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

Percentage Accurate: 77.8% → 99.5%

Time: 11.3s

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: 77.8% 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: 99.5% 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 74.9%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 76.4%

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

   3. Taylor expanded in x around 0 99.6%

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

      1. log-rec 99.6%

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

      2. sub-neg 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 87.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right)\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 4 \cdot 10^{+301}\right):\\ \;\;\;\;\left(-z\right) - x \cdot \log \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, t\_0, -z\right)\\ \end{array} \end{array} \]

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 4e+301)))
     (- (- z) (* x (log (* y x))))
     (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 <= 4e+301)) {
		tmp = -z - (x * log((y * x)));
	} 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 <= 4e+301))
		tmp = Float64(Float64(-z) - Float64(x * log(Float64(y * x))));
	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, 4e+301]], $MachinePrecision]], N[((-z) - N[(x * N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision]), $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 4.00000000000000021e301 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 3.7%

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

   3. Taylor expanded in x around 0 50.0%

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

      1. log-rec 50.0%

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

      2. sub-neg 50.0%

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

   5. Simplified 50.0%

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

      1. sub-neg 50.0%

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

      2. distribute-rgt-in 49.9%

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

   7. Applied egg-rr 49.9%

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

      1. +-commutative 49.9%

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

      2. distribute-rgt-out 50.0%

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

      3. neg-sub0 50.0%

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

      4. associate--r- 50.0%

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

      5. log-div 7.0%

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

      6. neg-sub0 7.0%

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

      7. distribute-rgt-neg-out 7.0%

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

      8. distribute-lft-neg-in 7.0%

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

      9. add-sqr-sqrt 1.0%

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

      10. sqrt-unprod 2.9%

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

      11. sqr-neg 2.9%

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

      12. neg-log 1.4%

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

      13. clear-num 1.4%

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

      14. neg-log 1.4%

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

      15. clear-num 1.4%

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

      16. sqrt-prod 0.4%

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

      17. add-sqr-sqrt 1.5%

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

      18. log-div 26.4%

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

      19. sub-neg 26.4%

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

      20. add-log-exp 26.4%

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

      21. sum-log 0.7%

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

   9. Applied egg-rr 54.3%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 4.00000000000000021e301

   1. Initial program 99.7%

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

      1. fma-neg 99.7%

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

   3. Simplified 99.7%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 4 \cdot 10^{+301}\right):\\ \;\;\;\;\left(-z\right) - x \cdot \log \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log \left(\frac{x}{y}\right), -z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 4 \cdot 10^{+301}\right):\\ \;\;\;\;\left(-z\right) - x \cdot \log \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 4e+301)))
     (- (- z) (* x (log (* y x))))
     (- 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 <= 4e+301)) {
		tmp = -z - (x * log((y * x)));
	} 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 <= 4e+301)) {
		tmp = -z - (x * Math.log((y * x)));
	} 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 <= 4e+301):
		tmp = -z - (x * math.log((y * x)))
	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 <= 4e+301))
		tmp = Float64(Float64(-z) - Float64(x * log(Float64(y * x))));
	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 <= 4e+301)))
		tmp = -z - (x * log((y * x)));
	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, 4e+301]], $MachinePrecision]], N[((-z) - N[(x * N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision]), $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 4.00000000000000021e301 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 3.7%

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

   3. Taylor expanded in x around 0 50.0%

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

      1. log-rec 50.0%

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

      2. sub-neg 50.0%

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

   5. Simplified 50.0%

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

      1. sub-neg 50.0%

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

      2. distribute-rgt-in 49.9%

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

   7. Applied egg-rr 49.9%

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

      1. +-commutative 49.9%

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

      2. distribute-rgt-out 50.0%

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

      3. neg-sub0 50.0%

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

      4. associate--r- 50.0%

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

      5. log-div 7.0%

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

      6. neg-sub0 7.0%

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

      7. distribute-rgt-neg-out 7.0%

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

      8. distribute-lft-neg-in 7.0%

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

      9. add-sqr-sqrt 1.0%

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

      10. sqrt-unprod 2.9%

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

      11. sqr-neg 2.9%

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

      12. neg-log 1.4%

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

      13. clear-num 1.4%

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

      14. neg-log 1.4%

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

      15. clear-num 1.4%

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

      16. sqrt-prod 0.4%

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

      17. add-sqr-sqrt 1.5%

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

      18. log-div 26.4%

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

      19. sub-neg 26.4%

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

      20. add-log-exp 26.4%

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

      21. sum-log 0.7%

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

   9. Applied egg-rr 54.3%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 4.00000000000000021e301

   1. Initial program 99.7%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 4 \cdot 10^{+301}\right):\\ \;\;\;\;\left(-z\right) - x \cdot \log \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 4 \cdot 10^{+301}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 4e+301))) (- 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 <= 4e+301)) {
		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 <= 4e+301)) {
		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 <= 4e+301):
		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 <= 4e+301))
		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 <= 4e+301)))
		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, 4e+301]], $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 4.00000000000000021e301 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 3.7%

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

   3. Taylor expanded in x around 0 49.6%

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

      1. mul-1-neg 49.6%

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

   5. Simplified 49.6%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 4.00000000000000021e301

   1. Initial program 99.7%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 4 \cdot 10^{+301}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 4 \cdot 10^{+301}\right):\\ \;\;\;\;x \cdot \log \left(y \cdot x\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 4e+301)))
     (- (* x (log (* y x))) 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 <= 4e+301)) {
		tmp = (x * log((y * x))) - 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 <= 4e+301)) {
		tmp = (x * Math.log((y * x))) - 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 <= 4e+301):
		tmp = (x * math.log((y * x))) - 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 <= 4e+301))
		tmp = Float64(Float64(x * log(Float64(y * x))) - 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 <= 4e+301)))
		tmp = (x * log((y * x))) - 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, 4e+301]], $MachinePrecision]], N[(N[(x * N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $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 4.00000000000000021e301 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 3.7%

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

   3. Taylor expanded in x around 0 50.0%

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

      1. log-rec 50.0%

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

      2. sub-neg 50.0%

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

   5. Simplified 50.0%

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

      1. sub-neg 50.0%

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

      2. distribute-rgt-in 49.9%

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

   7. Applied egg-rr 49.9%

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

      1. distribute-rgt-out 50.0%

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

      2. sub-neg 50.0%

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

      3. log-div 3.7%

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

      4. *-commutative 3.7%

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

      5. log-div 50.0%

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

      6. sub-neg 50.0%

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

      7. add-log-exp 50.0%

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

      8. sum-log 1.5%

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

      9. add-sqr-sqrt 0.9%

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

      10. sqrt-unprod 27.6%

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

      11. sqr-neg 27.6%

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

      12. sqrt-unprod 26.7%

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

      13. add-sqr-sqrt 27.3%

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

      14. add-exp-log 50.8%

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

   9. Applied egg-rr 50.8%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 4.00000000000000021e301

   1. Initial program 99.7%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 4 \cdot 10^{+301}\right):\\ \;\;\;\;x \cdot \log \left(y \cdot x\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{+105}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-170}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7.8e+105)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -1.05e-170)
     (- (* x (log (/ x y))) z)
     (if (<= x -1e-309) (- z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.8e+105) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -1.05e-170) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -1e-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 <= (-7.8d+105)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-1.05d-170)) then
        tmp = (x * log((x / y))) - z
    else if (x <= (-1d-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 <= -7.8e+105) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -1.05e-170) {
		tmp = (x * Math.log((x / y))) - z;
	} else if (x <= -1e-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 <= -7.8e+105:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -1.05e-170:
		tmp = (x * math.log((x / y))) - z
	elif x <= -1e-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 <= -7.8e+105)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -1.05e-170)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -1e-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 <= -7.8e+105)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -1.05e-170)
		tmp = (x * log((x / y))) - z;
	elseif (x <= -1e-309)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7.8e+105], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.05e-170], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -1e-309], (-z), N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.79999999999999957e105

   1. Initial program 61.6%

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

   3. Taylor expanded in y around -inf 99.4%

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

      1. metadata-eval 99.4%

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

      2. distribute-neg-frac 99.4%

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

      3. distribute-frac-neg2 99.4%

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

      4. neg-mul-1 99.4%

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

      5. log-rec 99.4%

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

      6. sub-neg 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in z around 0 83.5%

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

   if -7.79999999999999957e105 < x < -1.05e-170

   1. Initial program 94.0%

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

   if -1.05e-170 < x < -1.000000000000002e-309

   1. Initial program 62.2%

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

   3. Taylor expanded in x around 0 92.9%

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

      1. mul-1-neg 92.9%

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

   5. Simplified 92.9%

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

   if -1.000000000000002e-309 < x

   1. Initial program 76.4%

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

   3. Taylor expanded in x around 0 99.6%

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

      1. log-rec 99.6%

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

      2. sub-neg 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{+105}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-170}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-17} \lor \neg \left(x \leq 1.66 \cdot 10^{-71}\right) \land \left(x \leq 2.25 \cdot 10^{-24} \lor \neg \left(x \leq 760000000\right)\right):\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2e-17)
         (and (not (<= x 1.66e-71))
              (or (<= x 2.25e-24) (not (<= x 760000000.0)))))
   (* x (log (/ x y)))
   (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2e-17) || (!(x <= 1.66e-71) && ((x <= 2.25e-24) || !(x <= 760000000.0)))) {
		tmp = x * log((x / y));
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-2d-17)) .or. (.not. (x <= 1.66d-71)) .and. (x <= 2.25d-24) .or. (.not. (x <= 760000000.0d0))) then
        tmp = x * log((x / y))
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2e-17) || (!(x <= 1.66e-71) && ((x <= 2.25e-24) || !(x <= 760000000.0)))) {
		tmp = x * Math.log((x / y));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2e-17) or (not (x <= 1.66e-71) and ((x <= 2.25e-24) or not (x <= 760000000.0))):
		tmp = x * math.log((x / y))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2e-17) || (!(x <= 1.66e-71) && ((x <= 2.25e-24) || !(x <= 760000000.0))))
		tmp = Float64(x * log(Float64(x / y)));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2e-17) || (~((x <= 1.66e-71)) && ((x <= 2.25e-24) || ~((x <= 760000000.0)))))
		tmp = x * log((x / y));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2e-17], And[N[Not[LessEqual[x, 1.66e-71]], $MachinePrecision], Or[LessEqual[x, 2.25e-24], N[Not[LessEqual[x, 760000000.0]], $MachinePrecision]]]], N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -2.00000000000000014e-17 or 1.6599999999999999e-71 < x < 2.2499999999999999e-24 or 7.6e8 < x

   1. Initial program 77.5%

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

   3. Taylor expanded in z around 0 63.4%

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

   if -2.00000000000000014e-17 < x < 1.6599999999999999e-71 or 2.2499999999999999e-24 < x < 7.6e8

   1. Initial program 73.6%

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

   3. Taylor expanded in x around 0 83.6%

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

      1. mul-1-neg 83.6%

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

   5. Simplified 83.6%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-17} \lor \neg \left(x \leq 1.66 \cdot 10^{-71}\right) \land \left(x \leq 2.25 \cdot 10^{-24} \lor \neg \left(x \leq 760000000\right)\right):\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-17}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-73} \lor \neg \left(x \leq 7.5 \cdot 10^{-24}\right) \land x \leq 760000000:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.5e-17)
   (* (- x) (log (/ y x)))
   (if (or (<= x 1.7e-73) (and (not (<= x 7.5e-24)) (<= x 760000000.0)))
     (- z)
     (* x (log (/ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5e-17) {
		tmp = -x * log((y / x));
	} else if ((x <= 1.7e-73) || (!(x <= 7.5e-24) && (x <= 760000000.0))) {
		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 (x <= (-1.5d-17)) then
        tmp = -x * log((y / x))
    else if ((x <= 1.7d-73) .or. (.not. (x <= 7.5d-24)) .and. (x <= 760000000.0d0)) 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 (x <= -1.5e-17) {
		tmp = -x * Math.log((y / x));
	} else if ((x <= 1.7e-73) || (!(x <= 7.5e-24) && (x <= 760000000.0))) {
		tmp = -z;
	} else {
		tmp = x * Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.5e-17:
		tmp = -x * math.log((y / x))
	elif (x <= 1.7e-73) or (not (x <= 7.5e-24) and (x <= 760000000.0)):
		tmp = -z
	else:
		tmp = x * math.log((x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.5e-17)
		tmp = Float64(Float64(-x) * log(Float64(y / x)));
	elseif ((x <= 1.7e-73) || (!(x <= 7.5e-24) && (x <= 760000000.0)))
		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 (x <= -1.5e-17)
		tmp = -x * log((y / x));
	elseif ((x <= 1.7e-73) || (~((x <= 7.5e-24)) && (x <= 760000000.0)))
		tmp = -z;
	else
		tmp = x * log((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.5e-17], N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 1.7e-73], And[N[Not[LessEqual[x, 7.5e-24]], $MachinePrecision], LessEqual[x, 760000000.0]]], (-z), N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.50000000000000003e-17

   1. Initial program 71.7%

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

   3. Taylor expanded in z around 0 57.1%

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

      1. clear-num 57.1%

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

      2. log-div 58.7%

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

      3. metadata-eval 58.7%

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

   5. Applied egg-rr 58.7%

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

      1. neg-sub0 58.7%

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

   7. Simplified 58.7%

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

   if -1.50000000000000003e-17 < x < 1.7000000000000001e-73 or 7.50000000000000007e-24 < x < 7.6e8

   1. Initial program 73.6%

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

   3. Taylor expanded in x around 0 83.6%

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

      1. mul-1-neg 83.6%

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

   5. Simplified 83.6%

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

   if 1.7000000000000001e-73 < x < 7.50000000000000007e-24 or 7.6e8 < x

   1. Initial program 83.7%

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

   3. Taylor expanded in z around 0 70.1%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-17}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-73} \lor \neg \left(x \leq 7.5 \cdot 10^{-24}\right) \land x \leq 760000000:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.9% accurate, 53.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.7%

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

3. Taylor expanded in x around 0 49.1%

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

   1. mul-1-neg 49.1%

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

5. Simplified 49.1%

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

6. Final simplification 49.1%

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

Developer target: 88.1% 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 2024079 
(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))