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

Percentage Accurate: 77.2% → 99.5%

Time: 9.3s

Alternatives: 7

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: 77.2% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-311}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4e-311)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (- (* x (- (log x) (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e-311) {
		tmp = (x * (log(-x) - log(-y))) - z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-4d-311)) then
        tmp = (x * (log(-x) - log(-y))) - z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e-311) {
		tmp = (x * (Math.log(-x) - Math.log(-y))) - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4e-311:
		tmp = (x * (math.log(-x) - math.log(-y))) - z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4e-311)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-y)))) - z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4e-311)
		tmp = (x * (log(-x) - log(-y))) - z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4e-311], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.99999999999979e-311

   1. Initial program 75.0%

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

   3. Taylor expanded in y around -inf 99.7%

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

      1. neg-mul-1 99.7%

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

      2. metadata-eval 99.7%

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

      3. distribute-neg-frac 99.7%

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

      4. distribute-frac-neg2 99.7%

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

      5. log-rec 99.7%

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

      6. sub-neg 99.7%

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

   5. Simplified 99.7%

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

   if -3.99999999999979e-311 < y

   1. Initial program 75.5%

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

   3. Taylor expanded in x around 0 99.4%

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

      1. log-rec 99.4%

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

      2. sub-neg 99.4%

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

   5. Simplified 99.4%

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

Alternative 2: 86.1% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e+269)))
     (- (- 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 <= 5e+269)) {
		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 <= 5e+269)) {
		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 <= 5e+269):
		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 <= 5e+269))
		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 <= 5e+269)))
		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, 5e+269]], $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 5.0000000000000002e269 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 8.0%

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

      1. clear-num 8.0%

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

      2. log-div 11.6%

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

      3. metadata-eval 11.6%

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

   4. Applied egg-rr 11.6%

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

      1. neg-sub0 11.6%

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

   6. Simplified 11.6%

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

      1. add-sqr-sqrt 1.6%

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

      2. sqrt-unprod 4.6%

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

      3. sqr-neg 4.6%

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

      4. neg-log 1.9%

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

      5. clear-num 1.9%

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

      6. neg-log 1.9%

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

      7. clear-num 1.9%

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

      8. diff-log 0.9%

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

      9. diff-log 33.9%

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

      10. sqrt-unprod 1.5%

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

      11. add-sqr-sqrt 31.5%

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

      12. diff-log 1.6%

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

      13. *-un-lft-identity 1.6%

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

      14. log-prod 1.6%

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

      15. metadata-eval 1.6%

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

      16. div-inv 1.6%

          \[\leadsto x \cdot \left(-\left(0 + \log \color{blue}{\left(x \cdot \frac{1}{y}\right)}\right)\right) - z \]

      17. sum-log 31.5%

          \[\leadsto x \cdot \left(-\left(0 + \color{blue}{\left(\log x + \log \left(\frac{1}{y}\right)\right)}\right)\right) - z \]

      18. add-sqr-sqrt 1.5%

          \[\leadsto x \cdot \left(-\left(0 + \left(\log x + \color{blue}{\sqrt{\log \left(\frac{1}{y}\right)} \cdot \sqrt{\log \left(\frac{1}{y}\right)}}\right)\right)\right) - z \]

      19. sqrt-unprod 31.8%

          \[\leadsto x \cdot \left(-\left(0 + \left(\log x + \color{blue}{\sqrt{\log \left(\frac{1}{y}\right) \cdot \log \left(\frac{1}{y}\right)}}\right)\right)\right) - z \]

      20. log-rec 31.8%

          \[\leadsto x \cdot \left(-\left(0 + \left(\log x + \sqrt{\color{blue}{\left(-\log y\right)} \cdot \log \left(\frac{1}{y}\right)}\right)\right)\right) - z \]

      21. log-rec 31.8%

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

      22. sqr-neg 31.8%

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

   8. Applied egg-rr 62.1%

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

      1. +-lft-identity 62.1%

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

      2. *-commutative 62.1%

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

   10. Simplified 62.1%

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

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

   1. Initial program 99.5%

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

4. Final simplification 89.6%

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

Alternative 3: 86.4% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))) (t_1 (* x (log (* y x)))))
   (if (<= t_0 (- INFINITY))
     (- t_1 z)
     (if (<= t_0 5e+269) (- t_0 z) (- (- z) t_1)))))

C

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

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double t_1 = x * Math.log((y * x));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1 - z;
	} else if (t_0 <= 5e+269) {
		tmp = t_0 - z;
	} else {
		tmp = -z - t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	t_1 = x * math.log((y * x))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_1 - z
	elif t_0 <= 5e+269:
		tmp = t_0 - z
	else:
		tmp = -z - t_1
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	t_1 = x * log((y * x));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_1 - z;
	elseif (t_0 <= 5e+269)
		tmp = t_0 - z;
	else
		tmp = -z - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(t$95$1 - z), $MachinePrecision], If[LessEqual[t$95$0, 5e+269], N[(t$95$0 - z), $MachinePrecision], N[((-z) - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 11.3%

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

      1. clear-num 11.3%

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

      2. log-div 14.0%

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

      3. metadata-eval 14.0%

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

   4. Applied egg-rr 14.0%

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

      1. neg-sub0 14.0%

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

   6. Simplified 14.0%

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

      1. clear-num 11.3%

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

      2. log-div 11.3%

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

      3. metadata-eval 11.3%

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

      4. div-inv 11.3%

         \[\leadsto x \cdot \left(-\left(0 - \log \color{blue}{\left(x \cdot \frac{1}{y}\right)}\right)\right) - z \]

      5. sum-log 59.5%

         \[\leadsto x \cdot \left(-\left(0 - \color{blue}{\left(\log x + \log \left(\frac{1}{y}\right)\right)}\right)\right) - z \]

      6. add-sqr-sqrt 0.0%

         \[\leadsto x \cdot \left(-\left(0 - \left(\log x + \color{blue}{\sqrt{\log \left(\frac{1}{y}\right)} \cdot \sqrt{\log \left(\frac{1}{y}\right)}}\right)\right)\right) - z \]

      7. sqrt-unprod 55.0%

         \[\leadsto x \cdot \left(-\left(0 - \left(\log x + \color{blue}{\sqrt{\log \left(\frac{1}{y}\right) \cdot \log \left(\frac{1}{y}\right)}}\right)\right)\right) - z \]

      8. log-rec 55.0%

         \[\leadsto x \cdot \left(-\left(0 - \left(\log x + \sqrt{\color{blue}{\left(-\log y\right)} \cdot \log \left(\frac{1}{y}\right)}\right)\right)\right) - z \]

      9. log-rec 55.0%

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

      10. sqr-neg 55.0%

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

      11. sqrt-unprod 55.0%

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

      12. add-sqr-sqrt 55.0%

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

      13. sum-log 72.2%

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

   8. Applied egg-rr 72.2%

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

      1. neg-sub0 72.2%

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

      2. *-commutative 72.2%

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

   10. Simplified 72.2%

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

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

   1. Initial program 99.5%

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

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

   1. Initial program 4.0%

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

      1. clear-num 4.0%

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

      2. log-div 8.6%

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

      3. metadata-eval 8.6%

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

   4. Applied egg-rr 8.6%

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

      1. neg-sub0 8.6%

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

   6. Simplified 8.6%

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

      1. add-sqr-sqrt 1.8%

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

      2. sqrt-unprod 5.1%

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

      3. sqr-neg 5.1%

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

      4. neg-log 2.0%

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

      5. clear-num 2.0%

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

      6. neg-log 2.0%

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

      7. clear-num 2.0%

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

      8. diff-log 0.2%

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

      9. diff-log 3.3%

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

      10. sqrt-unprod 3.3%

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

      11. add-sqr-sqrt 3.3%

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

      12. diff-log 1.4%

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

      13. *-un-lft-identity 1.4%

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

      14. log-prod 1.4%

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

      15. metadata-eval 1.4%

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

      16. div-inv 1.4%

          \[\leadsto x \cdot \left(-\left(0 + \log \color{blue}{\left(x \cdot \frac{1}{y}\right)}\right)\right) - z \]

      17. sum-log 3.3%

          \[\leadsto x \cdot \left(-\left(0 + \color{blue}{\left(\log x + \log \left(\frac{1}{y}\right)\right)}\right)\right) - z \]

      18. add-sqr-sqrt 3.3%

          \[\leadsto x \cdot \left(-\left(0 + \left(\log x + \color{blue}{\sqrt{\log \left(\frac{1}{y}\right)} \cdot \sqrt{\log \left(\frac{1}{y}\right)}}\right)\right)\right) - z \]

      19. sqrt-unprod 3.3%

          \[\leadsto x \cdot \left(-\left(0 + \left(\log x + \color{blue}{\sqrt{\log \left(\frac{1}{y}\right) \cdot \log \left(\frac{1}{y}\right)}}\right)\right)\right) - z \]

      20. log-rec 3.3%

          \[\leadsto x \cdot \left(-\left(0 + \left(\log x + \sqrt{\color{blue}{\left(-\log y\right)} \cdot \log \left(\frac{1}{y}\right)}\right)\right)\right) - z \]

      21. log-rec 3.3%

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

      22. sqr-neg 3.3%

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

   8. Applied egg-rr 56.4%

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

      1. +-lft-identity 56.4%

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

      2. *-commutative 56.4%

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

   10. Simplified 56.4%

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

4. Final simplification 90.4%

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

Alternative 4: 85.5% accurate, 0.3× speedup

Math

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

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 8.0%

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

   3. Taylor expanded in x around 0 59.1%

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

      1. neg-mul-1 59.1%

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

   5. Simplified 59.1%

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

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

   1. Initial program 99.5%

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

4. Final simplification 88.8%

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

Alternative 5: 93.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+174}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-205}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7.6e+174)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -2.6e-205)
     (- (* (- x) (log (/ y x))) z)
     (if (<= x -1e-308) (- z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.6e+174) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -2.6e-205) {
		tmp = (-x * log((y / x))) - z;
	} else if (x <= -1e-308) {
		tmp = -z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-7.6d+174)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-2.6d-205)) then
        tmp = (-x * log((y / x))) - z
    else if (x <= (-1d-308)) then
        tmp = -z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.6e+174) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -2.6e-205) {
		tmp = (-x * Math.log((y / x))) - z;
	} else if (x <= -1e-308) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -7.6e+174:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -2.6e-205:
		tmp = (-x * math.log((y / x))) - z
	elif x <= -1e-308:
		tmp = -z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7.6e+174)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -2.6e-205)
		tmp = Float64(Float64(Float64(-x) * log(Float64(y / x))) - z);
	elseif (x <= -1e-308)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -7.6e+174)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -2.6e-205)
		tmp = (-x * log((y / x))) - z;
	elseif (x <= -1e-308)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7.6e+174], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.6e-205], N[(N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -1e-308], (-z), N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.6000000000000004e174

   1. Initial program 58.1%

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

   3. Taylor expanded in z around 0 57.4%

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

   4. Taylor expanded in y around -inf 93.4%

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

      1. neg-mul-1 99.1%

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

      2. metadata-eval 99.1%

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

      3. distribute-neg-frac 99.1%

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

      4. distribute-frac-neg2 99.1%

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

      5. log-rec 99.1%

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

      6. sub-neg 99.1%

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

   6. Simplified 93.4%

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

   if -7.6000000000000004e174 < x < -2.5999999999999998e-205

   1. Initial program 84.5%

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

      1. clear-num 84.5%

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

      2. log-div 85.7%

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

      3. metadata-eval 85.7%

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

   4. Applied egg-rr 85.7%

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

      1. neg-sub0 85.7%

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

   6. Simplified 85.7%

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

   if -2.5999999999999998e-205 < x < -9.9999999999999991e-309

   1. Initial program 58.1%

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

   3. Taylor expanded in x around 0 93.3%

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

      1. neg-mul-1 93.3%

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

   5. Simplified 93.3%

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

   if -9.9999999999999991e-309 < x

   1. Initial program 75.5%

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

   3. Taylor expanded in x around 0 99.4%

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

      1. log-rec 99.4%

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

      2. sub-neg 99.4%

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

   5. Simplified 99.4%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+174}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-205}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{-10} \lor \neg \left(z \leq 4.1 \cdot 10^{-73}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4.3e-10) (not (<= z 4.1e-73))) (- z) (* x (log (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.3e-10) || !(z <= 4.1e-73)) {
		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 <= (-4.3d-10)) .or. (.not. (z <= 4.1d-73))) 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 <= -4.3e-10) || !(z <= 4.1e-73)) {
		tmp = -z;
	} else {
		tmp = x * Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -4.3e-10) or not (z <= 4.1e-73):
		tmp = -z
	else:
		tmp = x * math.log((x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -4.3e-10) || !(z <= 4.1e-73))
		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 <= -4.3e-10) || ~((z <= 4.1e-73)))
		tmp = -z;
	else
		tmp = x * log((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -4.3e-10], N[Not[LessEqual[z, 4.1e-73]], $MachinePrecision]], (-z), N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.30000000000000014e-10 or 4.10000000000000016e-73 < z

   1. Initial program 72.0%

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

   3. Taylor expanded in x around 0 79.4%

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

      1. neg-mul-1 79.4%

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

   5. Simplified 79.4%

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

   if -4.30000000000000014e-10 < z < 4.10000000000000016e-73

   1. Initial program 79.0%

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

   3. Taylor expanded in z around 0 62.9%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.3 \cdot 10^{-10} \lor \neg \left(z \leq 4.1 \cdot 10^{-73}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.0% 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 75.2%

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

3. Taylor expanded in x around 0 54.4%

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

   1. neg-mul-1 54.4%

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

5. Simplified 54.4%

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

Developer target: 88.0% 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 2024108 
(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))