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

Percentage Accurate: 77.6% → 99.3%

Time: 11.9s

Alternatives: 11

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.6% 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: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(-x\right), x, \mathsf{fma}\left(-\log \left(-y\right), x, -z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x, -x \cdot \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-311)
   (fma (log (- x)) x (fma (- (log (- y))) x (- z)))
   (- (fma (log x) x (- (* x (log y)))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-311) {
		tmp = fma(log(-x), x, fma(-log(-y), x, -z));
	} else {
		tmp = fma(log(x), x, -(x * log(y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-311)
		tmp = fma(log(Float64(-x)), x, fma(Float64(-log(Float64(-y))), x, Float64(-z)));
	else
		tmp = Float64(fma(log(x), x, Float64(-Float64(x * log(y)))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5e-311], N[(N[Log[(-x)], $MachinePrecision] * x + N[((-N[Log[(-y)], $MachinePrecision]) * x + (-z)), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] * x + (-N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.00000000000023e-311

   1. Initial program 74.4%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. pow-lowering-pow.f64 N/A

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

      6. clear-num N/A

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

      7. inv-pow N/A

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

      8. pow-pow N/A

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

      9. pow-lowering-pow.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 74.4

          \[\leadsto x \cdot \log \left({\left({\left(\frac{x}{y}\right)}^{\color{blue}{0.5}}\right)}^{2}\right) - z \]

   4. Applied egg-rr 74.4%

      \[\leadsto x \cdot \log \color{blue}{\left({\left({\left(\frac{x}{y}\right)}^{0.5}\right)}^{2}\right)} - z \]
   5. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow1/2 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. unpow1/2 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 74.4

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

   6. Applied egg-rr 74.4%

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

      1. sub-neg N/A

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

      2. rem-square-sqrt N/A

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

      3. frac-2neg N/A

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

      4. diff-log N/A

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

      5. sub-neg N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-+l+ N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. log-lowering-log.f64 N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. neg-lowering-neg.f64 N/A

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

      13. log-lowering-log.f64 N/A

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

      14. neg-lowering-neg.f64 N/A

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

      15. neg-lowering-neg.f64 99.6

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

   8. Applied egg-rr 99.6%

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

   if -5.00000000000023e-311 < y

   1. Initial program 80.5%

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

      1. log-div N/A

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

      2. sub-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. log-lowering-log.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. log-lowering-log.f64 99.4

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

   4. Applied egg-rr 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(-x\right), x, \mathsf{fma}\left(-\log \left(-y\right), x, -z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x, -x \cdot \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ t_1 := \sqrt{\frac{x}{y}}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;x \cdot \log \left(t\_1 \cdot t\_1\right) - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))) (t_1 (sqrt (/ x y))))
   (if (<= t_0 (- INFINITY))
     (- z)
     (if (<= t_0 2e+301) (- (* x (log (* t_1 t_1))) z) (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double t_1 = sqrt((x / y));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_0 <= 2e+301) {
		tmp = (x * log((t_1 * t_1))) - 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 t_1 = Math.sqrt((x / y));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = -z;
	} else if (t_0 <= 2e+301) {
		tmp = (x * Math.log((t_1 * t_1))) - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	t_1 = math.sqrt((x / y))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -z
	elif t_0 <= 2e+301:
		tmp = (x * math.log((t_1 * t_1))) - z
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	t_1 = sqrt(Float64(x / y))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(-z);
	elseif (t_0 <= 2e+301)
		tmp = Float64(Float64(x * log(Float64(t_1 * t_1))) - z);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	t_1 = sqrt((x / y));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = -z;
	elseif (t_0 <= 2e+301)
		tmp = (x * log((t_1 * t_1))) - 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]}, Block[{t$95$1 = N[Sqrt[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], (-z), If[LessEqual[t$95$0, 2e+301], N[(N[(x * N[Log[N[(t$95$1 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], (-z)]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 2.00000000000000011e301 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 4.6%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. neg-lowering-neg.f64 45.3

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

   5. Simplified 45.3%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 2.00000000000000011e301

   1. Initial program 99.6%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. pow-lowering-pow.f64 N/A

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

      6. clear-num N/A

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

      7. inv-pow N/A

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

      8. pow-pow N/A

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

      9. pow-lowering-pow.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 99.6

          \[\leadsto x \cdot \log \left({\left({\left(\frac{x}{y}\right)}^{\color{blue}{0.5}}\right)}^{2}\right) - z \]

   4. Applied egg-rr 99.6%

      \[\leadsto x \cdot \log \color{blue}{\left({\left({\left(\frac{x}{y}\right)}^{0.5}\right)}^{2}\right)} - z \]
   5. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow1/2 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. unpow1/2 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 99.6

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

   6. Applied egg-rr 99.6%

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

Alternative 3: 87.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY)) (- z) (if (<= t_0 2e+301) (- 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+301) {
		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+301) {
		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+301:
		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+301)
		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+301)
		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+301], 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 2.00000000000000011e301 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 4.6%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. neg-lowering-neg.f64 45.3

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

   5. Simplified 45.3%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 2.00000000000000011e301

   1. Initial program 99.6%

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

Alternative 4: 91.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{x}{y}}\\ \mathbf{if}\;x \leq -5.7 \cdot 10^{+41}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-107}:\\ \;\;\;\;x \cdot \log \left(t\_0 \cdot t\_0\right) - z\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (sqrt (/ x y))))
   (if (<= x -5.7e+41)
     (* x (- (log (- x)) (log (- y))))
     (if (<= x -5.4e-107)
       (- (* x (log (* t_0 t_0))) z)
       (if (<= x -1e-308) (- z) (- (* x (- (log x) (log y))) z))))))

C

double code(double x, double y, double z) {
	double t_0 = sqrt((x / y));
	double tmp;
	if (x <= -5.7e+41) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -5.4e-107) {
		tmp = (x * log((t_0 * t_0))) - z;
	} else if (x <= -1e-308) {
		tmp = -z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((x / y))
    if (x <= (-5.7d+41)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-5.4d-107)) then
        tmp = (x * log((t_0 * t_0))) - z
    else if (x <= (-1d-308)) then
        tmp = -z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.sqrt((x / y));
	double tmp;
	if (x <= -5.7e+41) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -5.4e-107) {
		tmp = (x * Math.log((t_0 * t_0))) - z;
	} else if (x <= -1e-308) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.sqrt((x / y))
	tmp = 0
	if x <= -5.7e+41:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -5.4e-107:
		tmp = (x * math.log((t_0 * t_0))) - z
	elif x <= -1e-308:
		tmp = -z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	t_0 = sqrt(Float64(x / y))
	tmp = 0.0
	if (x <= -5.7e+41)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -5.4e-107)
		tmp = Float64(Float64(x * log(Float64(t_0 * t_0))) - z);
	elseif (x <= -1e-308)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = sqrt((x / y));
	tmp = 0.0;
	if (x <= -5.7e+41)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -5.4e-107)
		tmp = (x * log((t_0 * t_0))) - z;
	elseif (x <= -1e-308)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Sqrt[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -5.7e+41], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.4e-107], N[(N[(x * N[Log[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -1e-308], (-z), N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.70000000000000021e41

   1. Initial program 61.0%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. log-rec N/A

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

      9. unsub-neg N/A

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

      10. --lowering--.f64 N/A

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

      11. log-lowering-log.f64 N/A

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

      12. log-lowering-log.f64 0.0

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

   5. Simplified 0.0%

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

      1. diff-log N/A

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

      2. frac-2neg N/A

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

      3. log-div N/A

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

      4. --lowering--.f64 N/A

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

      5. log-lowering-log.f64 N/A

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

      6. neg-lowering-neg.f64 N/A

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

      7. log-lowering-log.f64 N/A

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

      8. neg-lowering-neg.f64 89.1

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

   7. Applied egg-rr 89.1%

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

   if -5.70000000000000021e41 < x < -5.3999999999999999e-107

   1. Initial program 98.6%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. pow-lowering-pow.f64 N/A

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

      6. clear-num N/A

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

      7. inv-pow N/A

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

      8. pow-pow N/A

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

      9. pow-lowering-pow.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 98.6

          \[\leadsto x \cdot \log \left({\left({\left(\frac{x}{y}\right)}^{\color{blue}{0.5}}\right)}^{2}\right) - z \]

   4. Applied egg-rr 98.6%

      \[\leadsto x \cdot \log \color{blue}{\left({\left({\left(\frac{x}{y}\right)}^{0.5}\right)}^{2}\right)} - z \]
   5. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow1/2 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. unpow1/2 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. /-lowering-/.f64 98.6

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

   6. Applied egg-rr 98.6%

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

   if -5.3999999999999999e-107 < x < -9.9999999999999991e-309

   1. Initial program 73.6%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. neg-lowering-neg.f64 94.1

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

   5. Simplified 94.1%

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

   if -9.9999999999999991e-309 < x

   1. Initial program 80.5%

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

      1. log-div N/A

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

      2. --lowering--.f64 N/A

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

      3. log-lowering-log.f64 N/A

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

      4. log-lowering-log.f64 99.4

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

   4. Applied egg-rr 99.4%

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

Alternative 5: 90.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-107}:\\ \;\;\;\;\left(-z\right) - x \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -9.8e-107)
   (- (- z) (* x (log (/ y x))))
   (if (<= x -2e-309) (- z) (- (* x (- (log x) (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9.8e-107) {
		tmp = -z - (x * log((y / x)));
	} else if (x <= -2e-309) {
		tmp = -z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-9.8d-107)) then
        tmp = -z - (x * log((y / x)))
    else if (x <= (-2d-309)) then
        tmp = -z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -9.8e-107) {
		tmp = -z - (x * Math.log((y / x)));
	} else if (x <= -2e-309) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -9.8e-107:
		tmp = -z - (x * math.log((y / x)))
	elif x <= -2e-309:
		tmp = -z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -9.8e-107)
		tmp = Float64(Float64(-z) - Float64(x * log(Float64(y / x))));
	elseif (x <= -2e-309)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -9.8e-107)
		tmp = -z - (x * log((y / x)));
	elseif (x <= -2e-309)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -9.8e-107], N[((-z) - N[(x * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2e-309], (-z), N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.79999999999999959e-107

   1. Initial program 74.9%

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

      1. clear-num N/A

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

      2. log-rec N/A

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

      3. neg-lowering-neg.f64 N/A

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

      4. log-lowering-log.f64 N/A

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

      5. /-lowering-/.f64 75.1

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

   4. Applied egg-rr 75.1%

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

   if -9.79999999999999959e-107 < x < -1.9999999999999988e-309

   1. Initial program 73.6%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. neg-lowering-neg.f64 94.1

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

   5. Simplified 94.1%

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

   if -1.9999999999999988e-309 < x

   1. Initial program 80.5%

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

      1. log-div N/A

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

      2. --lowering--.f64 N/A

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

      3. log-lowering-log.f64 N/A

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

      4. log-lowering-log.f64 99.4

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

   4. Applied egg-rr 99.4%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-107}:\\ \;\;\;\;\left(-z\right) - x \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-311}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x, -x \cdot \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-311)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (- (fma (log x) x (- (* x (log y)))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-311) {
		tmp = (x * (log(-x) - log(-y))) - z;
	} else {
		tmp = fma(log(x), x, -(x * log(y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-311)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-y)))) - z);
	else
		tmp = Float64(fma(log(x), x, Float64(-Float64(x * log(y)))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5e-311], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] * x + (-N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.00000000000023e-311

   1. Initial program 74.4%

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

      1. frac-2neg N/A

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

      2. log-div N/A

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

      3. --lowering--.f64 N/A

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

      4. log-lowering-log.f64 N/A

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

      5. neg-lowering-neg.f64 N/A

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

      6. log-lowering-log.f64 N/A

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

      7. neg-lowering-neg.f64 99.5

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

   4. Applied egg-rr 99.5%

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

   if -5.00000000000023e-311 < y

   1. Initial program 80.5%

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

      1. log-div N/A

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

      2. sub-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. log-lowering-log.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. log-lowering-log.f64 99.4

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

   4. Applied egg-rr 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-311}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x, -x \cdot \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-311}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-311)
   (- (* 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-311) {
		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-311)) 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-311) {
		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-311:
		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-311)
		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-311)
		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-311], 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 < -5.00000000000023e-311

   1. Initial program 74.4%

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

      1. frac-2neg N/A

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

      2. log-div N/A

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

      3. --lowering--.f64 N/A

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

      4. log-lowering-log.f64 N/A

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

      5. neg-lowering-neg.f64 N/A

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

      6. log-lowering-log.f64 N/A

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

      7. neg-lowering-neg.f64 99.5

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

   4. Applied egg-rr 99.5%

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

   if -5.00000000000023e-311 < y

   1. Initial program 80.5%

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

      1. log-div N/A

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

      2. --lowering--.f64 N/A

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

      3. log-lowering-log.f64 N/A

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

      4. log-lowering-log.f64 99.4

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

   4. Applied egg-rr 99.4%

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

Alternative 8: 65.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+23}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- x) (log (/ y x)))))
   (if (<= x -2.5e-32) t_0 (if (<= x 7.5e+23) (- z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -x * log((y / x));
	double tmp;
	if (x <= -2.5e-32) {
		tmp = t_0;
	} else if (x <= 7.5e+23) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = -x * log((y / x))
    if (x <= (-2.5d-32)) then
        tmp = t_0
    else if (x <= 7.5d+23) then
        tmp = -z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -x * Math.log((y / x));
	double tmp;
	if (x <= -2.5e-32) {
		tmp = t_0;
	} else if (x <= 7.5e+23) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -x * math.log((y / x))
	tmp = 0
	if x <= -2.5e-32:
		tmp = t_0
	elif x <= 7.5e+23:
		tmp = -z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-x) * log(Float64(y / x)))
	tmp = 0.0
	if (x <= -2.5e-32)
		tmp = t_0;
	elseif (x <= 7.5e+23)
		tmp = Float64(-z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -x * log((y / x));
	tmp = 0.0;
	if (x <= -2.5e-32)
		tmp = t_0;
	elseif (x <= 7.5e+23)
		tmp = -z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.5e-32], t$95$0, If[LessEqual[x, 7.5e+23], (-z), t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5e-32 or 7.49999999999999987e23 < x

   1. Initial program 73.9%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. log-rec N/A

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

      9. unsub-neg N/A

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

      10. --lowering--.f64 N/A

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

      11. log-lowering-log.f64 N/A

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

      12. log-lowering-log.f64 38.2

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

   5. Simplified 38.2%

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

      1. diff-log N/A

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

      2. clear-num N/A

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

      3. log-rec N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. log-lowering-log.f64 N/A

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

      6. /-lowering-/.f64 64.5

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

   7. Applied egg-rr 64.5%

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

   if -2.5e-32 < x < 7.49999999999999987e23

   1. Initial program 80.6%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. neg-lowering-neg.f64 81.4

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

   5. Simplified 81.4%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-32}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+23}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.42 \cdot 10^{+23}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= x -2.5e-32) t_0 (if (<= x 1.42e+23) (- z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if (x <= -2.5e-32) {
		tmp = t_0;
	} else if (x <= 1.42e+23) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = x * log((x / y))
    if (x <= (-2.5d-32)) then
        tmp = t_0
    else if (x <= 1.42d+23) then
        tmp = -z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if (x <= -2.5e-32) {
		tmp = t_0;
	} else if (x <= 1.42e+23) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if x <= -2.5e-32:
		tmp = t_0
	elif x <= 1.42e+23:
		tmp = -z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if (x <= -2.5e-32)
		tmp = t_0;
	elseif (x <= 1.42e+23)
		tmp = Float64(-z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if (x <= -2.5e-32)
		tmp = t_0;
	elseif (x <= 1.42e+23)
		tmp = -z;
	else
		tmp = t_0;
	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[x, -2.5e-32], t$95$0, If[LessEqual[x, 1.42e+23], (-z), t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5e-32 or 1.42000000000000004e23 < x

   1. Initial program 73.9%

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

   3. Taylor expanded in z around 0

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

      1. *-lowering-*.f64 N/A

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

      2. log-lowering-log.f64 N/A

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

      3. /-lowering-/.f64 62.7

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

   5. Simplified 62.7%

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

   if -2.5e-32 < x < 1.42000000000000004e23

   1. Initial program 80.6%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. neg-lowering-neg.f64 81.4

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

   5. Simplified 81.4%

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

Alternative 10: 49.9% accurate, 40.0× 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 Float64(-z)
end

MATLAB

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

Wolfram

code[x_, y_, z_] := (-z)

Derivation

1. Initial program 77.4%

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

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

   2. neg-lowering-neg.f64 50.1

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

5. Simplified 50.1%

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

Alternative 11: 2.7% accurate, 120.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 0.0)

C

double code(double x, double y, double z) {
	return 0.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 0.0;
}

Python

def code(x, y, z):
	return 0.0

Julia

function code(x, y, z)
	return 0.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.0;
end

Wolfram

code[x_, y_, z_] := 0.0

Derivation

1. Initial program 77.4%

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

3. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

   2. mul-1-neg N/A

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

   3. log-rec N/A

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

   4. remove-double-neg N/A

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

   5. distribute-rgt-in N/A

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

   6. +-commutative N/A

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

   7. *-lowering-*.f64 N/A

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

   8. log-rec N/A

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

   9. unsub-neg N/A

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

   10. --lowering--.f64 N/A

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

   11. log-lowering-log.f64 N/A

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

   12. log-lowering-log.f64 24.7

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

5. Simplified 24.7%

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

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

   3. diff-log N/A

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

   4. log-lowering-log.f64 N/A

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

   5. /-lowering-/.f64 39.8

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

7. Applied egg-rr 39.8%

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

9. Applied egg-rr 2.7%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Developer Target 1: 88.4% accurate, 0.6× 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 2024199 
(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))