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

Percentage Accurate: 76.9% → 99.7%

Time: 7.5s

Alternatives: 10

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.3%

   \[x \cdot \log \left(\frac{x}{y}\right) - z \]
2. 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. associate-*l* 78.3%

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

   3. log-prod 78.2%

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

   4. pow2 78.2%

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

   5. metadata-eval 78.2%

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

   6. log-pow 78.2%

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

   7. metadata-eval 78.2%

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

3. Applied egg-rr 78.2%

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

   1. distribute-rgt1-in 78.2%

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

   2. metadata-eval 78.2%

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

   3. *-commutative 78.2%

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

5. Simplified 78.2%

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

   1. cbrt-div 99.7%

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

   2. div-inv 99.7%

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

7. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 86.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right)\\ t_1 := x \cdot t_0\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(t_0, x, -z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (/ x y))) (t_1 (* x t_0)))
   (if (<= t_1 (- INFINITY))
     (- z)
     (if (<= t_1 2e+305) (fma t_0 x (- z)) (* x (- (log x) (log y)))))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y));
	double t_1 = x * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_1 <= 2e+305) {
		tmp = fma(t_0, x, -z);
	} else {
		tmp = x * (log(x) - log(y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = log(Float64(x / y))
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-z);
	elseif (t_1 <= 2e+305)
		tmp = fma(t_0, x, Float64(-z));
	else
		tmp = Float64(x * Float64(log(x) - log(y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-z), If[LessEqual[t$95$1, 2e+305], N[(t$95$0 * x + (-z)), $MachinePrecision], N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.2%

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

   2. Taylor expanded in x around 0 46.3%

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

      1. neg-mul-1 46.3%

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

   4. Simplified 46.3%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1.9999999999999999e305

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. fma-neg 99.8%

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

   3. Applied egg-rr 99.8%

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

   if 1.9999999999999999e305 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 7.7%

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

   2. Taylor expanded in z around 0 7.7%

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

      1. log-div 48.6%

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

   4. Applied egg-rr 48.6%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, -z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \]

Alternative 3: 86.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY))
     (- z)
     (if (<= t_0 2e+305) (- t_0 z) (* x (- (log x) (log y)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_0 <= 2e+305) {
		tmp = t_0 - z;
	} else {
		tmp = x * (log(x) - log(y));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = -z;
	} else if (t_0 <= 2e+305) {
		tmp = t_0 - z;
	} else {
		tmp = x * (Math.log(x) - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -z
	elif t_0 <= 2e+305:
		tmp = t_0 - z
	else:
		tmp = x * (math.log(x) - math.log(y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(-z);
	elseif (t_0 <= 2e+305)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(x * Float64(log(x) - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = -z;
	elseif (t_0 <= 2e+305)
		tmp = t_0 - z;
	else
		tmp = x * (log(x) - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], (-z), If[LessEqual[t$95$0, 2e+305], N[(t$95$0 - z), $MachinePrecision], N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.2%

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

   2. Taylor expanded in x around 0 46.3%

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

      1. neg-mul-1 46.3%

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

   4. Simplified 46.3%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1.9999999999999999e305

   1. Initial program 99.8%

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

   if 1.9999999999999999e305 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 7.7%

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

   2. Taylor expanded in z around 0 7.7%

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

      1. log-div 48.6%

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

   4. Applied egg-rr 48.6%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 2 \cdot 10^{+305}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \]

Alternative 4: 87.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY)) (- z) (if (<= t_0 2e+305) (- t_0 z) (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_0 <= 2e+305) {
		tmp = t_0 - z;
	} else {
		tmp = -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) {
		tmp = -z;
	} else if (t_0 <= 2e+305) {
		tmp = t_0 - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -z
	elif t_0 <= 2e+305:
		tmp = t_0 - z
	else:
		tmp = -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))
		tmp = Float64(-z);
	elseif (t_0 <= 2e+305)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(-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)
		tmp = -z;
	elseif (t_0 <= 2e+305)
		tmp = t_0 - z;
	else
		tmp = -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[LessEqual[t$95$0, (-Infinity)], (-z), If[LessEqual[t$95$0, 2e+305], N[(t$95$0 - z), $MachinePrecision], (-z)]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 1.9999999999999999e305 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 6.5%

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

   2. Taylor expanded in x around 0 44.4%

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

      1. neg-mul-1 44.4%

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

   4. Simplified 44.4%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1.9999999999999999e305

   1. Initial program 99.8%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 2 \cdot 10^{+305}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 5: 87.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+141}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.36 \cdot 10^{-149}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-146}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+232}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.7e+141)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -1.36e-149)
     (- (* x (log (/ x y))) z)
     (if (<= x 7.8e-146)
       (- z)
       (if (<= x 1.55e+232)
         (- (* (- x) (log (/ y x))) z)
         (* x (- (log x) (log y))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.7e+141) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -1.36e-149) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= 7.8e-146) {
		tmp = -z;
	} else if (x <= 1.55e+232) {
		tmp = (-x * log((y / x))) - z;
	} else {
		tmp = x * (log(x) - log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-4.7d+141)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-1.36d-149)) then
        tmp = (x * log((x / y))) - z
    else if (x <= 7.8d-146) then
        tmp = -z
    else if (x <= 1.55d+232) then
        tmp = (-x * log((y / x))) - z
    else
        tmp = x * (log(x) - log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.7e+141) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -1.36e-149) {
		tmp = (x * Math.log((x / y))) - z;
	} else if (x <= 7.8e-146) {
		tmp = -z;
	} else if (x <= 1.55e+232) {
		tmp = (-x * Math.log((y / x))) - z;
	} else {
		tmp = x * (Math.log(x) - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.7e+141:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -1.36e-149:
		tmp = (x * math.log((x / y))) - z
	elif x <= 7.8e-146:
		tmp = -z
	elif x <= 1.55e+232:
		tmp = (-x * math.log((y / x))) - z
	else:
		tmp = x * (math.log(x) - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.7e+141)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -1.36e-149)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= 7.8e-146)
		tmp = Float64(-z);
	elseif (x <= 1.55e+232)
		tmp = Float64(Float64(Float64(-x) * log(Float64(y / x))) - z);
	else
		tmp = Float64(x * Float64(log(x) - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -4.7e+141)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -1.36e-149)
		tmp = (x * log((x / y))) - z;
	elseif (x <= 7.8e-146)
		tmp = -z;
	elseif (x <= 1.55e+232)
		tmp = (-x * log((y / x))) - z;
	else
		tmp = x * (log(x) - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.7e+141], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.36e-149], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, 7.8e-146], (-z), If[LessEqual[x, 1.55e+232], N[(N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -4.69999999999999979e141

   1. Initial program 53.4%

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

   2. Taylor expanded in z around 0 47.1%

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

      1. frac-2neg 47.1%

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

      2. log-div 89.8%

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

   4. Applied egg-rr 89.8%

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

   if -4.69999999999999979e141 < x < -1.36e-149

   1. Initial program 96.0%

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

   if -1.36e-149 < x < 7.80000000000000005e-146

   1. Initial program 70.0%

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

   2. Taylor expanded in x around 0 90.5%

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

      1. neg-mul-1 90.5%

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

   4. Simplified 90.5%

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

   if 7.80000000000000005e-146 < x < 1.54999999999999992e232

   1. Initial program 88.1%

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

      1. clear-num 45.6%

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

      2. neg-log 46.7%

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

   3. Applied egg-rr 89.2%

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

   if 1.54999999999999992e232 < x

   1. Initial program 57.1%

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

   2. Taylor expanded in z around 0 53.7%

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

      1. log-div 95.2%

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

   4. Applied egg-rr 95.2%

      \[\leadsto \color{blue}{\left(\log x - \log y\right)} \cdot x \]
3. Recombined 5 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+141}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.36 \cdot 10^{-149}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-146}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+232}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \]

Alternative 6: 93.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{+139}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -5.6 \cdot 10^{-149}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.1e+139)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -5.6e-149)
     (- (* x (log (/ x y))) z)
     (if (<= x -5e-309) (- z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.1e+139) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -5.6e-149) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -5e-309) {
		tmp = -z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-4.1d+139)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-5.6d-149)) then
        tmp = (x * log((x / y))) - z
    else if (x <= (-5d-309)) then
        tmp = -z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.1e+139) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -5.6e-149) {
		tmp = (x * Math.log((x / y))) - z;
	} else if (x <= -5e-309) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.1e+139:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -5.6e-149:
		tmp = (x * math.log((x / y))) - z
	elif x <= -5e-309:
		tmp = -z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.1e+139)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -5.6e-149)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -5e-309)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -4.1e+139)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -5.6e-149)
		tmp = (x * log((x / y))) - z;
	elseif (x <= -5e-309)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.1e+139], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.6e-149], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -5e-309], (-z), N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.1000000000000002e139

   1. Initial program 53.4%

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

   2. Taylor expanded in z around 0 47.1%

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

      1. frac-2neg 47.1%

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

      2. log-div 89.8%

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

   4. Applied egg-rr 89.8%

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

   if -4.1000000000000002e139 < x < -5.5999999999999997e-149

   1. Initial program 96.0%

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

   if -5.5999999999999997e-149 < x < -4.9999999999999995e-309

   1. Initial program 69.9%

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

   2. Taylor expanded in x around 0 91.4%

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

      1. neg-mul-1 91.4%

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

   4. Simplified 91.4%

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

   if -4.9999999999999995e-309 < x

   1. Initial program 79.7%

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

      1. log-div 47.4%

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

   3. 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 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{+139}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -5.6 \cdot 10^{-149}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 7: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-309}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-309)
   (- (* 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 <= -1e-309) {
		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 <= (-1d-309)) 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 <= -1e-309) {
		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 <= -1e-309:
		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 <= -1e-309)
		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 <= -1e-309)
		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, -1e-309], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.000000000000002e-309

   1. Initial program 76.7%

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

      1. frac-2neg 30.6%

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

      2. log-div 42.5%

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

   3. Applied egg-rr 99.6%

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

   if -1.000000000000002e-309 < y

   1. Initial program 79.7%

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

      1. log-div 47.4%

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

   3. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-309}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 8: 67.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{-29}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-84}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.5e-29) (- z) (if (<= z 8.6e-84) (* (- x) (log (/ y x))) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.5e-29) {
		tmp = -z;
	} else if (z <= 8.6e-84) {
		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 (z <= (-3.5d-29)) then
        tmp = -z
    else if (z <= 8.6d-84) 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 (z <= -3.5e-29) {
		tmp = -z;
	} else if (z <= 8.6e-84) {
		tmp = -x * Math.log((y / x));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.5e-29:
		tmp = -z
	elif z <= 8.6e-84:
		tmp = -x * math.log((y / x))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.5e-29)
		tmp = Float64(-z);
	elseif (z <= 8.6e-84)
		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 (z <= -3.5e-29)
		tmp = -z;
	elseif (z <= 8.6e-84)
		tmp = -x * log((y / x));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.5e-29], (-z), If[LessEqual[z, 8.6e-84], N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -3.4999999999999997e-29 or 8.6000000000000007e-84 < z

   1. Initial program 80.7%

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

   2. Taylor expanded in x around 0 77.3%

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

      1. neg-mul-1 77.3%

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

   4. Simplified 77.3%

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

   if -3.4999999999999997e-29 < z < 8.6000000000000007e-84

   1. Initial program 74.3%

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

   2. Taylor expanded in z around 0 62.3%

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

      1. clear-num 62.2%

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

      2. neg-log 62.9%

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

   4. Applied egg-rr 62.9%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{-29}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-84}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 9: 67.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-29}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-84}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3e-29) (- z) (if (<= z 2.7e-84) (* x (log (/ x y))) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3e-29) {
		tmp = -z;
	} else if (z <= 2.7e-84) {
		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 (z <= (-3d-29)) then
        tmp = -z
    else if (z <= 2.7d-84) 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 (z <= -3e-29) {
		tmp = -z;
	} else if (z <= 2.7e-84) {
		tmp = x * Math.log((x / y));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3e-29:
		tmp = -z
	elif z <= 2.7e-84:
		tmp = x * math.log((x / y))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3e-29)
		tmp = Float64(-z);
	elseif (z <= 2.7e-84)
		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 (z <= -3e-29)
		tmp = -z;
	elseif (z <= 2.7e-84)
		tmp = x * log((x / y));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3e-29], (-z), If[LessEqual[z, 2.7e-84], N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -3.0000000000000003e-29 or 2.6999999999999999e-84 < z

   1. Initial program 80.7%

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

   2. Taylor expanded in x around 0 77.3%

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

      1. neg-mul-1 77.3%

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

   4. Simplified 77.3%

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

   if -3.0000000000000003e-29 < z < 2.6999999999999999e-84

   1. Initial program 74.3%

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

   2. Taylor expanded in z around 0 62.3%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-29}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-84}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 10: 49.9% 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. Taylor expanded in x around 0 55.8%

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

   1. neg-mul-1 55.8%

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

4. Simplified 55.8%

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

5. Final simplification 55.8%

   \[\leadsto -z \]

Developer target: 88.0% 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 2023268 
(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))