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

Percentage Accurate: 77.7% → 99.4%

Time: 11.4s

Alternatives: 10

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(-y\right)\\ t_1 := \log \left(-x\right)\\ \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \frac{{t\_1}^{3} - {t\_0}^{3}}{{t\_1}^{2} + \left({t\_0}^{2} + t\_1 \cdot t\_0\right)} - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\log y, x, x \cdot \log x\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (- y))) (t_1 (log (- x))))
   (if (<= y -5e-310)
     (-
      (*
       x
       (/
        (- (pow t_1 3.0) (pow t_0 3.0))
        (+ (pow t_1 2.0) (+ (pow t_0 2.0) (* t_1 t_0)))))
      z)
     (- (fma (- (log y)) x (* x (log x))) z))))

C

double code(double x, double y, double z) {
	double t_0 = log(-y);
	double t_1 = log(-x);
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * ((pow(t_1, 3.0) - pow(t_0, 3.0)) / (pow(t_1, 2.0) + (pow(t_0, 2.0) + (t_1 * t_0))))) - z;
	} else {
		tmp = fma(-log(y), x, (x * log(x))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = log(Float64(-y))
	t_1 = log(Float64(-x))
	tmp = 0.0
	if (y <= -5e-310)
		tmp = Float64(Float64(x * Float64(Float64((t_1 ^ 3.0) - (t_0 ^ 3.0)) / Float64((t_1 ^ 2.0) + Float64((t_0 ^ 2.0) + Float64(t_1 * t_0))))) - z);
	else
		tmp = Float64(fma(Float64(-log(y)), x, Float64(x * log(x))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Log[(-y)], $MachinePrecision]}, Block[{t$95$1 = N[Log[(-x)], $MachinePrecision]}, If[LessEqual[y, -5e-310], N[(N[(x * N[(N[(N[Power[t$95$1, 3.0], $MachinePrecision] - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$1, 2.0], $MachinePrecision] + N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[((-N[Log[y], $MachinePrecision]) * x + N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 76.4%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. log-div N/A

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

      5. flip3-- N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-+.f64 N/A

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

   4. Applied rewrites 99.2%

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

   if -4.999999999999985e-310 < y

   1. Initial program 74.5%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. log-rec N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 78.2

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

   4. Applied rewrites 78.2%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. neg-sub0 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. log-div N/A

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

      7. lift-log.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. associate--r- N/A

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

      10. neg-sub0 N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-rgt-out N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 99.7

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

Alternative 2: 87.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+296}:\\ \;\;\;\;x \cdot \log \left(\frac{\frac{1}{y}}{\frac{1}{x}}\right) - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY))
     (- z)
     (if (<= t_0 5e+296) (- (* x (log (/ (/ 1.0 y) (/ 1.0 x)))) 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 <= 5e+296) {
		tmp = (x * log(((1.0 / y) / (1.0 / x)))) - 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 <= 5e+296) {
		tmp = (x * Math.log(((1.0 / y) / (1.0 / x)))) - 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 <= 5e+296:
		tmp = (x * math.log(((1.0 / y) / (1.0 / x)))) - 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 <= 5e+296)
		tmp = Float64(Float64(x * log(Float64(Float64(1.0 / y) / Float64(1.0 / x)))) - 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 <= 5e+296)
		tmp = (x * log(((1.0 / y) / (1.0 / x)))) - 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, 5e+296], N[(N[(x * N[Log[N[(N[(1.0 / y), $MachinePrecision] / N[(1.0 / x), $MachinePrecision]), $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 5.0000000000000001e296 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 8.2%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. lower-neg.f64 52.0

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

   5. Applied rewrites 52.0%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 5.0000000000000001e296

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

      \[\leadsto x \cdot \log \color{blue}{\left(\frac{\frac{1}{y}}{\frac{1}{x}}\right)} - z \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 87.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+296}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY)) (- z) (if (<= t_0 5e+296) (- 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 <= 5e+296) {
		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 <= 5e+296) {
		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 <= 5e+296:
		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 <= 5e+296)
		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 <= 5e+296)
		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, 5e+296], 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 5.0000000000000001e296 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 8.2%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. lower-neg.f64 52.0

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

   5. Applied rewrites 52.0%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 5.0000000000000001e296

   1. Initial program 99.8%

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

Alternative 4: 96.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(-x\right) - \log \left(-y\right)\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{+167}:\\ \;\;\;\;x \cdot t\_0\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(t\_0, \frac{x}{z}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\log y, x, x \cdot \log x\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (log (- x)) (log (- y)))))
   (if (<= x -3.5e+167)
     (* x t_0)
     (if (<= x -2e-310)
       (* z (fma t_0 (/ x z) -1.0))
       (- (fma (- (log y)) x (* x (log x))) z)))))

C

double code(double x, double y, double z) {
	double t_0 = log(-x) - log(-y);
	double tmp;
	if (x <= -3.5e+167) {
		tmp = x * t_0;
	} else if (x <= -2e-310) {
		tmp = z * fma(t_0, (x / z), -1.0);
	} else {
		tmp = fma(-log(y), x, (x * log(x))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(log(Float64(-x)) - log(Float64(-y)))
	tmp = 0.0
	if (x <= -3.5e+167)
		tmp = Float64(x * t_0);
	elseif (x <= -2e-310)
		tmp = Float64(z * fma(t_0, Float64(x / z), -1.0));
	else
		tmp = Float64(fma(Float64(-log(y)), x, Float64(x * log(x))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.5e+167], N[(x * t$95$0), $MachinePrecision], If[LessEqual[x, -2e-310], N[(z * N[(t$95$0 * N[(x / z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[((-N[Log[y], $MachinePrecision]) * x + N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.49999999999999987e167

   1. Initial program 62.9%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. lower-*.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. lower--.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-log.f64 0.0

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

   5. Applied rewrites 0.0%

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

   7. Applied rewrites 94.0%

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

   if -3.49999999999999987e167 < x < -1.999999999999994e-310

   1. Initial program 80.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. lower-neg.f64 60.2

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

   5. Applied rewrites 60.2%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 79.3

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

   8. Applied rewrites 79.3%

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

   10. Applied rewrites 97.8%

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

   if -1.999999999999994e-310 < x

   1. Initial program 74.5%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. log-rec N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 78.2

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

   4. Applied rewrites 78.2%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. neg-sub0 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. log-div N/A

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

      7. lift-log.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. associate--r- N/A

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

      10. neg-sub0 N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-rgt-out N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 99.7

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

Alternative 5: 93.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+165}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-173}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-304}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\log y, x, x \cdot \log x\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7.2e+165)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -1.5e-173)
     (- (* (- x) (log (/ y x))) z)
     (if (<= x -2e-304) (- z) (- (fma (- (log y)) x (* x (log x))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.2e+165) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -1.5e-173) {
		tmp = (-x * log((y / x))) - z;
	} else if (x <= -2e-304) {
		tmp = -z;
	} else {
		tmp = fma(-log(y), x, (x * log(x))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7.2e+165)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -1.5e-173)
		tmp = Float64(Float64(Float64(-x) * log(Float64(y / x))) - z);
	elseif (x <= -2e-304)
		tmp = Float64(-z);
	else
		tmp = Float64(fma(Float64(-log(y)), x, Float64(x * log(x))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7.2e+165], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.5e-173], N[(N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -2e-304], (-z), N[(N[((-N[Log[y], $MachinePrecision]) * x + N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.1999999999999996e165

   1. Initial program 62.9%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. lower-*.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. lower--.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-log.f64 0.0

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

   5. Applied rewrites 0.0%

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

   7. Applied rewrites 94.0%

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

   if -7.1999999999999996e165 < x < -1.5000000000000001e-173

   1. Initial program 91.7%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. log-rec N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 92.5

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

   4. Applied rewrites 92.5%

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

   if -1.5000000000000001e-173 < x < -1.99999999999999994e-304

   1. Initial program 50.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. lower-neg.f64 96.8

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

   5. Applied rewrites 96.8%

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

   if -1.99999999999999994e-304 < x

   1. Initial program 74.5%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. log-rec N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 78.2

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

   4. Applied rewrites 78.2%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. neg-sub0 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. log-div N/A

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

      7. lift-log.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. associate--r- N/A

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

      10. neg-sub0 N/A

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

      11. lift-neg.f64 N/A

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

      12. distribute-rgt-out N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 99.7

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+165}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-173}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-304}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\log y, x, x \cdot \log x\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+165}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-173}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-304}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7.2e+165)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -1.5e-173)
     (- (* (- x) (log (/ y x))) z)
     (if (<= x -2e-304) (- z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.2e+165) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -1.5e-173) {
		tmp = (-x * log((y / x))) - z;
	} else if (x <= -2e-304) {
		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 <= (-7.2d+165)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-1.5d-173)) then
        tmp = (-x * log((y / x))) - z
    else if (x <= (-2d-304)) 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 <= -7.2e+165) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -1.5e-173) {
		tmp = (-x * Math.log((y / x))) - z;
	} else if (x <= -2e-304) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -7.2e+165:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -1.5e-173:
		tmp = (-x * math.log((y / x))) - z
	elif x <= -2e-304:
		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 <= -7.2e+165)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -1.5e-173)
		tmp = Float64(Float64(Float64(-x) * log(Float64(y / x))) - z);
	elseif (x <= -2e-304)
		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 <= -7.2e+165)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -1.5e-173)
		tmp = (-x * log((y / x))) - z;
	elseif (x <= -2e-304)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7.2e+165], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.5e-173], N[(N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -2e-304], (-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 < -7.1999999999999996e165

   1. Initial program 62.9%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. lower-*.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. lower--.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. lower-log.f64 0.0

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

   5. Applied rewrites 0.0%

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

   7. Applied rewrites 94.0%

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

   if -7.1999999999999996e165 < x < -1.5000000000000001e-173

   1. Initial program 91.7%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. log-rec N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 92.5

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

   4. Applied rewrites 92.5%

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

   if -1.5000000000000001e-173 < x < -1.99999999999999994e-304

   1. Initial program 50.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. lower-neg.f64 96.8

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

   5. Applied rewrites 96.8%

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

   if -1.99999999999999994e-304 < x

   1. Initial program 74.5%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 99.6

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

   4. Applied rewrites 99.6%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+165}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-173}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-304}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-173}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-304}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.5e-173)
   (- (* (- x) (log (/ y x))) z)
   (if (<= x -2e-304) (- z) (- (* x (- (log x) (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5e-173) {
		tmp = (-x * log((y / x))) - z;
	} else if (x <= -2e-304) {
		tmp = -z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.5d-173)) then
        tmp = (-x * log((y / x))) - z
    else if (x <= (-2d-304)) then
        tmp = -z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5e-173) {
		tmp = (-x * Math.log((y / x))) - z;
	} else if (x <= -2e-304) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.5e-173:
		tmp = (-x * math.log((y / x))) - z
	elif x <= -2e-304:
		tmp = -z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.5e-173)
		tmp = Float64(Float64(Float64(-x) * log(Float64(y / x))) - z);
	elseif (x <= -2e-304)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.5e-173)
		tmp = (-x * log((y / x))) - z;
	elseif (x <= -2e-304)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.5e-173], N[(N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -2e-304], (-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 < -1.5000000000000001e-173

   1. Initial program 83.4%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. log-rec N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 83.9

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

   4. Applied rewrites 83.9%

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

   if -1.5000000000000001e-173 < x < -1.99999999999999994e-304

   1. Initial program 50.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. lower-neg.f64 96.8

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

   5. Applied rewrites 96.8%

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

   if -1.99999999999999994e-304 < x

   1. Initial program 74.5%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 99.6

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

   4. Applied rewrites 99.6%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-173}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-304}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+26}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-62}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.9e+26) (- z) (if (<= z 3.2e-62) (* (- x) (log (/ y x))) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.9e+26) {
		tmp = -z;
	} else if (z <= 3.2e-62) {
		tmp = -x * log((y / x));
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2.9d+26)) then
        tmp = -z
    else if (z <= 3.2d-62) then
        tmp = -x * log((y / x))
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.9e+26) {
		tmp = -z;
	} else if (z <= 3.2e-62) {
		tmp = -x * Math.log((y / x));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.9e+26:
		tmp = -z
	elif z <= 3.2e-62:
		tmp = -x * math.log((y / x))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.9e+26)
		tmp = Float64(-z);
	elseif (z <= 3.2e-62)
		tmp = Float64(Float64(-x) * log(Float64(y / x)));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.9e+26)
		tmp = -z;
	elseif (z <= 3.2e-62)
		tmp = -x * log((y / x));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.9e+26], (-z), If[LessEqual[z, 3.2e-62], N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -2.9e26 or 3.20000000000000021e-62 < z

   1. Initial program 73.5%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. lower-neg.f64 76.7

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

   5. Applied rewrites 76.7%

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

   if -2.9e26 < z < 3.20000000000000021e-62

   1. Initial program 77.8%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. log-rec N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 80.6

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

   4. Applied rewrites 80.6%

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

   5. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 67.7

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

   7. Applied rewrites 67.7%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+26}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-62}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+26}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-62}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.9e+26) (- z) (if (<= z 3.2e-62) (* x (log (/ x y))) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.9e+26) {
		tmp = -z;
	} else if (z <= 3.2e-62) {
		tmp = x * log((x / y));
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2.9d+26)) then
        tmp = -z
    else if (z <= 3.2d-62) then
        tmp = x * log((x / y))
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.9e+26) {
		tmp = -z;
	} else if (z <= 3.2e-62) {
		tmp = x * Math.log((x / y));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.9e+26:
		tmp = -z
	elif z <= 3.2e-62:
		tmp = x * math.log((x / y))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.9e+26)
		tmp = Float64(-z);
	elseif (z <= 3.2e-62)
		tmp = Float64(x * log(Float64(x / y)));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.9e+26)
		tmp = -z;
	elseif (z <= 3.2e-62)
		tmp = x * log((x / y));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.9e+26], (-z), If[LessEqual[z, 3.2e-62], N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -2.9e26 or 3.20000000000000021e-62 < z

   1. Initial program 73.5%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. lower-neg.f64 76.7

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

   5. Applied rewrites 76.7%

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

   if -2.9e26 < z < 3.20000000000000021e-62

   1. Initial program 77.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 64.9

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

   5. Applied rewrites 64.9%

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

Alternative 10: 50.9% 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 75.5%

   \[x \cdot \log \left(\frac{x}{y}\right) - z \]
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. lower-neg.f64 50.4

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

5. Applied rewrites 50.4%

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

Developer Target 1: 88.8% 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 2024222 
(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))