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

Percentage Accurate: 77.0% → 99.5%

Time: 11.1s

Alternatives: 12

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: 77.0% 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 75.6%

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

   3. Taylor expanded in y around -inf 99.5%

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

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

   if -4.999999999999985e-310 < y

   1. Initial program 78.5%

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

   3. Taylor expanded in x around 0 99.5%

      \[\leadsto \color{blue}{x \cdot \left(\log x + \log \left(\frac{1}{y}\right)\right)} - z \]
   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. neg-mul-1 99.5%

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

      3. neg-mul-1 99.5%

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

      4. sub-neg 99.5%

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

   5. Simplified 99.5%

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

Alternative 2: 86.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;x \cdot \log \left(x \cdot \frac{1}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY))
     (* x (- (log (- x)) (log (- y))))
     (if (<= t_0 5e+305)
       (- (* x (log (* x (/ 1.0 y)))) z)
       (* x (- (log x) (log y)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = x * (log(-x) - log(-y));
	} else if (t_0 <= 5e+305) {
		tmp = (x * log((x * (1.0 / y)))) - z;
	} else {
		tmp = x * (log(x) - log(y));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (t_0 <= 5e+305) {
		tmp = (x * Math.log((x * (1.0 / y)))) - z;
	} else {
		tmp = x * (Math.log(x) - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = x * (math.log(-x) - math.log(-y))
	elif t_0 <= 5e+305:
		tmp = (x * math.log((x * (1.0 / y)))) - z
	else:
		tmp = x * (math.log(x) - math.log(y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (t_0 <= 5e+305)
		tmp = Float64(Float64(x * log(Float64(x * Float64(1.0 / y)))) - z);
	else
		tmp = Float64(x * Float64(log(x) - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = x * (log(-x) - log(-y));
	elseif (t_0 <= 5e+305)
		tmp = (x * log((x * (1.0 / y)))) - z;
	else
		tmp = x * (log(x) - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+305], N[(N[(x * N[Log[N[(x * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.8%

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

   3. Taylor expanded in z around 0 5.8%

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

   4. Taylor expanded in y around -inf 69.8%

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

      1. metadata-eval 74.5%

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

      2. distribute-neg-frac 74.5%

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

      3. distribute-frac-neg2 74.5%

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

      4. neg-mul-1 74.5%

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

      5. log-rec 74.5%

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

      6. sub-neg 74.5%

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

   6. Simplified 69.8%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 5.00000000000000009e305

   1. Initial program 99.6%

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

      1. clear-num 97.8%

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

      2. associate-/r/ 99.6%

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

   4. Applied egg-rr 99.6%

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

   if 5.00000000000000009e305 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 11.7%

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

   3. Taylor expanded in z around 0 11.7%

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

   4. Taylor expanded in x around 0 56.5%

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

      1. log-rec 64.7%

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

      2. neg-mul-1 64.7%

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

      3. neg-mul-1 64.7%

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

      4. sub-neg 64.7%

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

   6. Simplified 56.5%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;x \cdot \log \left(x \cdot \frac{1}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(-x\right) \cdot \log \left(y \cdot x\right) - z\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;x \cdot \log \left(x \cdot \frac{1}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY))
     (- (* (- x) (log (* y x))) z)
     (if (<= t_0 5e+305)
       (- (* x (log (* x (/ 1.0 y)))) z)
       (* x (- (log x) (log y)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (-x * log((y * x))) - z;
	} else if (t_0 <= 5e+305) {
		tmp = (x * log((x * (1.0 / y)))) - z;
	} else {
		tmp = x * (log(x) - log(y));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (-x * Math.log((y * x))) - z;
	} else if (t_0 <= 5e+305) {
		tmp = (x * Math.log((x * (1.0 / y)))) - z;
	} else {
		tmp = x * (Math.log(x) - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (-x * math.log((y * x))) - z
	elif t_0 <= 5e+305:
		tmp = (x * math.log((x * (1.0 / y)))) - z
	else:
		tmp = x * (math.log(x) - math.log(y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-x) * log(Float64(y * x))) - z);
	elseif (t_0 <= 5e+305)
		tmp = Float64(Float64(x * log(Float64(x * Float64(1.0 / y)))) - z);
	else
		tmp = Float64(x * Float64(log(x) - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (-x * log((y * x))) - z;
	elseif (t_0 <= 5e+305)
		tmp = (x * log((x * (1.0 / y)))) - z;
	else
		tmp = x * (log(x) - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[((-x) * N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[t$95$0, 5e+305], N[(N[(x * N[Log[N[(x * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.8%

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

   3. Taylor expanded in y around -inf 74.5%

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

      1. metadata-eval 74.5%

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

      2. distribute-neg-frac 74.5%

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

      3. distribute-frac-neg2 74.5%

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

      4. neg-mul-1 74.5%

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

      5. log-rec 74.5%

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

      6. sub-neg 74.5%

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

   5. Simplified 74.5%

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

      1. sub-neg 74.5%

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

      2. distribute-rgt-in 74.7%

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

      3. add-sqr-sqrt 74.7%

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

      4. sqrt-unprod 25.0%

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

      5. sqr-neg 25.0%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.6%

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

      10. sqr-neg 0.6%

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

      11. sqrt-unprod 25.0%

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

      12. add-sqr-sqrt 25.0%

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

   7. Applied egg-rr 25.0%

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

      1. +-commutative 25.0%

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

      2. distribute-rgt-out 25.0%

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

      3. neg-sub0 25.0%

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

      4. associate--r- 25.0%

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

      5. log-div 15.2%

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

      6. neg-sub0 15.2%

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

      7. distribute-rgt-neg-out 15.2%

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

      8. distribute-lft-neg-in 15.2%

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

      9. add-sqr-sqrt 0.6%

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

      10. sqrt-unprod 0.7%

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

      11. sqr-neg 0.7%

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

      12. neg-log 0.7%

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

      13. clear-num 0.7%

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

      14. neg-log 0.7%

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

      15. clear-num 0.7%

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

      16. sqrt-unprod 0.1%

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

      17. add-sqr-sqrt 0.6%

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

      18. log-div 20.9%

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

      19. sub-neg 20.9%

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

      20. add-log-exp 20.9%

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

      21. sum-log 0.5%

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

   9. Applied egg-rr 32.5%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 5.00000000000000009e305

   1. Initial program 99.6%

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

      1. clear-num 97.8%

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

      2. associate-/r/ 99.6%

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

   4. Applied egg-rr 99.6%

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

   if 5.00000000000000009e305 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 11.7%

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

   3. Taylor expanded in z around 0 11.7%

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

   4. Taylor expanded in x around 0 56.5%

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

      1. log-rec 64.7%

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

      2. neg-mul-1 64.7%

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

      3. neg-mul-1 64.7%

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

      4. sub-neg 64.7%

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

   6. Simplified 56.5%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;\left(-x\right) \cdot \log \left(y \cdot x\right) - z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;x \cdot \log \left(x \cdot \frac{1}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ t_1 := \log \left(y \cdot x\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(-x\right) \cdot t\_1 - z\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;x \cdot \log \left(x \cdot \frac{1}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_1 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))) (t_1 (log (* y x))))
   (if (<= t_0 (- INFINITY))
     (- (* (- x) t_1) z)
     (if (<= t_0 5e+305) (- (* x (log (* x (/ 1.0 y)))) z) (- (* x t_1) z)))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double t_1 = log((y * x));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (-x * t_1) - z;
	} else if (t_0 <= 5e+305) {
		tmp = (x * log((x * (1.0 / y)))) - z;
	} else {
		tmp = (x * t_1) - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double t_1 = Math.log((y * x));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (-x * t_1) - z;
	} else if (t_0 <= 5e+305) {
		tmp = (x * Math.log((x * (1.0 / y)))) - z;
	} else {
		tmp = (x * t_1) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	t_1 = math.log((y * x))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (-x * t_1) - z
	elif t_0 <= 5e+305:
		tmp = (x * math.log((x * (1.0 / y)))) - z
	else:
		tmp = (x * t_1) - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	t_1 = log(Float64(y * x))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-x) * t_1) - z);
	elseif (t_0 <= 5e+305)
		tmp = Float64(Float64(x * log(Float64(x * Float64(1.0 / y)))) - z);
	else
		tmp = Float64(Float64(x * t_1) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	t_1 = log((y * x));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (-x * t_1) - z;
	elseif (t_0 <= 5e+305)
		tmp = (x * log((x * (1.0 / y)))) - z;
	else
		tmp = (x * t_1) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[((-x) * t$95$1), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[t$95$0, 5e+305], N[(N[(x * N[Log[N[(x * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x * t$95$1), $MachinePrecision] - z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.8%

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

   3. Taylor expanded in y around -inf 74.5%

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

      1. metadata-eval 74.5%

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

      2. distribute-neg-frac 74.5%

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

      3. distribute-frac-neg2 74.5%

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

      4. neg-mul-1 74.5%

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

      5. log-rec 74.5%

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

      6. sub-neg 74.5%

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

   5. Simplified 74.5%

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

      1. sub-neg 74.5%

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

      2. distribute-rgt-in 74.7%

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

      3. add-sqr-sqrt 74.7%

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

      4. sqrt-unprod 25.0%

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

      5. sqr-neg 25.0%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.6%

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

      10. sqr-neg 0.6%

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

      11. sqrt-unprod 25.0%

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

      12. add-sqr-sqrt 25.0%

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

   7. Applied egg-rr 25.0%

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

      1. +-commutative 25.0%

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

      2. distribute-rgt-out 25.0%

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

      3. neg-sub0 25.0%

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

      4. associate--r- 25.0%

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

      5. log-div 15.2%

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

      6. neg-sub0 15.2%

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

      7. distribute-rgt-neg-out 15.2%

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

      8. distribute-lft-neg-in 15.2%

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

      9. add-sqr-sqrt 0.6%

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

      10. sqrt-unprod 0.7%

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

      11. sqr-neg 0.7%

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

      12. neg-log 0.7%

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

      13. clear-num 0.7%

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

      14. neg-log 0.7%

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

      15. clear-num 0.7%

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

      16. sqrt-unprod 0.1%

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

      17. add-sqr-sqrt 0.6%

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

      18. log-div 20.9%

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

      19. sub-neg 20.9%

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

      20. add-log-exp 20.9%

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

      21. sum-log 0.5%

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

   9. Applied egg-rr 32.5%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 5.00000000000000009e305

   1. Initial program 99.6%

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

      1. clear-num 97.8%

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

      2. associate-/r/ 99.6%

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

   4. Applied egg-rr 99.6%

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

   if 5.00000000000000009e305 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 11.7%

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

   3. Taylor expanded in y around -inf 35.0%

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

      1. metadata-eval 35.0%

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

      2. distribute-neg-frac 35.0%

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

      3. distribute-frac-neg2 35.0%

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

      4. neg-mul-1 35.0%

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

      5. log-rec 35.0%

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

      6. sub-neg 35.0%

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

   5. Simplified 35.0%

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

      1. sub-neg 35.0%

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

      2. distribute-rgt-in 35.0%

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

      3. add-sqr-sqrt 35.0%

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

      4. sqrt-unprod 8.1%

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

      5. sqr-neg 8.1%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 24.0%

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

      10. sqr-neg 24.0%

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

      11. sqrt-unprod 62.4%

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

      12. add-sqr-sqrt 62.4%

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

   7. Applied egg-rr 62.4%

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

      1. distribute-rgt-out 64.7%

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

      2. sub-neg 64.7%

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

      3. log-div 11.7%

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

      4. *-commutative 11.7%

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

      5. log-div 64.7%

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

      6. sub-neg 64.7%

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

      7. add-log-exp 64.7%

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

      8. sum-log 11.1%

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

      9. add-sqr-sqrt 6.1%

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

      10. sqrt-unprod 11.1%

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

      11. sqr-neg 11.1%

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

      12. sqrt-unprod 5.0%

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

      13. add-sqr-sqrt 16.7%

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

      14. add-exp-log 49.3%

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

   9. Applied egg-rr 49.3%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;\left(-x\right) \cdot \log \left(y \cdot x\right) - z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;x \cdot \log \left(x \cdot \frac{1}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(y \cdot x\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 5 \cdot 10^{+305}\right):\\ \;\;\;\;x \cdot \log \left(y \cdot x\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e+305)))
     (- (* 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 <= 5e+305)) {
		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 <= 5e+305)) {
		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 <= 5e+305):
		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 <= 5e+305))
		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 <= 5e+305)))
		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, 5e+305]], $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 5.00000000000000009e305 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 9.5%

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

   3. Taylor expanded in y around -inf 49.8%

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

      1. metadata-eval 49.8%

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

      2. distribute-neg-frac 49.8%

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

      3. distribute-frac-neg2 49.8%

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

      4. neg-mul-1 49.8%

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

      5. log-rec 49.8%

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

      6. sub-neg 49.8%

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

   5. Simplified 49.8%

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

      1. sub-neg 49.8%

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

      2. distribute-rgt-in 49.9%

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

      3. add-sqr-sqrt 49.9%

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

      4. sqrt-unprod 14.4%

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

      5. sqr-neg 14.4%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 15.2%

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

      10. sqr-neg 15.2%

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

      11. sqrt-unprod 48.4%

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

      12. add-sqr-sqrt 48.4%

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

   7. Applied egg-rr 48.4%

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

      1. distribute-rgt-out 49.8%

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

      2. sub-neg 49.8%

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

      3. log-div 9.5%

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

      4. *-commutative 9.5%

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

      5. log-div 49.8%

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

      6. sub-neg 49.8%

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

      7. add-log-exp 49.8%

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

      8. sum-log 7.1%

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

      9. add-sqr-sqrt 3.8%

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

      10. sqrt-unprod 14.8%

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

      11. sqr-neg 14.8%

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

      12. sqrt-unprod 11.0%

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

      13. add-sqr-sqrt 18.3%

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

      14. add-exp-log 42.4%

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

   9. Applied egg-rr 42.4%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 5.00000000000000009e305

   1. Initial program 99.6%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+305}\right):\\ \;\;\;\;x \cdot \log \left(y \cdot x\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 5 \cdot 10^{+305}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e+305))) (- 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+305)) {
		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+305)) {
		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+305):
		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+305))
		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+305)))
		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+305]], $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 5.00000000000000009e305 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 9.5%

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

   3. Taylor expanded in x around 0 36.0%

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

      1. neg-mul-1 36.0%

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

   5. Simplified 36.0%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 5.00000000000000009e305

   1. Initial program 99.6%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+305}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ t_1 := \log \left(y \cdot x\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(-x\right) \cdot t\_1 - z\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_1 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))) (t_1 (log (* y x))))
   (if (<= t_0 (- INFINITY))
     (- (* (- x) t_1) z)
     (if (<= t_0 5e+305) (- t_0 z) (- (* x t_1) z)))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double t_1 = log((y * x));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (-x * t_1) - z;
	} else if (t_0 <= 5e+305) {
		tmp = t_0 - z;
	} else {
		tmp = (x * t_1) - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double t_1 = Math.log((y * x));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (-x * t_1) - z;
	} else if (t_0 <= 5e+305) {
		tmp = t_0 - z;
	} else {
		tmp = (x * t_1) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	t_1 = math.log((y * x))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (-x * t_1) - z
	elif t_0 <= 5e+305:
		tmp = t_0 - z
	else:
		tmp = (x * t_1) - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	t_1 = log(Float64(y * x))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-x) * t_1) - z);
	elseif (t_0 <= 5e+305)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(Float64(x * t_1) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	t_1 = log((y * x));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (-x * t_1) - z;
	elseif (t_0 <= 5e+305)
		tmp = t_0 - z;
	else
		tmp = (x * t_1) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[((-x) * t$95$1), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[t$95$0, 5e+305], N[(t$95$0 - z), $MachinePrecision], N[(N[(x * t$95$1), $MachinePrecision] - z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.8%

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

   3. Taylor expanded in y around -inf 74.5%

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

      1. metadata-eval 74.5%

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

      2. distribute-neg-frac 74.5%

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

      3. distribute-frac-neg2 74.5%

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

      4. neg-mul-1 74.5%

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

      5. log-rec 74.5%

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

      6. sub-neg 74.5%

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

   5. Simplified 74.5%

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

      1. sub-neg 74.5%

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

      2. distribute-rgt-in 74.7%

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

      3. add-sqr-sqrt 74.7%

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

      4. sqrt-unprod 25.0%

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

      5. sqr-neg 25.0%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 0.6%

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

      10. sqr-neg 0.6%

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

      11. sqrt-unprod 25.0%

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

      12. add-sqr-sqrt 25.0%

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

   7. Applied egg-rr 25.0%

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

      1. +-commutative 25.0%

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

      2. distribute-rgt-out 25.0%

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

      3. neg-sub0 25.0%

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

      4. associate--r- 25.0%

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

      5. log-div 15.2%

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

      6. neg-sub0 15.2%

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

      7. distribute-rgt-neg-out 15.2%

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

      8. distribute-lft-neg-in 15.2%

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

      9. add-sqr-sqrt 0.6%

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

      10. sqrt-unprod 0.7%

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

      11. sqr-neg 0.7%

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

      12. neg-log 0.7%

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

      13. clear-num 0.7%

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

      14. neg-log 0.7%

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

      15. clear-num 0.7%

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

      16. sqrt-unprod 0.1%

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

      17. add-sqr-sqrt 0.6%

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

      18. log-div 20.9%

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

      19. sub-neg 20.9%

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

      20. add-log-exp 20.9%

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

      21. sum-log 0.5%

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

   9. Applied egg-rr 32.5%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 5.00000000000000009e305

   1. Initial program 99.6%

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

   if 5.00000000000000009e305 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 11.7%

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

   3. Taylor expanded in y around -inf 35.0%

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

      1. metadata-eval 35.0%

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

      2. distribute-neg-frac 35.0%

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

      3. distribute-frac-neg2 35.0%

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

      4. neg-mul-1 35.0%

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

      5. log-rec 35.0%

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

      6. sub-neg 35.0%

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

   5. Simplified 35.0%

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

      1. sub-neg 35.0%

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

      2. distribute-rgt-in 35.0%

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

      3. add-sqr-sqrt 35.0%

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

      4. sqrt-unprod 8.1%

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

      5. sqr-neg 8.1%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 24.0%

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

      10. sqr-neg 24.0%

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

      11. sqrt-unprod 62.4%

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

      12. add-sqr-sqrt 62.4%

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

   7. Applied egg-rr 62.4%

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

      1. distribute-rgt-out 64.7%

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

      2. sub-neg 64.7%

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

      3. log-div 11.7%

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

      4. *-commutative 11.7%

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

      5. log-div 64.7%

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

      6. sub-neg 64.7%

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

      7. add-log-exp 64.7%

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

      8. sum-log 11.1%

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

      9. add-sqr-sqrt 6.1%

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

      10. sqrt-unprod 11.1%

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

      11. sqr-neg 11.1%

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

      12. sqrt-unprod 5.0%

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

      13. add-sqr-sqrt 16.7%

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

      14. add-exp-log 49.3%

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

   9. Applied egg-rr 49.3%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;\left(-x\right) \cdot \log \left(y \cdot x\right) - z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+305}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(y \cdot x\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+142}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-128}:\\ \;\;\;\;x \cdot \log \left(x \cdot \frac{1}{y}\right) - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.8e+142)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -5.8e-128)
     (- (* x (log (* x (/ 1.0 y)))) z)
     (if (<= x -5e-310) (- z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.8e+142) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -5.8e-128) {
		tmp = (x * log((x * (1.0 / y)))) - z;
	} else if (x <= -5e-310) {
		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 <= (-4.8d+142)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-5.8d-128)) then
        tmp = (x * log((x * (1.0d0 / y)))) - z
    else if (x <= (-5d-310)) 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 <= -4.8e+142) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -5.8e-128) {
		tmp = (x * Math.log((x * (1.0 / y)))) - z;
	} else if (x <= -5e-310) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.8e+142:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -5.8e-128:
		tmp = (x * math.log((x * (1.0 / y)))) - z
	elif x <= -5e-310:
		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 <= -4.8e+142)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -5.8e-128)
		tmp = Float64(Float64(x * log(Float64(x * Float64(1.0 / y)))) - z);
	elseif (x <= -5e-310)
		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 <= -4.8e+142)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -5.8e-128)
		tmp = (x * log((x * (1.0 / y)))) - z;
	elseif (x <= -5e-310)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.8e+142], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.8e-128], N[(N[(x * N[Log[N[(x * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -5e-310], (-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 < -4.7999999999999998e142

   1. Initial program 55.1%

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

   3. Taylor expanded in z around 0 54.2%

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

   4. Taylor expanded in y around -inf 97.6%

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

      1. metadata-eval 99.1%

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

      2. distribute-neg-frac 99.1%

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

      3. distribute-frac-neg2 99.1%

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

      4. neg-mul-1 99.1%

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

      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 97.6%

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

   if -4.7999999999999998e142 < x < -5.8000000000000001e-128

   1. Initial program 93.7%

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

      1. clear-num 93.7%

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

      2. associate-/r/ 93.7%

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

   4. Applied egg-rr 93.7%

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

   if -5.8000000000000001e-128 < x < -4.999999999999985e-310

   1. Initial program 58.8%

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

   3. Taylor expanded in x around 0 90.9%

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

      1. neg-mul-1 90.9%

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

   5. Simplified 90.9%

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

   if -4.999999999999985e-310 < x

   1. Initial program 78.5%

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

   3. Taylor expanded in x around 0 99.5%

      \[\leadsto \color{blue}{x \cdot \left(\log x + \log \left(\frac{1}{y}\right)\right)} - z \]
   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. neg-mul-1 99.5%

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

      3. neg-mul-1 99.5%

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

      4. sub-neg 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+142}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-128}:\\ \;\;\;\;x \cdot \log \left(x \cdot \frac{1}{y}\right) - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -49000000000000 \lor \neg \left(z \leq 1.78 \cdot 10^{-155}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\log \left(\frac{y}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -49000000000000.0) (not (<= z 1.78e-155)))
   (- z)
   (* x (- (log (/ y x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -49000000000000.0) || !(z <= 1.78e-155)) {
		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 <= (-49000000000000.0d0)) .or. (.not. (z <= 1.78d-155))) 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 <= -49000000000000.0) || !(z <= 1.78e-155)) {
		tmp = -z;
	} else {
		tmp = x * -Math.log((y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -49000000000000.0) or not (z <= 1.78e-155):
		tmp = -z
	else:
		tmp = x * -math.log((y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -49000000000000.0) || !(z <= 1.78e-155))
		tmp = Float64(-z);
	else
		tmp = Float64(x * Float64(-log(Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -49000000000000.0) || ~((z <= 1.78e-155)))
		tmp = -z;
	else
		tmp = x * -log((y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -49000000000000.0], N[Not[LessEqual[z, 1.78e-155]], $MachinePrecision]], (-z), N[(x * (-N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.9e13 or 1.78e-155 < z

   1. Initial program 76.5%

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

   3. Taylor expanded in x around 0 68.6%

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

      1. neg-mul-1 68.6%

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

   5. Simplified 68.6%

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

   if -4.9e13 < z < 1.78e-155

   1. Initial program 78.0%

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

      1. clear-num 76.3%

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

      2. neg-log 79.8%

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

   4. Applied egg-rr 79.8%

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

   5. Taylor expanded in z around 0 66.4%

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

      1. associate-*r* 66.4%

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

      2. neg-mul-1 66.4%

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

   7. Simplified 66.4%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -49000000000000 \lor \neg \left(z \leq 1.78 \cdot 10^{-155}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\log \left(\frac{y}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -130000000 \lor \neg \left(z \leq 1.78 \cdot 10^{-155}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -130000000.0) (not (<= z 1.78e-155)))
   (- z)
   (* x (log (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -130000000.0) || !(z <= 1.78e-155)) {
		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 <= (-130000000.0d0)) .or. (.not. (z <= 1.78d-155))) 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 <= -130000000.0) || !(z <= 1.78e-155)) {
		tmp = -z;
	} else {
		tmp = x * Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -130000000.0) or not (z <= 1.78e-155):
		tmp = -z
	else:
		tmp = x * math.log((x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -130000000.0) || !(z <= 1.78e-155))
		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 <= -130000000.0) || ~((z <= 1.78e-155)))
		tmp = -z;
	else
		tmp = x * log((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -130000000.0], N[Not[LessEqual[z, 1.78e-155]], $MachinePrecision]], (-z), N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e8 or 1.78e-155 < z

   1. Initial program 76.5%

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

   3. Taylor expanded in x around 0 68.6%

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

      1. neg-mul-1 68.6%

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

   5. Simplified 68.6%

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

   if -1.3e8 < z < 1.78e-155

   1. Initial program 78.0%

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

   3. Taylor expanded in z around 0 63.7%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -130000000 \lor \neg \left(z \leq 1.78 \cdot 10^{-155}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 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 77.1%

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

3. Taylor expanded in x around 0 48.8%

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

   1. neg-mul-1 48.8%

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

5. Simplified 48.8%

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

Alternative 12: 2.3% accurate, 107.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 77.1%

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

3. Taylor expanded in x around 0 48.8%

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

   1. neg-mul-1 48.8%

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

5. Simplified 48.8%

   \[\leadsto \color{blue}{-z} \]
6. Step-by-step derivation

   1. neg-sub0 48.8%

      \[\leadsto \color{blue}{0 - z} \]

   2. sub-neg 48.8%

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

   3. add-sqr-sqrt 23.3%

      \[\leadsto 0 + \color{blue}{\sqrt{-z} \cdot \sqrt{-z}} \]

   4. sqrt-unprod 15.1%

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

   5. sqr-neg 15.1%

      \[\leadsto 0 + \sqrt{\color{blue}{z \cdot z}} \]

   6. sqrt-unprod 1.2%

      \[\leadsto 0 + \color{blue}{\sqrt{z} \cdot \sqrt{z}} \]

   7. add-sqr-sqrt 2.3%

      \[\leadsto 0 + \color{blue}{z} \]

7. Applied egg-rr 2.3%

   \[\leadsto \color{blue}{0 + z} \]
8. Step-by-step derivation

   1. +-lft-identity 2.3%

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

9. Simplified 2.3%

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

Developer Target 1: 88.6% 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 2024139 
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y 7595077799083773/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* x (log (/ x y))) z) (- (* x (- (log x) (log y))) z)))

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