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

Percentage Accurate: 76.6% → 99.5%

Time: 9.5s

Alternatives: 8

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.6% 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 79.3%

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

      1. frac-2neg 79.3%

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

      2. log-div 99.5%

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

   4. Applied egg-rr 99.5%

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

   if -4.999999999999985e-310 < y

   1. Initial program 72.2%

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

      1. log-div 99.5%

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

   4. Applied egg-rr 99.5%

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

4. Final simplification 99.5%

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

Alternative 2: 87.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;x \cdot \log \left(y \cdot x\right) - z\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+297}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

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 2e+297) (- t_0 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 <= 2e+297) {
		tmp = t_0 - 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 <= 2e+297) {
		tmp = t_0 - 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 <= 2e+297:
		tmp = t_0 - 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(x * log(Float64(y * x))) - z);
	elseif (t_0 <= 2e+297)
		tmp = Float64(t_0 - 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 <= 2e+297)
		tmp = t_0 - 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, 2e+297], N[(t$95$0 - 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 5.4%

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

      1. log-div 43.2%

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

      2. add-cube-cbrt 43.2%

         \[\leadsto x \cdot \left(\color{blue}{\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}\right) \cdot \sqrt[3]{\log x}} - \log y\right) - z \]

      3. *-un-lft-identity 43.2%

         \[\leadsto x \cdot \left(\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}\right) \cdot \sqrt[3]{\log x} - \color{blue}{1 \cdot \log y}\right) - z \]

      4. prod-diff 43.2%

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}, \sqrt[3]{\log x}, -\log y \cdot 1\right) + \mathsf{fma}\left(-\log y, 1, \log y \cdot 1\right)\right)} - z \]

   4. Applied egg-rr 43.2%

      \[\leadsto x \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}, \sqrt[3]{\log x}, -\log y \cdot 1\right) + \mathsf{fma}\left(-\log y, 1, \log y \cdot 1\right)\right)} - z \]
   5. Step-by-step derivation

      1. prod-diff 43.2%

         \[\leadsto x \cdot \color{blue}{\left(\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}\right) \cdot \sqrt[3]{\log x} - 1 \cdot \log y\right)} - z \]

      2. *-un-lft-identity 43.2%

         \[\leadsto x \cdot \left(\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}\right) \cdot \sqrt[3]{\log x} - \color{blue}{\log y}\right) - z \]

      3. fma-neg 43.2%

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}, \sqrt[3]{\log x}, -\log y\right)} - z \]

      4. *-rgt-identity 43.2%

         \[\leadsto x \cdot \mathsf{fma}\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}, \sqrt[3]{\log x}, -\color{blue}{\log y \cdot 1}\right) - z \]

      5. fma-undefine 43.2%

         \[\leadsto x \cdot \color{blue}{\left(\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}\right) \cdot \sqrt[3]{\log x} + \left(-\log y \cdot 1\right)\right)} - z \]

      6. add-cube-cbrt 43.2%

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

      7. distribute-lft-in 43.2%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 40.9%

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

   6. Applied egg-rr 40.9%

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

      1. distribute-lft-out 40.9%

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

      2. log-prod 65.5%

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

   8. Simplified 65.5%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 2e297

   1. Initial program 99.8%

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

   if 2e297 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 11.4%

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

      1. remove-double-neg 11.4%

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

      2. sub-neg 11.4%

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

      3. distribute-neg-in 11.4%

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

      4. sub-neg 11.4%

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

      5. distribute-rgt-neg-in 11.4%

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

      6. fma-neg 11.4%

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

      7. log-div 73.0%

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

      8. sub-neg 73.0%

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

      9. distribute-neg-in 73.0%

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

      10. remove-double-neg 73.0%

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

      11. +-commutative 73.0%

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

      12. sub-neg 73.0%

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

      13. log-div 14.7%

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

      14. remove-double-neg 14.7%

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

   3. Simplified 14.7%

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

   5. Taylor expanded in x around inf 56.8%

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

      1. log-rec 56.8%

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

      2. sub-neg 56.8%

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

      3. log-div 11.4%

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

   7. Simplified 11.4%

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

      1. log-div 56.8%

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

   9. Applied egg-rr 56.8%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;x \cdot \log \left(y \cdot x\right) - z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 2 \cdot 10^{+297}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.9% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 1e+301)))
     (- (* x (log (* y x))) z)
     (- t_0 z))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 1e+301)) {
		tmp = (x * log((y * x))) - z;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 1e+301)) {
		tmp = (x * Math.log((y * x))) - z;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 1e+301):
		tmp = (x * math.log((y * x))) - z
	else:
		tmp = t_0 - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 1e+301))
		tmp = Float64(Float64(x * log(Float64(y * x))) - z);
	else
		tmp = Float64(t_0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 1e+301)))
		tmp = (x * log((y * x))) - z;
	else
		tmp = t_0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 1e+301]], $MachinePrecision]], N[(N[(x * N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(t$95$0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 1.00000000000000005e301 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 6.7%

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

      1. log-div 55.9%

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

      2. add-cube-cbrt 55.8%

         \[\leadsto x \cdot \left(\color{blue}{\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}\right) \cdot \sqrt[3]{\log x}} - \log y\right) - z \]

      3. *-un-lft-identity 55.8%

         \[\leadsto x \cdot \left(\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}\right) \cdot \sqrt[3]{\log x} - \color{blue}{1 \cdot \log y}\right) - z \]

      4. prod-diff 55.8%

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}, \sqrt[3]{\log x}, -\log y \cdot 1\right) + \mathsf{fma}\left(-\log y, 1, \log y \cdot 1\right)\right)} - z \]

   4. Applied egg-rr 55.8%

      \[\leadsto x \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}, \sqrt[3]{\log x}, -\log y \cdot 1\right) + \mathsf{fma}\left(-\log y, 1, \log y \cdot 1\right)\right)} - z \]
   5. Step-by-step derivation

      1. prod-diff 55.8%

         \[\leadsto x \cdot \color{blue}{\left(\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}\right) \cdot \sqrt[3]{\log x} - 1 \cdot \log y\right)} - z \]

      2. *-un-lft-identity 55.8%

         \[\leadsto x \cdot \left(\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}\right) \cdot \sqrt[3]{\log x} - \color{blue}{\log y}\right) - z \]

      3. fma-neg 55.8%

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}, \sqrt[3]{\log x}, -\log y\right)} - z \]

      4. *-rgt-identity 55.8%

         \[\leadsto x \cdot \mathsf{fma}\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}, \sqrt[3]{\log x}, -\color{blue}{\log y \cdot 1}\right) - z \]

      5. fma-undefine 55.8%

         \[\leadsto x \cdot \color{blue}{\left(\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}\right) \cdot \sqrt[3]{\log x} + \left(-\log y \cdot 1\right)\right)} - z \]

      6. add-cube-cbrt 55.9%

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

      7. distribute-lft-in 55.9%

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

      8. add-sqr-sqrt 31.7%

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

      9. sqrt-unprod 54.7%

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

   6. Applied egg-rr 32.7%

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

      1. distribute-lft-out 34.2%

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

      2. log-prod 60.0%

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

   8. Simplified 60.0%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1.00000000000000005e301

   1. Initial program 99.8%

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

4. Final simplification 89.6%

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

Alternative 4: 93.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+163}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-139}:\\ \;\;\;\;-\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -8e+163)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -1.2e-139)
     (- (fma 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 <= -8e+163) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -1.2e-139) {
		tmp = -fma(x, log((y / x)), z);
	} else if (x <= -1e-308) {
		tmp = -z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -8e+163)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -1.2e-139)
		tmp = Float64(-fma(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

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -8e+163], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.2e-139], (-N[(x * N[Log[N[(y / x), $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 < -7.9999999999999995e163

   1. Initial program 59.1%

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

      1. remove-double-neg 59.1%

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

      2. sub-neg 59.1%

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

      3. distribute-neg-in 59.1%

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

      4. sub-neg 59.1%

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

      5. distribute-rgt-neg-in 59.1%

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

      6. fma-neg 59.1%

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

      7. log-div 0.0%

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

      8. sub-neg 0.0%

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

      9. distribute-neg-in 0.0%

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

      10. remove-double-neg 0.0%

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

      11. +-commutative 0.0%

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

      12. sub-neg 0.0%

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

      13. log-div 60.9%

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

      14. remove-double-neg 60.9%

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

   3. Simplified 60.9%

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

   5. Taylor expanded in x around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 57.3%

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

   7. Simplified 57.3%

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

      1. frac-2neg 57.3%

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

      2. log-div 87.8%

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

   9. Applied egg-rr 87.8%

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

   if -7.9999999999999995e163 < x < -1.20000000000000007e-139

   1. Initial program 88.7%

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

      1. remove-double-neg 88.7%

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

      2. sub-neg 88.7%

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

      3. distribute-neg-in 88.7%

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

      4. sub-neg 88.7%

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

      5. distribute-rgt-neg-in 88.7%

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

      6. fma-neg 88.7%

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

      7. log-div 0.0%

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

      8. sub-neg 0.0%

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

      9. distribute-neg-in 0.0%

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

      10. remove-double-neg 0.0%

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

      11. +-commutative 0.0%

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

      12. sub-neg 0.0%

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

      13. log-div 91.2%

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

      14. remove-double-neg 91.2%

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

   3. Simplified 91.2%

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

   if -1.20000000000000007e-139 < x < -9.9999999999999991e-309

   1. Initial program 71.7%

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

      1. remove-double-neg 71.7%

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

      2. sub-neg 71.7%

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

      3. distribute-neg-in 71.7%

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

      4. sub-neg 71.7%

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

      5. distribute-rgt-neg-in 71.7%

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

      6. fma-neg 71.7%

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

      7. log-div 0.0%

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

      8. sub-neg 0.0%

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

      9. distribute-neg-in 0.0%

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

      10. remove-double-neg 0.0%

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

      11. +-commutative 0.0%

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

      12. sub-neg 0.0%

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

      13. log-div 71.7%

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

      14. remove-double-neg 71.7%

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

   3. Simplified 71.7%

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

   5. Taylor expanded in x around 0 96.2%

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

   if -9.9999999999999991e-309 < x

   1. Initial program 72.2%

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

      1. log-div 99.5%

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

   4. Applied egg-rr 99.5%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+163}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-139}:\\ \;\;\;\;-\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-139}:\\ \;\;\;\;-\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.5e-139)
   (- (fma 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 <= -1.5e-139) {
		tmp = -fma(x, log((y / x)), z);
	} else if (x <= -1e-308) {
		tmp = -z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.5e-139)
		tmp = Float64(-fma(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

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.5e-139], (-N[(x * N[Log[N[(y / x), $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 3 regimes

2. if x < -1.5e-139

   1. Initial program 81.0%

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

      1. remove-double-neg 81.0%

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

      2. sub-neg 81.0%

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

      3. distribute-neg-in 81.0%

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

      4. sub-neg 81.0%

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

      5. distribute-rgt-neg-in 81.0%

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

      6. fma-neg 81.0%

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

      7. log-div 0.0%

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

      8. sub-neg 0.0%

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

      9. distribute-neg-in 0.0%

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

      10. remove-double-neg 0.0%

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

      11. +-commutative 0.0%

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

      12. sub-neg 0.0%

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

      13. log-div 83.2%

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

      14. remove-double-neg 83.2%

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

   3. Simplified 83.2%

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

   if -1.5e-139 < x < -9.9999999999999991e-309

   1. Initial program 71.7%

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

      1. remove-double-neg 71.7%

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

      2. sub-neg 71.7%

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

      3. distribute-neg-in 71.7%

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

      4. sub-neg 71.7%

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

      5. distribute-rgt-neg-in 71.7%

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

      6. fma-neg 71.7%

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

      7. log-div 0.0%

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

      8. sub-neg 0.0%

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

      9. distribute-neg-in 0.0%

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

      10. remove-double-neg 0.0%

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

      11. +-commutative 0.0%

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

      12. sub-neg 0.0%

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

      13. log-div 71.7%

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

      14. remove-double-neg 71.7%

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

   3. Simplified 71.7%

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

   5. Taylor expanded in x around 0 96.2%

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

   if -9.9999999999999991e-309 < x

   1. Initial program 72.2%

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

      1. log-div 99.5%

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

   4. Applied egg-rr 99.5%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-139}:\\ \;\;\;\;-\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{-53} \lor \neg \left(z \leq 2.1 \cdot 10^{-19}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.85e-53) (not (<= z 2.1e-19))) (- z) (* (- x) (log (/ y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.85e-53) || !(z <= 2.1e-19)) {
		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 <= (-1.85d-53)) .or. (.not. (z <= 2.1d-19))) 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 <= -1.85e-53) || !(z <= 2.1e-19)) {
		tmp = -z;
	} else {
		tmp = -x * Math.log((y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.85e-53) or not (z <= 2.1e-19):
		tmp = -z
	else:
		tmp = -x * math.log((y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.85e-53) || !(z <= 2.1e-19))
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(-x) * log(Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.85e-53) || ~((z <= 2.1e-19)))
		tmp = -z;
	else
		tmp = -x * log((y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.85e-53], N[Not[LessEqual[z, 2.1e-19]], $MachinePrecision]], (-z), N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.84999999999999991e-53 or 2.0999999999999999e-19 < z

   1. Initial program 72.9%

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

      1. remove-double-neg 72.9%

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

      2. sub-neg 72.9%

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

      3. distribute-neg-in 72.9%

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

      4. sub-neg 72.9%

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

      5. distribute-rgt-neg-in 72.9%

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

      6. fma-neg 72.9%

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

      7. log-div 52.0%

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

      8. sub-neg 52.0%

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

      9. distribute-neg-in 52.0%

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

      10. remove-double-neg 52.0%

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

      11. +-commutative 52.0%

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

      12. sub-neg 52.0%

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

      13. log-div 74.1%

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

      14. remove-double-neg 74.1%

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

   3. Simplified 74.1%

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

   5. Taylor expanded in x around 0 74.9%

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

   if -1.84999999999999991e-53 < z < 2.0999999999999999e-19

   1. Initial program 80.4%

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

      1. remove-double-neg 80.4%

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

      2. sub-neg 80.4%

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

      3. distribute-neg-in 80.4%

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

      4. sub-neg 80.4%

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

      5. distribute-rgt-neg-in 80.4%

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

      6. fma-neg 80.4%

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

      7. log-div 43.1%

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

      8. sub-neg 43.1%

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

      9. distribute-neg-in 43.1%

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

      10. remove-double-neg 43.1%

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

      11. +-commutative 43.1%

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

      12. sub-neg 43.1%

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

      13. log-div 80.9%

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

      14. remove-double-neg 80.9%

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

   3. Simplified 80.9%

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

   5. Taylor expanded in x around inf 34.6%

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

      1. log-rec 34.6%

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

      2. sub-neg 34.6%

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

      3. log-div 70.2%

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

   7. Simplified 70.2%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{-53} \lor \neg \left(z \leq 2.1 \cdot 10^{-19}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{-54} \lor \neg \left(z \leq 2.5 \cdot 10^{-25}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4.6e-54) (not (<= z 2.5e-25))) (- z) (* x (log (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.6e-54) || !(z <= 2.5e-25)) {
		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 <= (-4.6d-54)) .or. (.not. (z <= 2.5d-25))) 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 <= -4.6e-54) || !(z <= 2.5e-25)) {
		tmp = -z;
	} else {
		tmp = x * Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -4.6e-54) or not (z <= 2.5e-25):
		tmp = -z
	else:
		tmp = x * math.log((x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -4.6e-54) || !(z <= 2.5e-25))
		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 <= -4.6e-54) || ~((z <= 2.5e-25)))
		tmp = -z;
	else
		tmp = x * log((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -4.6e-54], N[Not[LessEqual[z, 2.5e-25]], $MachinePrecision]], (-z), N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.5999999999999998e-54 or 2.49999999999999981e-25 < z

   1. Initial program 72.9%

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

      1. remove-double-neg 72.9%

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

      2. sub-neg 72.9%

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

      3. distribute-neg-in 72.9%

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

      4. sub-neg 72.9%

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

      5. distribute-rgt-neg-in 72.9%

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

      6. fma-neg 72.9%

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

      7. log-div 52.0%

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

      8. sub-neg 52.0%

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

      9. distribute-neg-in 52.0%

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

      10. remove-double-neg 52.0%

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

      11. +-commutative 52.0%

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

      12. sub-neg 52.0%

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

      13. log-div 74.1%

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

      14. remove-double-neg 74.1%

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

   3. Simplified 74.1%

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

   5. Taylor expanded in x around 0 74.9%

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

   if -4.5999999999999998e-54 < z < 2.49999999999999981e-25

   1. Initial program 80.4%

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

   3. Taylor expanded in z around 0 69.8%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{-54} \lor \neg \left(z \leq 2.5 \cdot 10^{-25}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.1% 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 75.8%

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

   1. remove-double-neg 75.8%

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

   2. sub-neg 75.8%

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

   3. distribute-neg-in 75.8%

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

   4. sub-neg 75.8%

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

   5. distribute-rgt-neg-in 75.8%

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

   6. fma-neg 75.8%

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

   7. log-div 48.6%

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

   8. sub-neg 48.6%

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

   9. distribute-neg-in 48.6%

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

   10. remove-double-neg 48.6%

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

   11. +-commutative 48.6%

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

   12. sub-neg 48.6%

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

   13. log-div 76.8%

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

   14. remove-double-neg 76.8%

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

3. Simplified 76.8%

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

5. Taylor expanded in x around 0 53.0%

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

6. Final simplification 53.0%

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

Developer target: 88.2% 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 2024043 
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64

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

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