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

Percentage Accurate: 76.3% → 99.4%

Time: 14.3s

Alternatives: 9

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-310)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (- (- (* x (log x)) (* x (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * (log(-x) - log(-y))) - z;
	} else {
		tmp = ((x * log(x)) - (x * log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-5d-310)) then
        tmp = (x * (log(-x) - log(-y))) - z
    else
        tmp = ((x * log(x)) - (x * log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * (Math.log(-x) - Math.log(-y))) - z;
	} else {
		tmp = ((x * Math.log(x)) - (x * Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5e-310:
		tmp = (x * (math.log(-x) - math.log(-y))) - z
	else:
		tmp = ((x * math.log(x)) - (x * math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-310)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-y)))) - z);
	else
		tmp = Float64(Float64(Float64(x * log(x)) - Float64(x * log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5e-310)
		tmp = (x * (log(-x) - log(-y))) - z;
	else
		tmp = ((x * log(x)) - (x * log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5e-310], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 84.9%

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

   3. Taylor expanded in y around -inf 99.5%

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

      1. metadata-eval 99.5%

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

      2. distribute-neg-frac 99.5%

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

      3. distribute-frac-neg2 99.5%

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

      4. neg-mul-1 99.5%

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

      5. log-rec 99.5%

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

      6. sub-neg 99.5%

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

   5. Simplified 99.5%

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

   if -4.999999999999985e-310 < y

   1. Initial program 83.0%

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

      1. add-sqr-sqrt 30.4%

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

      2. pow2 30.4%

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

   4. Applied egg-rr 30.4%

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

      1. unpow2 30.4%

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

      2. add-sqr-sqrt 83.0%

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

      3. diff-log 99.5%

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

      4. sub-neg 99.5%

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

      5. distribute-rgt-in 99.5%

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

   6. Applied egg-rr 99.5%

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

4. Final simplification 99.5%

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

Alternative 2: 86.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right)\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 5 \cdot 10^{+269}\right):\\ \;\;\;\;z \cdot \left(\frac{x}{z} \cdot \log \left(y \cdot x\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, t\_0, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (/ x y))) (t_1 (* x t_0)))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 5e+269)))
     (* z (+ (* (/ x z) (log (* y x))) -1.0))
     (fma x t_0 (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y));
	double t_1 = x * t_0;
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 5e+269)) {
		tmp = z * (((x / z) * log((y * x))) + -1.0);
	} else {
		tmp = fma(x, t_0, -z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = log(Float64(x / y))
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 5e+269))
		tmp = Float64(z * Float64(Float64(Float64(x / z) * log(Float64(y * x))) + -1.0));
	else
		tmp = fma(x, t_0, Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 5e+269]], $MachinePrecision]], N[(z * N[(N[(N[(x / z), $MachinePrecision] * N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(x * t$95$0 + (-z)), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 5.0000000000000002e269 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 4.5%

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

   3. Taylor expanded in z around inf 4.5%

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

      1. sub-neg 4.5%

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

      2. associate-/l* 4.5%

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

      3. metadata-eval 4.5%

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

   5. Simplified 4.5%

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

      1. clear-num 4.5%

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

      2. un-div-inv 4.5%

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

   7. Applied egg-rr 4.5%

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

      1. frac-2neg 4.5%

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

      2. div-inv 4.5%

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

      3. add-sqr-sqrt 1.8%

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

      4. sqrt-unprod 1.9%

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

      5. sqr-neg 1.9%

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

      6. sqrt-unprod 1.1%

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

      7. add-sqr-sqrt 1.8%

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

      8. diff-log 23.6%

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

      9. distribute-neg-frac2 23.6%

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

      10. diff-log 1.8%

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

      11. neg-log 1.8%

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

      12. clear-num 1.8%

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

      13. log-div 23.6%

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

      14. sub-neg 23.6%

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

      15. add-sqr-sqrt 18.9%

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

      16. sqrt-unprod 27.2%

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

      17. sqr-neg 27.2%

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

      18. sqrt-unprod 8.3%

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

      19. add-sqr-sqrt 27.3%

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

      20. +-commutative 27.3%

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

      21. sum-log 52.6%

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

   9. Applied egg-rr 52.6%

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

       1. associate-*r/ 52.6%

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

       2. *-rgt-identity 52.6%

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

       3. associate-/r/ 52.6%

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

   11. Simplified 52.6%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 5.0000000000000002e269

   1. Initial program 99.8%

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

      1. fma-neg 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+269}\right):\\ \;\;\;\;z \cdot \left(\frac{x}{z} \cdot \log \left(y \cdot x\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log \left(\frac{x}{y}\right), -z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 5 \cdot 10^{+269}\right):\\ \;\;\;\;z \cdot \left(\frac{x}{z} \cdot \log \left(y \cdot x\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e+269)))
     (* z (+ (* (/ x z) (log (* y x))) -1.0))
     (- t_0 z))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 5e+269)) {
		tmp = z * (((x / z) * log((y * x))) + -1.0);
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 5e+269)) {
		tmp = z * (((x / z) * Math.log((y * x))) + -1.0);
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 5e+269):
		tmp = z * (((x / z) * math.log((y * x))) + -1.0)
	else:
		tmp = t_0 - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 5e+269))
		tmp = Float64(z * Float64(Float64(Float64(x / z) * log(Float64(y * x))) + -1.0));
	else
		tmp = Float64(t_0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 5e+269)))
		tmp = z * (((x / z) * log((y * x))) + -1.0);
	else
		tmp = t_0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 5e+269]], $MachinePrecision]], N[(z * N[(N[(N[(x / z), $MachinePrecision] * N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 5.0000000000000002e269 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 4.5%

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

   3. Taylor expanded in z around inf 4.5%

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

      1. sub-neg 4.5%

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

      2. associate-/l* 4.5%

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

      3. metadata-eval 4.5%

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

   5. Simplified 4.5%

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

      1. clear-num 4.5%

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

      2. un-div-inv 4.5%

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

   7. Applied egg-rr 4.5%

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

      1. frac-2neg 4.5%

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

      2. div-inv 4.5%

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

      3. add-sqr-sqrt 1.8%

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

      4. sqrt-unprod 1.9%

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

      5. sqr-neg 1.9%

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

      6. sqrt-unprod 1.1%

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

      7. add-sqr-sqrt 1.8%

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

      8. diff-log 23.6%

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

      9. distribute-neg-frac2 23.6%

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

      10. diff-log 1.8%

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

      11. neg-log 1.8%

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

      12. clear-num 1.8%

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

      13. log-div 23.6%

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

      14. sub-neg 23.6%

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

      15. add-sqr-sqrt 18.9%

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

      16. sqrt-unprod 27.2%

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

      17. sqr-neg 27.2%

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

      18. sqrt-unprod 8.3%

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

      19. add-sqr-sqrt 27.3%

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

      20. +-commutative 27.3%

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

      21. sum-log 52.6%

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

   9. Applied egg-rr 52.6%

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

       1. associate-*r/ 52.6%

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

       2. *-rgt-identity 52.6%

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

       3. associate-/r/ 52.6%

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

   11. Simplified 52.6%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 5.0000000000000002e269

   1. Initial program 99.8%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+269}\right):\\ \;\;\;\;z \cdot \left(\frac{x}{z} \cdot \log \left(y \cdot x\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 5 \cdot 10^{+269}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e+269))) (- z) (- t_0 z))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 5e+269)) {
		tmp = -z;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 5e+269)) {
		tmp = -z;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 5e+269):
		tmp = -z
	else:
		tmp = t_0 - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 5e+269))
		tmp = Float64(-z);
	else
		tmp = Float64(t_0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 5e+269)))
		tmp = -z;
	else
		tmp = t_0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 5e+269]], $MachinePrecision]], (-z), N[(t$95$0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 5.0000000000000002e269 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 4.5%

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

   3. Taylor expanded in x around 0 48.1%

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

      1. mul-1-neg 48.1%

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

   5. Simplified 48.1%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 5.0000000000000002e269

   1. Initial program 99.8%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+269}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.3 \cdot 10^{-107}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.3e-107)
   (- (* (- x) (log (/ y x))) z)
   (if (<= x -5e-309) (- z) (- (* x (- (log x) (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.3e-107) {
		tmp = (-x * log((y / x))) - z;
	} else if (x <= -5e-309) {
		tmp = -z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-5.3d-107)) then
        tmp = (-x * log((y / x))) - z
    else if (x <= (-5d-309)) then
        tmp = -z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.3e-107) {
		tmp = (-x * Math.log((y / x))) - z;
	} else if (x <= -5e-309) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.3e-107:
		tmp = (-x * math.log((y / x))) - z
	elif x <= -5e-309:
		tmp = -z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.3e-107)
		tmp = Float64(Float64(Float64(-x) * log(Float64(y / x))) - z);
	elseif (x <= -5e-309)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5.3e-107)
		tmp = (-x * log((y / x))) - z;
	elseif (x <= -5e-309)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.3e-107], N[(N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -5e-309], (-z), N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.3e-107

   1. Initial program 92.1%

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

      1. clear-num 92.1%

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

      2. log-div 92.6%

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

      3. metadata-eval 92.6%

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

   4. Applied egg-rr 92.6%

      \[\leadsto x \cdot \color{blue}{\left(0 - \log \left(\frac{y}{x}\right)\right)} - z \]
   5. Step-by-step derivation

      1. neg-sub0 92.6%

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

   6. Simplified 92.6%

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

   if -5.3e-107 < x < -4.9999999999999995e-309

   1. Initial program 67.1%

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

   3. Taylor expanded in x around 0 88.7%

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

      1. mul-1-neg 88.7%

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

   5. Simplified 88.7%

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

   if -4.9999999999999995e-309 < x

   1. Initial program 83.0%

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

   3. Taylor expanded in x around 0 99.5%

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

      1. log-rec 99.5%

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

      2. sub-neg 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.3 \cdot 10^{-107}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-310)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (- (* x (- (log x) (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * (log(-x) - log(-y))) - z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-5d-310)) then
        tmp = (x * (log(-x) - log(-y))) - z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * (Math.log(-x) - Math.log(-y))) - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5e-310:
		tmp = (x * (math.log(-x) - math.log(-y))) - z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-310)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-y)))) - z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5e-310)
		tmp = (x * (log(-x) - log(-y))) - z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5e-310], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 84.9%

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

   3. Taylor expanded in y around -inf 99.5%

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

      1. metadata-eval 99.5%

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

      2. distribute-neg-frac 99.5%

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

      3. distribute-frac-neg2 99.5%

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

      4. neg-mul-1 99.5%

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

      5. log-rec 99.5%

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

      6. sub-neg 99.5%

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

   5. Simplified 99.5%

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

   if -4.999999999999985e-310 < y

   1. Initial program 83.0%

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

   3. Taylor expanded in x around 0 99.5%

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

      1. log-rec 99.5%

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

      2. sub-neg 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 7: 67.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+30}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -2.7 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-92} \lor \neg \left(z \leq 1.45 \cdot 10^{-55}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.5e+30)
   (- z)
   (if (<= z -2.7e-9)
     (* x (log (/ x y)))
     (if (or (<= z -1.8e-92) (not (<= z 1.45e-55)))
       (- z)
       (* (- x) (log (/ y x)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.5e+30) {
		tmp = -z;
	} else if (z <= -2.7e-9) {
		tmp = x * log((x / y));
	} else if ((z <= -1.8e-92) || !(z <= 1.45e-55)) {
		tmp = -z;
	} else {
		tmp = -x * log((y / x));
	}
	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 <= (-4.5d+30)) then
        tmp = -z
    else if (z <= (-2.7d-9)) then
        tmp = x * log((x / y))
    else if ((z <= (-1.8d-92)) .or. (.not. (z <= 1.45d-55))) then
        tmp = -z
    else
        tmp = -x * log((y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.5e+30) {
		tmp = -z;
	} else if (z <= -2.7e-9) {
		tmp = x * Math.log((x / y));
	} else if ((z <= -1.8e-92) || !(z <= 1.45e-55)) {
		tmp = -z;
	} else {
		tmp = -x * Math.log((y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4.5e+30:
		tmp = -z
	elif z <= -2.7e-9:
		tmp = x * math.log((x / y))
	elif (z <= -1.8e-92) or not (z <= 1.45e-55):
		tmp = -z
	else:
		tmp = -x * math.log((y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.5e+30)
		tmp = Float64(-z);
	elseif (z <= -2.7e-9)
		tmp = Float64(x * log(Float64(x / y)));
	elseif ((z <= -1.8e-92) || !(z <= 1.45e-55))
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(-x) * log(Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -4.5e+30)
		tmp = -z;
	elseif (z <= -2.7e-9)
		tmp = x * log((x / y));
	elseif ((z <= -1.8e-92) || ~((z <= 1.45e-55)))
		tmp = -z;
	else
		tmp = -x * log((y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.5e+30], (-z), If[LessEqual[z, -2.7e-9], N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, -1.8e-92], N[Not[LessEqual[z, 1.45e-55]], $MachinePrecision]], (-z), N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.49999999999999995e30 or -2.7000000000000002e-9 < z < -1.80000000000000008e-92 or 1.45e-55 < z

   1. Initial program 82.1%

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

   3. Taylor expanded in x around 0 78.3%

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

      1. mul-1-neg 78.3%

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

   5. Simplified 78.3%

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

   if -4.49999999999999995e30 < z < -2.7000000000000002e-9

   1. Initial program 80.8%

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

   3. Taylor expanded in z around 0 79.3%

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

   if -1.80000000000000008e-92 < z < 1.45e-55

   1. Initial program 86.6%

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

      1. clear-num 86.6%

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

      2. log-div 87.0%

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

      3. metadata-eval 87.0%

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

   4. Applied egg-rr 87.0%

      \[\leadsto x \cdot \color{blue}{\left(0 - \log \left(\frac{y}{x}\right)\right)} - z \]
   5. Step-by-step derivation

      1. neg-sub0 87.0%

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

   6. Simplified 87.0%

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

   7. Taylor expanded in z around 0 79.9%

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

      1. neg-mul-1 79.9%

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

      2. distribute-lft-neg-in 79.9%

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

      3. *-commutative 79.9%

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

   9. Simplified 79.9%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+30}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -2.7 \cdot 10^{-9}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-92} \lor \neg \left(z \leq 1.45 \cdot 10^{-55}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{+29} \lor \neg \left(z \leq -2.3 \cdot 10^{-12} \lor \neg \left(z \leq -1.8 \cdot 10^{-92}\right) \land z \leq 7.2 \cdot 10^{-56}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -9.8e+29)
         (not (or (<= z -2.3e-12) (and (not (<= z -1.8e-92)) (<= z 7.2e-56)))))
   (- z)
   (* x (log (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -9.8e+29) || !((z <= -2.3e-12) || (!(z <= -1.8e-92) && (z <= 7.2e-56)))) {
		tmp = -z;
	} else {
		tmp = x * log((x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-9.8d+29)) .or. (.not. (z <= (-2.3d-12)) .or. (.not. (z <= (-1.8d-92))) .and. (z <= 7.2d-56))) then
        tmp = -z
    else
        tmp = x * log((x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -9.8e+29) || !((z <= -2.3e-12) || (!(z <= -1.8e-92) && (z <= 7.2e-56)))) {
		tmp = -z;
	} else {
		tmp = x * Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -9.8e+29) or not ((z <= -2.3e-12) or (not (z <= -1.8e-92) and (z <= 7.2e-56))):
		tmp = -z
	else:
		tmp = x * math.log((x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -9.8e+29) || !((z <= -2.3e-12) || (!(z <= -1.8e-92) && (z <= 7.2e-56))))
		tmp = Float64(-z);
	else
		tmp = Float64(x * log(Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -9.8e+29) || ~(((z <= -2.3e-12) || (~((z <= -1.8e-92)) && (z <= 7.2e-56)))))
		tmp = -z;
	else
		tmp = x * log((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -9.8e+29], N[Not[Or[LessEqual[z, -2.3e-12], And[N[Not[LessEqual[z, -1.8e-92]], $MachinePrecision], LessEqual[z, 7.2e-56]]]], $MachinePrecision]], (-z), N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.8000000000000003e29 or -2.29999999999999989e-12 < z < -1.80000000000000008e-92 or 7.19999999999999956e-56 < z

   1. Initial program 82.1%

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

   3. Taylor expanded in x around 0 78.3%

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

      1. mul-1-neg 78.3%

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

   5. Simplified 78.3%

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

   if -9.8000000000000003e29 < z < -2.29999999999999989e-12 or -1.80000000000000008e-92 < z < 7.19999999999999956e-56

   1. Initial program 86.0%

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

   3. Taylor expanded in z around 0 79.4%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{+29} \lor \neg \left(z \leq -2.3 \cdot 10^{-12} \lor \neg \left(z \leq -1.8 \cdot 10^{-92}\right) \land z \leq 7.2 \cdot 10^{-56}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.8% accurate, 53.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

3. Taylor expanded in x around 0 49.3%

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

   1. mul-1-neg 49.3%

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

5. Simplified 49.3%

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

6. Final simplification 49.3%

   \[\leadsto -z \]
7. Add Preprocessing

Developer target: 87.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y < 7.595077799083773 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< y 7.595077799083773e-308)
   (- (* x (log (/ x y))) z)
   (- (* x (- (log x) (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y < 7.595077799083773e-308) {
		tmp = (x * log((x / y))) - z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y < 7.595077799083773d-308) then
        tmp = (x * log((x / y))) - z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y < 7.595077799083773e-308) {
		tmp = (x * Math.log((x / y))) - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y < 7.595077799083773e-308:
		tmp = (x * math.log((x / y))) - z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y < 7.595077799083773e-308)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y < 7.595077799083773e-308)
		tmp = (x * log((x / y))) - z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[y, 7.595077799083773e-308], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Reproduce

herbie shell --seed 2024076 
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (if (< y 7.595077799083773e-308) (- (* x (log (/ x y))) z) (- (* x (- (log x) (log y))) z))

  (- (* x (log (/ x y))) z))