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

Percentage Accurate: 77.3% → 99.4%

Time: 8.9s

Alternatives: 11

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: 77.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(-x\right), x, \log \left(-y\right) \cdot \left(-x\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-310)
   (- (fma (log (- x)) x (* (log (- y)) (- x))) z)
   (- (* (- (log x) (log y)) x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-310) {
		tmp = fma(log(-x), x, (log(-y) * -x)) - z;
	} else {
		tmp = ((log(x) - log(y)) * x) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1e-310)
		tmp = Float64(fma(log(Float64(-x)), x, Float64(log(Float64(-y)) * Float64(-x))) - z);
	else
		tmp = Float64(Float64(Float64(log(x) - log(y)) * x) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1e-310], N[(N[(N[Log[(-x)], $MachinePrecision] * x + N[(N[Log[(-y)], $MachinePrecision] * (-x)), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.999999999999969e-311

   1. Initial program 71.9%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-neg.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. distribute-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 99.6

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

   6. Applied rewrites 99.6%

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

   if -9.999999999999969e-311 < y

   1. Initial program 78.7%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 99.6

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

   4. Applied rewrites 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(-x\right), x, \log \left(-y\right) \cdot \left(-x\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right) \cdot x\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t\_0 \leq 10^{+293}:\\ \;\;\;\;\log \left(\frac{--1}{y} \cdot x\right) \cdot x - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log (/ x y)) x)))
   (if (<= t_0 (- INFINITY))
     (- z)
     (if (<= t_0 1e+293) (- (* (log (* (/ (- -1.0) y) x)) x) z) (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y)) * x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_0 <= 1e+293) {
		tmp = (log(((-(-1.0) / y) * x)) * x) - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.log((x / y)) * x;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = -z;
	} else if (t_0 <= 1e+293) {
		tmp = (Math.log(((-(-1.0) / y) * x)) * x) - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.log((x / y)) * x
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -z
	elif t_0 <= 1e+293:
		tmp = (math.log(((-(-1.0) / y) * x)) * x) - z
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(log(Float64(x / y)) * x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(-z);
	elseif (t_0 <= 1e+293)
		tmp = Float64(Float64(log(Float64(Float64(Float64(-(-1.0)) / y) * x)) * x) - z);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = log((x / y)) * x;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = -z;
	elseif (t_0 <= 1e+293)
		tmp = (log(((-(-1.0) / y) * x)) * x) - z;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], (-z), If[LessEqual[t$95$0, 1e+293], N[(N[(N[Log[N[(N[((--1.0) / y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] - z), $MachinePrecision], (-z)]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 9.9999999999999992e292 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 3.9%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. lower-neg.f64 66.7

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

   5. Applied rewrites 66.7%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 9.9999999999999992e292

   1. Initial program 99.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 99.4

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

   4. Applied rewrites 99.4%

      \[\leadsto x \cdot \log \color{blue}{\left(\frac{-1}{y} \cdot \left(-x\right)\right)} - z \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(\frac{x}{y}\right) \cdot x \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;\log \left(\frac{x}{y}\right) \cdot x \leq 10^{+293}:\\ \;\;\;\;\log \left(\frac{--1}{y} \cdot x\right) \cdot x - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right) \cdot x\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t\_0 \leq 10^{+293}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log (/ x y)) x)))
   (if (<= t_0 (- INFINITY)) (- z) (if (<= t_0 1e+293) (- t_0 z) (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y)) * x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_0 <= 1e+293) {
		tmp = t_0 - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.log((x / y)) * x;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = -z;
	} else if (t_0 <= 1e+293) {
		tmp = t_0 - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.log((x / y)) * x
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -z
	elif t_0 <= 1e+293:
		tmp = t_0 - z
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(log(Float64(x / y)) * x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(-z);
	elseif (t_0 <= 1e+293)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = log((x / y)) * x;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = -z;
	elseif (t_0 <= 1e+293)
		tmp = t_0 - z;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], (-z), If[LessEqual[t$95$0, 1e+293], N[(t$95$0 - z), $MachinePrecision], (-z)]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 9.9999999999999992e292 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 3.9%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. lower-neg.f64 66.7

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

   5. Applied rewrites 66.7%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 9.9999999999999992e292

   1. Initial program 99.4%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(\frac{x}{y}\right) \cdot x \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;\log \left(\frac{x}{y}\right) \cdot x \leq 10^{+293}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right) \cdot x\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t\_0 \leq 10^{+293}:\\ \;\;\;\;-\mathsf{fma}\left(\log \left(\frac{y}{x}\right), x, z\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log (/ x y)) x)))
   (if (<= t_0 (- INFINITY))
     (- z)
     (if (<= t_0 1e+293) (- (fma (log (/ y x)) x z)) (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y)) * x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_0 <= 1e+293) {
		tmp = -fma(log((y / x)), x, z);
	} else {
		tmp = -z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(log(Float64(x / y)) * x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(-z);
	elseif (t_0 <= 1e+293)
		tmp = Float64(-fma(log(Float64(y / x)), x, z));
	else
		tmp = Float64(-z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], (-z), If[LessEqual[t$95$0, 1e+293], (-N[(N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision] * x + z), $MachinePrecision]), (-z)]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 9.9999999999999992e292 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 3.9%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. lower-neg.f64 66.7

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

   5. Applied rewrites 66.7%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 9.9999999999999992e292

   1. Initial program 99.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. log-pow-rev N/A

         \[\leadsto \color{blue}{\log \left({\left(\frac{x}{y}\right)}^{x}\right)} + \left(\mathsf{neg}\left(z\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \log \left({\color{blue}{\left(\frac{x}{y}\right)}}^{x}\right) + \left(\mathsf{neg}\left(z\right)\right) \]

      7. clear-num N/A

         \[\leadsto \log \left({\color{blue}{\left(\frac{1}{\frac{y}{x}}\right)}}^{x}\right) + \left(\mathsf{neg}\left(z\right)\right) \]

      8. inv-pow N/A

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

      9. pow-pow N/A

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

      10. neg-mul-1 N/A

          \[\leadsto \log \left({\left(\frac{y}{x}\right)}^{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}\right) + \left(\mathsf{neg}\left(z\right)\right) \]

      11. log-pow N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 98.2

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

   4. Applied rewrites 98.2%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. cancel-sub-sign N/A

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

      5. *-commutative N/A

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

      6. lift-log.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. lift-/.f64 N/A

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

      10. log-rec N/A

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

      11. lift-log.f64 N/A

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

      12. cancel-sign-sub-inv N/A

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

      13. +-commutative N/A

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

      14. lift-fma.f64 99.4

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

      15. remove-double-div N/A

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

      16. unpow-1 N/A

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

      17. lift-pow.f64 N/A

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

      18. frac-2neg N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left({\left(\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, -z\right)\right)}^{-1}\right)}} \]

      19. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left({\left(\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, -z\right)\right)}^{-1}\right)} \]

      20. distribute-frac-neg2 N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{-1}{{\left(\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, -z\right)\right)}^{-1}}\right)} \]

   6. Applied rewrites 98.2%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(\frac{x}{y}\right) \cdot x \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;\log \left(\frac{x}{y}\right) \cdot x \leq 10^{+293}:\\ \;\;\;\;-\mathsf{fma}\left(\log \left(\frac{y}{x}\right), x, z\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+176}:\\ \;\;\;\;\left(\log \left(-x\right) - \log \left(-y\right)\right) \cdot x\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-125}:\\ \;\;\;\;\frac{\log \left(\frac{y}{x}\right)}{\frac{-1}{x}} - z\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.8e+176)
   (* (- (log (- x)) (log (- y))) x)
   (if (<= x -5.8e-125)
     (- (/ (log (/ y x)) (/ -1.0 x)) z)
     (if (<= x -1e-309) (- z) (- (* (- (log x) (log y)) x) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.8e+176) {
		tmp = (log(-x) - log(-y)) * x;
	} else if (x <= -5.8e-125) {
		tmp = (log((y / x)) / (-1.0 / x)) - z;
	} else if (x <= -1e-309) {
		tmp = -z;
	} else {
		tmp = ((log(x) - log(y)) * x) - 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+176)) then
        tmp = (log(-x) - log(-y)) * x
    else if (x <= (-5.8d-125)) then
        tmp = (log((y / x)) / ((-1.0d0) / x)) - z
    else if (x <= (-1d-309)) then
        tmp = -z
    else
        tmp = ((log(x) - log(y)) * x) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.8e+176) {
		tmp = (Math.log(-x) - Math.log(-y)) * x;
	} else if (x <= -5.8e-125) {
		tmp = (Math.log((y / x)) / (-1.0 / x)) - z;
	} else if (x <= -1e-309) {
		tmp = -z;
	} else {
		tmp = ((Math.log(x) - Math.log(y)) * x) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.8e+176:
		tmp = (math.log(-x) - math.log(-y)) * x
	elif x <= -5.8e-125:
		tmp = (math.log((y / x)) / (-1.0 / x)) - z
	elif x <= -1e-309:
		tmp = -z
	else:
		tmp = ((math.log(x) - math.log(y)) * x) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.8e+176)
		tmp = Float64(Float64(log(Float64(-x)) - log(Float64(-y))) * x);
	elseif (x <= -5.8e-125)
		tmp = Float64(Float64(log(Float64(y / x)) / Float64(-1.0 / x)) - z);
	elseif (x <= -1e-309)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(Float64(log(x) - log(y)) * x) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -4.8e+176)
		tmp = (log(-x) - log(-y)) * x;
	elseif (x <= -5.8e-125)
		tmp = (log((y / x)) / (-1.0 / x)) - z;
	elseif (x <= -1e-309)
		tmp = -z;
	else
		tmp = ((log(x) - log(y)) * x) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.8e+176], N[(N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, -5.8e-125], N[(N[(N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision] / N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -1e-309], (-z), N[(N[(N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.8000000000000003e176

   1. Initial program 71.2%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-/.f64 66.1

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

   5. Applied rewrites 66.1%

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

   7. Applied rewrites 90.5%

      \[\leadsto \left(\log \left(-x\right) - \log \left(-y\right)\right) \cdot x \]

   if -4.8000000000000003e176 < x < -5.8000000000000004e-125

   1. Initial program 90.7%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-neg.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. diff-log N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. /-rgt-identity N/A

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

      13. clear-num N/A

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

      14. lift-/.f64 N/A

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

      15. div-inv N/A

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

      16. frac-2neg N/A

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

      17. lift-log.f64 N/A

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

      18. neg-log N/A

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

      19. lift-/.f64 N/A

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

      20. clear-num N/A

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

      21. lift-/.f64 N/A

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

      22. lift-log.f64 N/A

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

      23. lower-/.f64 N/A

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

   6. Applied rewrites 95.7%

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

   if -5.8000000000000004e-125 < x < -1.000000000000002e-309

   1. Initial program 50.2%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. lower-neg.f64 90.9

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

   5. Applied rewrites 90.9%

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

   if -1.000000000000002e-309 < x

   1. Initial program 78.7%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 99.6

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

   4. Applied rewrites 99.6%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+176}:\\ \;\;\;\;\left(\log \left(-x\right) - \log \left(-y\right)\right) \cdot x\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-125}:\\ \;\;\;\;\frac{\log \left(\frac{y}{x}\right)}{\frac{-1}{x}} - z\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-125}:\\ \;\;\;\;\frac{\log \left(\frac{y}{x}\right)}{\frac{-1}{x}} - z\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.8e-125)
   (- (/ (log (/ y x)) (/ -1.0 x)) z)
   (if (<= x -1e-309) (- z) (- (* (- (log x) (log y)) x) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.8e-125) {
		tmp = (log((y / x)) / (-1.0 / x)) - z;
	} else if (x <= -1e-309) {
		tmp = -z;
	} else {
		tmp = ((log(x) - log(y)) * x) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-5.8d-125)) then
        tmp = (log((y / x)) / ((-1.0d0) / x)) - z
    else if (x <= (-1d-309)) then
        tmp = -z
    else
        tmp = ((log(x) - log(y)) * x) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.8e-125) {
		tmp = (Math.log((y / x)) / (-1.0 / x)) - z;
	} else if (x <= -1e-309) {
		tmp = -z;
	} else {
		tmp = ((Math.log(x) - Math.log(y)) * x) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.8e-125:
		tmp = (math.log((y / x)) / (-1.0 / x)) - z
	elif x <= -1e-309:
		tmp = -z
	else:
		tmp = ((math.log(x) - math.log(y)) * x) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.8e-125)
		tmp = Float64(Float64(log(Float64(y / x)) / Float64(-1.0 / x)) - z);
	elseif (x <= -1e-309)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(Float64(log(x) - log(y)) * x) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5.8e-125)
		tmp = (log((y / x)) / (-1.0 / x)) - z;
	elseif (x <= -1e-309)
		tmp = -z;
	else
		tmp = ((log(x) - log(y)) * x) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.8e-125], N[(N[(N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision] / N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -1e-309], (-z), N[(N[(N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.8000000000000004e-125

   1. Initial program 83.4%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-neg.f64 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. diff-log N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. /-rgt-identity N/A

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

      13. clear-num N/A

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

      14. lift-/.f64 N/A

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

      15. div-inv N/A

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

      16. frac-2neg N/A

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

      17. lift-log.f64 N/A

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

      18. neg-log N/A

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

      19. lift-/.f64 N/A

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

      20. clear-num N/A

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

      21. lift-/.f64 N/A

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

      22. lift-log.f64 N/A

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

      23. lower-/.f64 N/A

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

   6. Applied rewrites 86.5%

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

   if -5.8000000000000004e-125 < x < -1.000000000000002e-309

   1. Initial program 50.2%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. lower-neg.f64 90.9

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

   5. Applied rewrites 90.9%

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

   if -1.000000000000002e-309 < x

   1. Initial program 78.7%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 99.6

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

   4. Applied rewrites 99.6%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-125}:\\ \;\;\;\;\frac{\log \left(\frac{y}{x}\right)}{\frac{-1}{x}} - z\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1e-309)
   (- (* (- (log (- x)) (log (- y))) x) z)
   (- (* (- (log x) (log y)) x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1e-309) {
		tmp = ((log(-x) - log(-y)) * x) - z;
	} else {
		tmp = ((log(x) - log(y)) * x) - 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 <= (-1d-309)) then
        tmp = ((log(-x) - log(-y)) * x) - z
    else
        tmp = ((log(x) - log(y)) * x) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1e-309) {
		tmp = ((Math.log(-x) - Math.log(-y)) * x) - z;
	} else {
		tmp = ((Math.log(x) - Math.log(y)) * x) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1e-309:
		tmp = ((math.log(-x) - math.log(-y)) * x) - z
	else:
		tmp = ((math.log(x) - math.log(y)) * x) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1e-309)
		tmp = Float64(Float64(Float64(log(Float64(-x)) - log(Float64(-y))) * x) - z);
	else
		tmp = Float64(Float64(Float64(log(x) - log(y)) * x) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1e-309)
		tmp = ((log(-x) - log(-y)) * x) - z;
	else
		tmp = ((log(x) - log(y)) * x) - z;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.000000000000002e-309

   1. Initial program 71.9%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-neg.f64 99.5

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

   4. Applied rewrites 99.5%

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

   if -1.000000000000002e-309 < x

   1. Initial program 78.7%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 99.6

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

   4. Applied rewrites 99.6%

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

4. Final simplification 99.5%

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

Alternative 8: 64.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{y}{x}\right) \cdot \left(-x\right)\\ \mathbf{if}\;x \leq -2.45 \cdot 10^{+70}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+22}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log (/ y x)) (- x))))
   (if (<= x -2.45e+70) t_0 (if (<= x 2.2e+22) (- z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = log((y / x)) * -x;
	double tmp;
	if (x <= -2.45e+70) {
		tmp = t_0;
	} else if (x <= 2.2e+22) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = log((y / x)) * -x
    if (x <= (-2.45d+70)) then
        tmp = t_0
    else if (x <= 2.2d+22) then
        tmp = -z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.log((y / x)) * -x;
	double tmp;
	if (x <= -2.45e+70) {
		tmp = t_0;
	} else if (x <= 2.2e+22) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.log((y / x)) * -x
	tmp = 0
	if x <= -2.45e+70:
		tmp = t_0
	elif x <= 2.2e+22:
		tmp = -z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(log(Float64(y / x)) * Float64(-x))
	tmp = 0.0
	if (x <= -2.45e+70)
		tmp = t_0;
	elseif (x <= 2.2e+22)
		tmp = Float64(-z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = log((y / x)) * -x;
	tmp = 0.0;
	if (x <= -2.45e+70)
		tmp = t_0;
	elseif (x <= 2.2e+22)
		tmp = -z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision] * (-x)), $MachinePrecision]}, If[LessEqual[x, -2.45e+70], t$95$0, If[LessEqual[x, 2.2e+22], (-z), t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.45000000000000014e70 or 2.2e22 < x

   1. Initial program 78.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. log-pow-rev N/A

         \[\leadsto \color{blue}{\log \left({\left(\frac{x}{y}\right)}^{x}\right)} + \left(\mathsf{neg}\left(z\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \log \left({\color{blue}{\left(\frac{x}{y}\right)}}^{x}\right) + \left(\mathsf{neg}\left(z\right)\right) \]

      7. clear-num N/A

         \[\leadsto \log \left({\color{blue}{\left(\frac{1}{\frac{y}{x}}\right)}}^{x}\right) + \left(\mathsf{neg}\left(z\right)\right) \]

      8. inv-pow N/A

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

      9. pow-pow N/A

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

      10. neg-mul-1 N/A

          \[\leadsto \log \left({\left(\frac{y}{x}\right)}^{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}\right) + \left(\mathsf{neg}\left(z\right)\right) \]

      11. log-pow N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 80.4

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

   4. Applied rewrites 80.4%

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

   5. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 64.8

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

   7. Applied rewrites 64.8%

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

   if -2.45000000000000014e70 < x < 2.2e22

   1. Initial program 73.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. lower-neg.f64 78.4

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

   5. Applied rewrites 78.4%

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

Alternative 9: 64.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right) \cdot x\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{+75}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+22}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log (/ x y)) x)))
   (if (<= x -2.7e+75) t_0 (if (<= x 2.2e+22) (- z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y)) * x;
	double tmp;
	if (x <= -2.7e+75) {
		tmp = t_0;
	} else if (x <= 2.2e+22) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = log((x / y)) * x
    if (x <= (-2.7d+75)) then
        tmp = t_0
    else if (x <= 2.2d+22) then
        tmp = -z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.log((x / y)) * x;
	double tmp;
	if (x <= -2.7e+75) {
		tmp = t_0;
	} else if (x <= 2.2e+22) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.log((x / y)) * x
	tmp = 0
	if x <= -2.7e+75:
		tmp = t_0
	elif x <= 2.2e+22:
		tmp = -z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(log(Float64(x / y)) * x)
	tmp = 0.0
	if (x <= -2.7e+75)
		tmp = t_0;
	elseif (x <= 2.2e+22)
		tmp = Float64(-z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = log((x / y)) * x;
	tmp = 0.0;
	if (x <= -2.7e+75)
		tmp = t_0;
	elseif (x <= 2.2e+22)
		tmp = -z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.7e+75], t$95$0, If[LessEqual[x, 2.2e+22], (-z), t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.69999999999999998e75 or 2.2e22 < x

   1. Initial program 78.4%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-/.f64 63.8

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

   5. Applied rewrites 63.8%

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

   if -2.69999999999999998e75 < x < 2.2e22

   1. Initial program 73.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. lower-neg.f64 78.4

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

   5. Applied rewrites 78.4%

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

Alternative 10: 49.1% accurate, 40.0× 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 Float64(-z)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.2%

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

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

   2. lower-neg.f64 54.0

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

5. Applied rewrites 54.0%

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

Alternative 11: 2.3% accurate, 120.0× 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 75.2%

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

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

   2. lower-neg.f64 54.0

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

5. Applied rewrites 54.0%

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

7. Applied rewrites 2.3%

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

Developer Target 1: 88.3% accurate, 0.6× 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 2024298 
(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))