z) (- (fabs (* x t_1)) z)))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double t_1 = log((y * x));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (x * fabs(t_1)) - z;
	} else if (t_0 <= 5e+277) {
		tmp = t_0 - z;
	} else {
		tmp = fabs((x * t_1)) - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double t_1 = Math.log((y * x));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (x * Math.abs(t_1)) - z;
	} else if (t_0 <= 5e+277) {
		tmp = t_0 - z;
	} else {
		tmp = Math.abs((x * t_1)) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	t_1 = math.log((y * x))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (x * math.fabs(t_1)) - z
	elif t_0 <= 5e+277:
		tmp = t_0 - z
	else:
		tmp = math.fabs((x * t_1)) - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	t_1 = log(Float64(y * x))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(x * abs(t_1)) - z);
	elseif (t_0 <= 5e+277)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(abs(Float64(x * t_1)) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	t_1 = log((y * x));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (x * abs(t_1)) - z;
	elseif (t_0 <= 5e+277)
		tmp = t_0 - z;
	else
		tmp = abs((x * t_1)) - 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]}, Block[{t$95$1 = N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(x * N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[t$95$0, 5e+277], N[(t$95$0 - z), $MachinePrecision], N[(N[Abs[N[(x * t$95$1), $MachinePrecision]], $MachinePrecision] - z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 7.4%

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

      1. add-cube-cbrt 7.4%

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

      2. associate-/l* 7.4%

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

      3. log-prod 79.0%

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

      4. pow2 79.0%

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

   4. Applied egg-rr 79.0%

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

      1. sum-log 7.4%

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

      2. associate-*r/ 7.4%

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

      3. unpow2 7.4%

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

      4. add-cube-cbrt 7.4%

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

      5. add-cbrt-cube 7.4%

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

      6. unpow3 7.4%

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

      7. add-sqr-sqrt 6.0%

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

      8. sqrt-unprod 6.7%

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

      9. pow2 6.7%

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

   6. Applied egg-rr 57.0%

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

      1. unpow2 57.0%

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

      2. rem-sqrt-square 57.0%

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

   8. Simplified 57.0%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 4.99999999999999982e277

   1. Initial program 99.8%

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

   if 4.99999999999999982e277 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 7.7%

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

   3. Taylor expanded in y around -inf 32.1%

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

      1. metadata-eval 32.1%

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

      2. distribute-neg-frac 32.1%

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

      3. distribute-frac-neg2 32.1%

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

      4. neg-mul-1 32.1%

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

      5. log-rec 32.1%

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

      6. sub-neg 32.1%

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

   5. Simplified 32.1%

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

      1. sub-neg 32.1%

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

      2. distribute-rgt-in 32.1%

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

      3. add-sqr-sqrt 32.1%

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

      4. sqrt-unprod 0.5%

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

      5. sqr-neg 0.5%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 24.1%

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

      10. sqr-neg 24.1%

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

      11. sqrt-unprod 67.7%

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

      12. add-sqr-sqrt 67.7%

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

   7. Applied egg-rr 67.7%

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

      1. distribute-rgt-out 67.6%

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

      2. sub-neg 67.6%

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

      3. add-cube-cbrt 67.6%

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

      4. unpow2 67.6%

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

      5. log-prod 67.5%

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

      6. associate-+r- 67.5%

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

      7. log-div 54.9%

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

      8. add-sqr-sqrt 54.9%

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

      9. sqrt-unprod 28.0%

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

      10. pow2 28.0%

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

   9. Applied egg-rr 45.4%

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

       1. unpow2 45.4%

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

       2. rem-sqrt-square 59.1%

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

   11. Simplified 59.1%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;x \cdot \left|\log \left(y \cdot x\right)\right| - z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+277}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \log \left(y \cdot x\right)\right| - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ t_1 := x \cdot \log \left(y \cdot x\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1 - z\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+277}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1\right| - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))) (t_1 (* x (log (* y x)))))
   (if (<= t_0 (- INFINITY))
     (- t_1 z)
     (if (<= t_0 5e+277) (- t_0 z) (- (fabs t_1) z)))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double t_1 = x * log((y * x));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1 - z;
	} else if (t_0 <= 5e+277) {
		tmp = t_0 - z;
	} else {
		tmp = fabs(t_1) - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double t_1 = x * Math.log((y * x));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1 - z;
	} else if (t_0 <= 5e+277) {
		tmp = t_0 - z;
	} else {
		tmp = Math.abs(t_1) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	t_1 = x * math.log((y * x))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_1 - z
	elif t_0 <= 5e+277:
		tmp = t_0 - z
	else:
		tmp = math.fabs(t_1) - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	t_1 = Float64(x * log(Float64(y * x)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_1 - z);
	elseif (t_0 <= 5e+277)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(abs(t_1) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	t_1 = x * log((y * x));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_1 - z;
	elseif (t_0 <= 5e+277)
		tmp = t_0 - z;
	else
		tmp = abs(t_1) - 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]}, Block[{t$95$1 = N[(x * N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(t$95$1 - z), $MachinePrecision], If[LessEqual[t$95$0, 5e+277], N[(t$95$0 - z), $MachinePrecision], N[(N[Abs[t$95$1], $MachinePrecision] - z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 7.4%

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

   3. Taylor expanded in y around -inf 43.7%

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

      1. metadata-eval 43.7%

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

      2. distribute-neg-frac 43.7%

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

      3. distribute-frac-neg2 43.7%

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

      4. neg-mul-1 43.7%

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

      5. log-rec 43.7%

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

      6. sub-neg 43.7%

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

   5. Simplified 43.7%

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

      1. sub-neg 43.7%

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

      2. distribute-rgt-in 43.7%

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

      3. add-sqr-sqrt 43.7%

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

      4. sqrt-unprod 14.8%

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

      5. sqr-neg 14.8%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 22.8%

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

      10. sqr-neg 22.8%

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

      11. sqrt-unprod 56.2%

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

      12. add-sqr-sqrt 56.2%

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

   7. Applied egg-rr 56.2%

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

      1. distribute-rgt-out 56.1%

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

      2. sub-neg 56.1%

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

      3. log-div 7.4%

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

      4. add-cbrt-cube 7.4%

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

      5. unpow3 7.4%

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

      6. *-commutative 7.4%

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

   9. Applied egg-rr 55.8%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 4.99999999999999982e277

   1. Initial program 99.8%

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

   if 4.99999999999999982e277 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 7.7%

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

   3. Taylor expanded in y around -inf 32.1%

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

      1. metadata-eval 32.1%

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

      2. distribute-neg-frac 32.1%

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

      3. distribute-frac-neg2 32.1%

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

      4. neg-mul-1 32.1%

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

      5. log-rec 32.1%

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

      6. sub-neg 32.1%

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

   5. Simplified 32.1%

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

      1. sub-neg 32.1%

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

      2. distribute-rgt-in 32.1%

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

      3. add-sqr-sqrt 32.1%

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

      4. sqrt-unprod 0.5%

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

      5. sqr-neg 0.5%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 24.1%

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

      10. sqr-neg 24.1%

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

      11. sqrt-unprod 67.7%

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

      12. add-sqr-sqrt 67.7%

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

   7. Applied egg-rr 67.7%

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

      1. distribute-rgt-out 67.6%

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

      2. sub-neg 67.6%

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

      3. add-cube-cbrt 67.6%

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

      4. unpow2 67.6%

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

      5. log-prod 67.5%

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

      6. associate-+r- 67.5%

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

      7. log-div 54.9%

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

      8. add-sqr-sqrt 54.9%

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

      9. sqrt-unprod 28.0%

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

      10. pow2 28.0%

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

   9. Applied egg-rr 45.4%

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

       1. unpow2 45.4%

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

       2. rem-sqrt-square 59.1%

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

   11. Simplified 59.1%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;x \cdot \log \left(y \cdot x\right) - z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+277}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \log \left(y \cdot x\right)\right| - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 5 \cdot 10^{+277}\right):\\ \;\;\;\;x \cdot \log \left(y \cdot x\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e+277)))
     (- (* x (log (* y x))) z)
     (- t_0 z))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 5e+277)) {
		tmp = (x * log((y * x))) - z;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 5e+277)) {
		tmp = (x * Math.log((y * x))) - z;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 5e+277):
		tmp = (x * math.log((y * x))) - z
	else:
		tmp = t_0 - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 5e+277))
		tmp = Float64(Float64(x * log(Float64(y * x))) - z);
	else
		tmp = Float64(t_0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 5e+277)))
		tmp = (x * log((y * x))) - z;
	else
		tmp = t_0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 5e+277]], $MachinePrecision]], N[(N[(x * N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(t$95$0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 4.99999999999999982e277 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 7.5%

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

   3. Taylor expanded in y around -inf 38.3%

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

      1. metadata-eval 38.3%

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

      2. distribute-neg-frac 38.3%

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

      3. distribute-frac-neg2 38.3%

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

      4. neg-mul-1 38.3%

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

      5. log-rec 38.3%

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

      6. sub-neg 38.3%

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

   5. Simplified 38.3%

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

      1. sub-neg 38.3%

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

      2. distribute-rgt-in 38.3%

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

      3. add-sqr-sqrt 38.3%

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

      4. sqrt-unprod 8.1%

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

      5. sqr-neg 8.1%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 23.4%

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

      10. sqr-neg 23.4%

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

      11. sqrt-unprod 61.6%

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

      12. add-sqr-sqrt 61.6%

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

   7. Applied egg-rr 61.6%

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

      1. distribute-rgt-out 61.5%

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

      2. sub-neg 61.5%

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

      3. log-div 7.5%

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

      4. add-cbrt-cube 7.5%

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

      5. unpow3 7.5%

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

      6. *-commutative 7.5%

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

   9. Applied egg-rr 55.9%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 4.99999999999999982e277

   1. Initial program 99.8%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+277}\right):\\ \;\;\;\;x \cdot \log \left(y \cdot x\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 5 \cdot 10^{+277}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e+277))) (- 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 <= 5e+277)) {
		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 <= 5e+277)) {
		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 <= 5e+277):
		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 <= 5e+277))
		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 <= 5e+277)))
		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, 5e+277]], $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 4.99999999999999982e277 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 7.5%

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

   3. Taylor expanded in x around 0 49.5%

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

      1. neg-mul-1 49.5%

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

   5. Simplified 49.5%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 4.99999999999999982e277

   1. Initial program 99.8%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+277}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+125}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-166}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.55e+125)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -4.6e-166)
     (- (* (- x) (log (/ y x))) 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 <= -1.55e+125) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -4.6e-166) {
		tmp = (-x * log((y / x))) - 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 <= (-1.55d+125)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-4.6d-166)) then
        tmp = (-x * log((y / x))) - 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 <= -1.55e+125) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -4.6e-166) {
		tmp = (-x * Math.log((y / x))) - 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 <= -1.55e+125:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -4.6e-166:
		tmp = (-x * math.log((y / x))) - 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 <= -1.55e+125)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -4.6e-166)
		tmp = Float64(Float64(Float64(-x) * log(Float64(y / x))) - 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 <= -1.55e+125)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -4.6e-166)
		tmp = (-x * log((y / x))) - 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, -1.55e+125], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.6e-166], N[(N[((-x) * N[Log[N[(y / x), $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 < -1.55e125

   1. Initial program 63.8%

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

   3. Taylor expanded in z around 0 58.0%

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

   4. Taylor expanded in y around -inf 88.3%

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

      1. metadata-eval 99.1%

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

      2. distribute-neg-frac 99.1%

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

      3. distribute-frac-neg2 99.1%

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

      4. neg-mul-1 99.1%

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

      5. log-rec 99.1%

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

      6. sub-neg 99.1%

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

   6. Simplified 88.3%

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

   if -1.55e125 < x < -4.59999999999999997e-166

   1. Initial program 98.1%

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

      1. clear-num 98.1%

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

      2. neg-log 99.8%

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

   4. Applied egg-rr 99.8%

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

   if -4.59999999999999997e-166 < x < -4.99999999999999955e-308

   1. Initial program 71.4%

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

   3. Taylor expanded in x around 0 91.9%

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

      1. neg-mul-1 91.9%

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

   5. Simplified 91.9%

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

   if -4.99999999999999955e-308 < x

   1. Initial program 74.8%

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

   3. Taylor expanded in x around 0 99.6%

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

      1. log-rec 99.6%

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

      2. sub-neg 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+125}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-166}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 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 82.1%

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

   3. Taylor expanded in y around -inf 99.6%

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

      1. metadata-eval 99.6%

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

      2. distribute-neg-frac 99.6%

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

      3. distribute-frac-neg2 99.6%

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

      4. neg-mul-1 99.6%

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

      5. log-rec 99.6%

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

      6. sub-neg 99.6%

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

   5. Simplified 99.6%

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

   if -4.999999999999985e-310 < y

   1. Initial program 74.8%

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

   3. Taylor expanded in x around 0 99.6%

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

      1. log-rec 99.6%

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

      2. sub-neg 99.6%

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

   5. Simplified 99.6%

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

Alternative 8: 66.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -410 \lor \neg \left(x \leq 2.8 \cdot 10^{-61}\right):\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -410.0) (not (<= x 2.8e-61))) (* (- x) (log (/ y x))) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -410.0) || !(x <= 2.8e-61)) {
		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 <= (-410.0d0)) .or. (.not. (x <= 2.8d-61))) 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 <= -410.0) || !(x <= 2.8e-61)) {
		tmp = -x * Math.log((y / x));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -410.0) or not (x <= 2.8e-61):
		tmp = -x * math.log((y / x))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -410.0) || !(x <= 2.8e-61))
		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 <= -410.0) || ~((x <= 2.8e-61)))
		tmp = -x * log((y / x));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -410.0], N[Not[LessEqual[x, 2.8e-61]], $MachinePrecision]], N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -410 or 2.8000000000000001e-61 < x

   1. Initial program 78.1%

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

   3. Taylor expanded in z around 0 59.4%

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

      1. clear-num 76.7%

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

      2. neg-log 78.7%

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

   5. Applied egg-rr 60.0%

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

   if -410 < x < 2.8000000000000001e-61

   1. Initial program 78.2%

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

   3. Taylor expanded in x around 0 81.4%

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

      1. neg-mul-1 81.4%

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

   5. Simplified 81.4%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -410 \lor \neg \left(x \leq 2.8 \cdot 10^{-61}\right):\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -24 \lor \neg \left(x \leq 1.75 \cdot 10^{-61}\right):\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -24.0) (not (<= x 1.75e-61))) (* x (log (/ x y))) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -24.0) || !(x <= 1.75e-61)) {
		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 <= (-24.0d0)) .or. (.not. (x <= 1.75d-61))) 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 <= -24.0) || !(x <= 1.75e-61)) {
		tmp = x * Math.log((x / y));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -24.0) or not (x <= 1.75e-61):
		tmp = x * math.log((x / y))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -24.0) || !(x <= 1.75e-61))
		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 <= -24.0) || ~((x <= 1.75e-61)))
		tmp = x * log((x / y));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -24.0], N[Not[LessEqual[x, 1.75e-61]], $MachinePrecision]], N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -24 or 1.7500000000000002e-61 < x

   1. Initial program 78.1%

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

   3. Taylor expanded in z around 0 59.4%

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

   if -24 < x < 1.7500000000000002e-61

   1. Initial program 78.2%

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

   3. Taylor expanded in x around 0 81.4%

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

      1. neg-mul-1 81.4%

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

   5. Simplified 81.4%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -24 \lor \neg \left(x \leq 1.75 \cdot 10^{-61}\right):\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.7% 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 78.2%

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

3. Taylor expanded in x around 0 51.4%

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

   1. neg-mul-1 51.4%

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

5. Simplified 51.4%

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

Alternative 11: 2.3% accurate, 107.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 78.2%

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

3. Taylor expanded in x around 0 51.4%

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

   1. neg-mul-1 51.4%

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

5. Simplified 51.4%

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

   1. neg-sub0 51.4%

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

   2. sub-neg 51.4%

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

   3. add-sqr-sqrt 22.5%

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

   4. sqrt-unprod 12.8%

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

   5. sqr-neg 12.8%

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

   6. sqrt-unprod 1.3%

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

   7. add-sqr-sqrt 2.4%

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

7. Applied egg-rr 2.4%

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

   1. +-lft-identity 2.4%

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

9. Simplified 2.4%

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

Developer Target 1: 88.5% 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 2024135 
(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))