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

Percentage Accurate: 77.3% → 99.7%

Time: 9.1s

Alternatives: 9

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.3% 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

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. add-cube-cbrt 73.4%

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

   2. log-prod 73.4%

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

   3. pow2 73.4%

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

3. Applied egg-rr 73.4%

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

   1. log-pow 73.4%

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

   2. distribute-lft1-in 73.4%

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

   3. metadata-eval 73.4%

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

5. Simplified 73.4%

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

   1. cbrt-div 99.7%

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

   2. div-inv 99.7%

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

7. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 86.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY))
     (- z)
     (if (<= t_0 5e+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 <= 5e+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 <= 5e+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 <= 5e+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 <= 5e+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 <= 5e+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, 5e+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 8.1%

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

      1. remove-double-neg 8.1%

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

      2. sub0-neg 8.1%

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

      3. associate--r- 8.1%

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

      4. neg-sub0 8.1%

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

      5. distribute-rgt-neg-in 8.1%

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

      6. neg-sub0 8.1%

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

      7. log-div 39.9%

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

      8. associate-+l- 39.9%

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

      9. neg-sub0 39.9%

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

      10. +-commutative 39.9%

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

      11. sub-neg 39.9%

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

      12. log-div 12.9%

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

      13. fma-udef 12.9%

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

   3. Simplified 12.9%

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

   4. Taylor expanded in x around 0 39.1%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 5.00000000000000009e305

   1. Initial program 99.8%

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

   if 5.00000000000000009e305 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 3.9%

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

   2. Taylor expanded in z around 0 3.9%

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

      1. log-div 48.9%

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

   4. Applied egg-rr 48.9%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \]

Alternative 3: 86.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY))
     (* x (- (log (- x)) (log (- y))))
     (if (<= t_0 5e+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 = x * (log(-x) - log(-y));
	} else if (t_0 <= 5e+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 = x * (Math.log(-x) - Math.log(-y));
	} else if (t_0 <= 5e+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 = x * (math.log(-x) - math.log(-y))
	elif t_0 <= 5e+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(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (t_0 <= 5e+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 = x * (log(-x) - log(-y));
	elseif (t_0 <= 5e+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)], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+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 8.1%

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

   2. Taylor expanded in z around 0 8.1%

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

      1. frac-2neg 8.1%

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

      2. log-div 54.2%

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

   4. Applied egg-rr 54.2%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 5.00000000000000009e305

   1. Initial program 99.8%

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

   if 5.00000000000000009e305 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 3.9%

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

   2. Taylor expanded in z around 0 3.9%

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

      1. log-div 48.9%

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

   4. Applied egg-rr 48.9%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \]

Alternative 4: 87.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 5 \cdot 10^{+305}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e+305))) (- 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+305)) {
		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+305)) {
		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+305):
		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+305))
		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+305)))
		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+305]], $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 5.00000000000000009e305 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 6.0%

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

      1. remove-double-neg 6.0%

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

      2. sub0-neg 6.0%

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

      3. associate--r- 6.0%

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

      4. neg-sub0 6.0%

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

      5. distribute-rgt-neg-in 6.0%

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

      6. neg-sub0 6.0%

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

      7. log-div 48.5%

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

      8. associate-+l- 48.5%

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

      9. neg-sub0 48.5%

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

      10. +-commutative 48.5%

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

      11. sub-neg 48.5%

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

      12. log-div 10.6%

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

      13. fma-udef 10.6%

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

   3. Simplified 10.6%

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

   4. Taylor expanded in x around 0 39.4%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 5.00000000000000009e305

   1. Initial program 99.8%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+305}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]

Alternative 5: 92.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+129}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-126}:\\ \;\;\;\;-\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-305}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -6.8e+129)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -1.75e-126)
     (- (fma x (log (/ y x)) z))
     (if (<= x -1e-305) (- z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.8e+129) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -1.75e-126) {
		tmp = -fma(x, log((y / x)), z);
	} else if (x <= -1e-305) {
		tmp = -z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -6.8e+129)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -1.75e-126)
		tmp = Float64(-fma(x, log(Float64(y / x)), z));
	elseif (x <= -1e-305)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -6.8e+129], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.75e-126], (-N[(x * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision] + z), $MachinePrecision]), If[LessEqual[x, -1e-305], (-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 < -6.80000000000000036e129

   1. Initial program 60.4%

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

   2. Taylor expanded in z around 0 51.5%

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

      1. frac-2neg 51.5%

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

      2. log-div 90.4%

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

   4. Applied egg-rr 90.4%

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

   if -6.80000000000000036e129 < x < -1.75e-126

   1. Initial program 93.8%

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

      1. remove-double-neg 93.8%

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

      2. sub0-neg 93.8%

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

      3. associate--r- 93.8%

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

      4. neg-sub0 93.8%

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

      5. distribute-rgt-neg-in 93.8%

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

      6. neg-sub0 93.8%

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

      7. log-div 0.0%

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

      8. associate-+l- 0.0%

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

      9. neg-sub0 0.0%

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

      10. +-commutative 0.0%

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

      11. sub-neg 0.0%

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

      12. log-div 93.7%

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

      13. fma-udef 93.8%

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

   3. Simplified 93.8%

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

   if -1.75e-126 < x < -9.99999999999999996e-306

   1. Initial program 59.2%

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

      1. remove-double-neg 59.2%

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

      2. sub0-neg 59.2%

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

      3. associate--r- 59.2%

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

      4. neg-sub0 59.2%

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

      5. distribute-rgt-neg-in 59.2%

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

      6. neg-sub0 59.2%

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

      7. log-div 0.0%

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

      8. associate-+l- 0.0%

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

      9. neg-sub0 0.0%

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

      10. +-commutative 0.0%

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

      11. sub-neg 0.0%

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

      12. log-div 56.5%

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

      13. fma-udef 56.5%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in x around 0 84.4%

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

   if -9.99999999999999996e-306 < x

   1. Initial program 74.0%

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

      1. log-div 52.7%

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

   3. Applied egg-rr 99.3%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+129}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-126}:\\ \;\;\;\;-\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-305}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 6: 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 72.8%

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

      1. frac-2neg 35.9%

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

      2. log-div 52.0%

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

   3. Applied egg-rr 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 74.0%

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

      1. log-div 52.7%

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

   3. Applied egg-rr 99.3%

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

4. Final simplification 99.5%

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

Alternative 7: 65.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+77} \lor \neg \left(x \leq 2 \cdot 10^{+67}\right):\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.5e+77) (not (<= x 2e+67))) (* (- x) (log (/ y x))) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.5e+77) || !(x <= 2e+67)) {
		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 <= (-1.5d+77)) .or. (.not. (x <= 2d+67))) 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 <= -1.5e+77) || !(x <= 2e+67)) {
		tmp = -x * Math.log((y / x));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.5e+77) or not (x <= 2e+67):
		tmp = -x * math.log((y / x))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.5e+77) || !(x <= 2e+67))
		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 <= -1.5e+77) || ~((x <= 2e+67)))
		tmp = -x * log((y / x));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.5e+77], N[Not[LessEqual[x, 2e+67]], $MachinePrecision]], N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -1.4999999999999999e77 or 1.99999999999999997e67 < x

   1. Initial program 71.6%

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

      1. remove-double-neg 71.6%

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

      2. sub0-neg 71.6%

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

      3. associate--r- 71.6%

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

      4. neg-sub0 71.6%

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

      5. distribute-rgt-neg-in 71.6%

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

      6. neg-sub0 71.6%

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

      7. log-div 54.9%

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

      8. associate-+l- 54.9%

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

      9. neg-sub0 54.9%

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

      10. +-commutative 54.9%

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

      11. sub-neg 54.9%

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

      12. log-div 72.8%

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

      13. fma-udef 72.8%

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

   3. Simplified 72.8%

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

   4. Taylor expanded in x around inf 45.8%

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

      1. log-rec 45.8%

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

      2. neg-mul-1 45.8%

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

      3. neg-mul-1 45.8%

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

      4. sub-neg 45.8%

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

      5. log-div 57.2%

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

   6. Simplified 57.2%

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

   if -1.4999999999999999e77 < x < 1.99999999999999997e67

   1. Initial program 75.0%

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

      1. remove-double-neg 75.0%

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

      2. sub0-neg 75.0%

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

      3. associate--r- 75.0%

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

      4. neg-sub0 75.0%

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

      5. distribute-rgt-neg-in 75.0%

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

      6. neg-sub0 75.0%

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

      7. log-div 46.6%

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

      8. associate-+l- 46.6%

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

      9. neg-sub0 46.6%

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

      10. +-commutative 46.6%

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

      11. sub-neg 46.6%

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

      12. log-div 75.0%

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

      13. fma-udef 75.0%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in x around 0 75.4%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+77} \lor \neg \left(x \leq 2 \cdot 10^{+67}\right):\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 8: 64.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+77} \lor \neg \left(x \leq 1.3 \cdot 10^{+76}\right):\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -7e+77) (not (<= x 1.3e+76))) (* x (log (/ x y))) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -7e+77) || !(x <= 1.3e+76)) {
		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 <= (-7d+77)) .or. (.not. (x <= 1.3d+76))) 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 <= -7e+77) || !(x <= 1.3e+76)) {
		tmp = x * Math.log((x / y));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -7e+77) or not (x <= 1.3e+76):
		tmp = x * math.log((x / y))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -7e+77) || !(x <= 1.3e+76))
		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 <= -7e+77) || ~((x <= 1.3e+76)))
		tmp = x * log((x / y));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -7e+77], N[Not[LessEqual[x, 1.3e+76]], $MachinePrecision]], N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -7.0000000000000003e77 or 1.3e76 < x

   1. Initial program 71.6%

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

   2. Taylor expanded in z around 0 56.0%

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

   if -7.0000000000000003e77 < x < 1.3e76

   1. Initial program 75.0%

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

      1. remove-double-neg 75.0%

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

      2. sub0-neg 75.0%

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

      3. associate--r- 75.0%

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

      4. neg-sub0 75.0%

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

      5. distribute-rgt-neg-in 75.0%

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

      6. neg-sub0 75.0%

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

      7. log-div 46.6%

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

      8. associate-+l- 46.6%

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

      9. neg-sub0 46.6%

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

      10. +-commutative 46.6%

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

      11. sub-neg 46.6%

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

      12. log-div 75.0%

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

      13. fma-udef 75.0%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in x around 0 75.4%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+77} \lor \neg \left(x \leq 1.3 \cdot 10^{+76}\right):\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 9: 50.2% 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 73.4%

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

   1. remove-double-neg 73.4%

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

   2. sub0-neg 73.4%

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

   3. associate--r- 73.4%

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

   4. neg-sub0 73.4%

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

   5. distribute-rgt-neg-in 73.4%

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

   6. neg-sub0 73.4%

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

   7. log-div 50.4%

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

   8. associate-+l- 50.4%

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

   9. neg-sub0 50.4%

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

   10. +-commutative 50.4%

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

   11. sub-neg 50.4%

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

   12. log-div 73.9%

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

   13. fma-udef 74.0%

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

3. Simplified 74.0%

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

4. Taylor expanded in x around 0 48.3%

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

5. Final simplification 48.3%

   \[\leadsto -z \]

Developer target: 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 2023308 
(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))