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

Percentage Accurate: 76.5% → 99.4%

Time: 8.4s

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: 76.5% 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

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-310)
   (- (fma (log (- x)) x (* (log (- y)) (- x))) z)
   (fma (- (log x) (log y)) x (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = fma(log(-x), x, (log(-y) * -x)) - z;
	} else {
		tmp = fma((log(x) - log(y)), x, -z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-310)
		tmp = Float64(fma(log(Float64(-x)), x, Float64(log(Float64(-y)) * Float64(-x))) - z);
	else
		tmp = fma(Float64(log(x) - log(y)), x, Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5e-310], N[(N[(N[Log[(-x)], $MachinePrecision] * x + N[(N[Log[(-y)], $MachinePrecision] * (-x)), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x + (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 76.6%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 99.5

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

   6. Applied rewrites 99.5%

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

   if -4.999999999999985e-310 < y

   1. Initial program 84.6%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 84.6

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

   4. Applied rewrites 84.6%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 99.6

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

   6. Applied rewrites 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 85.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+303} \lor \neg \left(t\_0 \leq 10^{+303}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\log \left(\frac{y}{x}\right), x, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* x (log (/ x y))) z)))
   (if (or (<= t_0 -1e+303) (not (<= t_0 1e+303)))
     (- z)
     (- (fma (log (/ y x)) x z)))))

C

double code(double x, double y, double z) {
	double t_0 = (x * log((x / y))) - z;
	double tmp;
	if ((t_0 <= -1e+303) || !(t_0 <= 1e+303)) {
		tmp = -z;
	} else {
		tmp = -fma(log((y / x)), x, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * log(Float64(x / y))) - z)
	tmp = 0.0
	if ((t_0 <= -1e+303) || !(t_0 <= 1e+303))
		tmp = Float64(-z);
	else
		tmp = Float64(-fma(log(Float64(y / x)), x, z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e+303], N[Not[LessEqual[t$95$0, 1e+303]], $MachinePrecision]], (-z), (-N[(N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision] * x + z), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x (log.f64 (/.f64 x y))) z) < -1e303 or 1e303 < (-.f64 (*.f64 x (log.f64 (/.f64 x y))) z)

   1. Initial program 12.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 45.9

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

   5. Applied rewrites 45.9%

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

   if -1e303 < (-.f64 (*.f64 x (log.f64 (/.f64 x y))) z) < 1e303

   1. Initial program 99.5%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. clear-num N/A

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

      4. log-div N/A

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

      5. flip-- N/A

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

      6. log-prod N/A

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

      7. metadata-eval N/A

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

      8. associate-/r/ N/A

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

      9. remove-double-div N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 97.8%

      \[\leadsto x \cdot \color{blue}{\frac{0 - {\log \left(\frac{y}{x}\right)}^{2}}{\log \left(\frac{y}{x}\right)}} - z \]

   5. Taylor expanded in z around 0

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

      1. distribute-lft-out N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-/.f64 97.8

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

   7. Applied rewrites 97.8%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) - z \leq -1 \cdot 10^{+303} \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) - z \leq 10^{+303}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\log \left(\frac{y}{x}\right), x, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right)\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 10^{+303}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, x, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (/ x y))) (t_1 (* x t_0)))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 1e+303)))
     (- z)
     (fma t_0 x (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y));
	double t_1 = x * t_0;
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 1e+303)) {
		tmp = -z;
	} else {
		tmp = fma(t_0, x, -z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = log(Float64(x / y))
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 1e+303))
		tmp = Float64(-z);
	else
		tmp = fma(t_0, x, Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 1e+303]], $MachinePrecision]], (-z), N[(t$95$0 * x + (-z)), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 1e303 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 9.2%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 45.7

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

   5. Applied rewrites 45.7%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1e303

   1. Initial program 99.5%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 99.6

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

   4. Applied rewrites 99.6%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 10^{+303}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, -z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 10^{+303}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 1e+303))) (- 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+303)) {
		tmp = -z;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 1e+303)) {
		tmp = -z;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 1e+303):
		tmp = -z
	else:
		tmp = t_0 - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 1e+303))
		tmp = Float64(-z);
	else
		tmp = Float64(t_0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 1e+303)))
		tmp = -z;
	else
		tmp = t_0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 1e+303]], $MachinePrecision]], (-z), N[(t$95$0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 1e303 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 9.2%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 45.7

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

   5. Applied rewrites 45.7%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1e303

   1. Initial program 99.5%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 10^{+303}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{+155}:\\ \;\;\;\;\left(\log \left(-x\right) - \log \left(-y\right)\right) \cdot x\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-157}:\\ \;\;\;\;\frac{x}{{\log \left(\frac{x}{y}\right)}^{-1}} - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x - \log y, x, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.7e+155)
   (* (- (log (- x)) (log (- y))) x)
   (if (<= x -7.5e-157)
     (- (/ x (pow (log (/ x y)) -1.0)) z)
     (if (<= x -2e-308) (- z) (fma (- (log x) (log y)) x (- z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.7e+155) {
		tmp = (log(-x) - log(-y)) * x;
	} else if (x <= -7.5e-157) {
		tmp = (x / pow(log((x / y)), -1.0)) - z;
	} else if (x <= -2e-308) {
		tmp = -z;
	} else {
		tmp = fma((log(x) - log(y)), x, -z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.7e+155)
		tmp = Float64(Float64(log(Float64(-x)) - log(Float64(-y))) * x);
	elseif (x <= -7.5e-157)
		tmp = Float64(Float64(x / (log(Float64(x / y)) ^ -1.0)) - z);
	elseif (x <= -2e-308)
		tmp = Float64(-z);
	else
		tmp = fma(Float64(log(x) - log(y)), x, Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.7e+155], N[(N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, -7.5e-157], N[(N[(x / N[Power[N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -2e-308], (-z), N[(N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x + (-z)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.6999999999999996e155

   1. Initial program 56.5%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-neg.f64 83.4

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

   7. Applied rewrites 83.4%

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

   if -5.6999999999999996e155 < x < -7.500000000000001e-157

   1. Initial program 93.2%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. lift-log.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. diff-log N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. frac-2neg N/A

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

      9. clear-num N/A

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

      10. lift-/.f64 N/A

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

      11. neg-log N/A

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

      12. lift-log.f64 N/A

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

      13. unpow1 N/A

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

      14. metadata-eval N/A

          \[\leadsto x \cdot \left(\mathsf{neg}\left({\log \left(\frac{y}{x}\right)}^{\color{blue}{\left(2 + -1\right)}}\right)\right) - z \]

      15. pow-prod-up N/A

          \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{{\log \left(\frac{y}{x}\right)}^{2} \cdot {\log \left(\frac{y}{x}\right)}^{-1}}\right)\right) - z \]

      16. lift-pow.f64 N/A

          \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{{\log \left(\frac{y}{x}\right)}^{2}} \cdot {\log \left(\frac{y}{x}\right)}^{-1}\right)\right) - z \]

      17. inv-pow N/A

          \[\leadsto x \cdot \left(\mathsf{neg}\left({\log \left(\frac{y}{x}\right)}^{2} \cdot \color{blue}{\frac{1}{\log \left(\frac{y}{x}\right)}}\right)\right) - z \]

      18. div-inv N/A

          \[\leadsto x \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{{\log \left(\frac{y}{x}\right)}^{2}}{\log \left(\frac{y}{x}\right)}}\right)\right) - z \]

      19. distribute-frac-neg N/A

          \[\leadsto x \cdot \color{blue}{\frac{\mathsf{neg}\left({\log \left(\frac{y}{x}\right)}^{2}\right)}{\log \left(\frac{y}{x}\right)}} - z \]

      20. sub0-neg N/A

          \[\leadsto x \cdot \frac{\color{blue}{0 - {\log \left(\frac{y}{x}\right)}^{2}}}{\log \left(\frac{y}{x}\right)} - z \]

      21. lift--.f64 N/A

          \[\leadsto x \cdot \frac{\color{blue}{0 - {\log \left(\frac{y}{x}\right)}^{2}}}{\log \left(\frac{y}{x}\right)} - z \]

   6. Applied rewrites 93.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. diff-log N/A

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

      9. lift-log.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower-/.f64 99.5

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

      13. lift--.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift-log.f64 N/A

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

      16. diff-log N/A

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

      17. lift-neg.f64 N/A

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

      18. lift-neg.f64 N/A

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

      19. frac-2neg N/A

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

      20. lift-/.f64 N/A

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

      21. lift-log.f64 93.2

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

   8. Applied rewrites 93.2%

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

   if -7.500000000000001e-157 < x < -1.9999999999999998e-308

   1. Initial program 63.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 93.1

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

   5. Applied rewrites 93.1%

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

   if -1.9999999999999998e-308 < x

   1. Initial program 84.6%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 84.6

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

   4. Applied rewrites 84.6%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 99.6

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

   6. Applied rewrites 99.6%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{+155}:\\ \;\;\;\;\left(\log \left(-x\right) - \log \left(-y\right)\right) \cdot x\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-157}:\\ \;\;\;\;\frac{x}{{\log \left(\frac{x}{y}\right)}^{-1}} - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x - \log y, x, -z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-156}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, -z\right)\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x - \log y, x, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -6.8e-156)
   (fma (log (/ x y)) x (- z))
   (if (<= x -2e-308) (- z) (fma (- (log x) (log y)) x (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.8e-156) {
		tmp = fma(log((x / y)), x, -z);
	} else if (x <= -2e-308) {
		tmp = -z;
	} else {
		tmp = fma((log(x) - log(y)), x, -z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -6.8e-156)
		tmp = fma(log(Float64(x / y)), x, Float64(-z));
	elseif (x <= -2e-308)
		tmp = Float64(-z);
	else
		tmp = fma(Float64(log(x) - log(y)), x, Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -6.8e-156], N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x + (-z)), $MachinePrecision], If[LessEqual[x, -2e-308], (-z), N[(N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x + (-z)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.79999999999999981e-156

   1. Initial program 80.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 80.4

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

   4. Applied rewrites 80.4%

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

   if -6.79999999999999981e-156 < x < -1.9999999999999998e-308

   1. Initial program 63.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 93.1

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

   5. Applied rewrites 93.1%

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

   if -1.9999999999999998e-308 < x

   1. Initial program 84.6%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 84.6

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

   4. Applied rewrites 84.6%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 99.6

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

   6. Applied rewrites 99.6%

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

Alternative 7: 91.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-156}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, -z\right)\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -6.8e-156)
   (fma (log (/ x y)) x (- z))
   (if (<= x -2e-308) (- z) (- (* x (- (log x) (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.8e-156) {
		tmp = fma(log((x / y)), x, -z);
	} else if (x <= -2e-308) {
		tmp = -z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -6.8e-156)
		tmp = fma(log(Float64(x / y)), x, Float64(-z));
	elseif (x <= -2e-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, -6.8e-156], N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x + (-z)), $MachinePrecision], If[LessEqual[x, -2e-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 < -6.79999999999999981e-156

   1. Initial program 80.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 80.4

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

   4. Applied rewrites 80.4%

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

   if -6.79999999999999981e-156 < x < -1.9999999999999998e-308

   1. Initial program 63.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 93.1

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

   5. Applied rewrites 93.1%

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

   if -1.9999999999999998e-308 < x

   1. Initial program 84.6%

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

Alternative 8: 99.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-310)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (fma (- (log x) (log y)) x (- 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 = fma((log(x) - log(y)), x, -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 = fma(Float64(log(x) - log(y)), x, Float64(-z));
	end
	return 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[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x + (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 76.6%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 -4.999999999999985e-310 < y

   1. Initial program 84.6%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 84.6

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

   4. Applied rewrites 84.6%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 99.6

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

   6. Applied rewrites 99.6%

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

Alternative 9: 67.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-82} \lor \neg \left(z \leq 7.2 \cdot 10^{-9}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y}{x}\right) \cdot \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.5e-82) (not (<= z 7.2e-9))) (- z) (* (log (/ y x)) (- x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.5e-82) || !(z <= 7.2e-9)) {
		tmp = -z;
	} else {
		tmp = log((y / x)) * -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 <= (-5.5d-82)) .or. (.not. (z <= 7.2d-9))) then
        tmp = -z
    else
        tmp = log((y / x)) * -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.5e-82) || !(z <= 7.2e-9)) {
		tmp = -z;
	} else {
		tmp = Math.log((y / x)) * -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -5.5e-82) or not (z <= 7.2e-9):
		tmp = -z
	else:
		tmp = math.log((y / x)) * -x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.5e-82) || !(z <= 7.2e-9))
		tmp = Float64(-z);
	else
		tmp = Float64(log(Float64(y / x)) * Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -5.5e-82) || ~((z <= 7.2e-9)))
		tmp = -z;
	else
		tmp = log((y / x)) * -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.5e-82], N[Not[LessEqual[z, 7.2e-9]], $MachinePrecision]], (-z), N[(N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision] * (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4999999999999998e-82 or 7.2e-9 < z

   1. Initial program 76.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 74.4

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

   5. Applied rewrites 74.4%

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

   if -5.4999999999999998e-82 < z < 7.2e-9

   1. Initial program 86.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. clear-num N/A

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

      4. log-div N/A

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

      5. flip-- N/A

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

      6. log-prod N/A

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

      7. metadata-eval N/A

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

      8. associate-/r/ N/A

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

      9. remove-double-div N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 83.9%

      \[\leadsto x \cdot \color{blue}{\frac{0 - {\log \left(\frac{y}{x}\right)}^{2}}{\log \left(\frac{y}{x}\right)}} - z \]

   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. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 74.4

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

   7. Applied rewrites 74.4%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-82} \lor \neg \left(z \leq 7.2 \cdot 10^{-9}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y}{x}\right) \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-82} \lor \neg \left(z \leq 7.2 \cdot 10^{-9}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.5e-82) (not (<= z 7.2e-9))) (- z) (* (log (/ x y)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.5e-82) || !(z <= 7.2e-9)) {
		tmp = -z;
	} else {
		tmp = log((x / 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 <= (-5.5d-82)) .or. (.not. (z <= 7.2d-9))) then
        tmp = -z
    else
        tmp = log((x / y)) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.5e-82) || !(z <= 7.2e-9)) {
		tmp = -z;
	} else {
		tmp = Math.log((x / y)) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -5.5e-82) or not (z <= 7.2e-9):
		tmp = -z
	else:
		tmp = math.log((x / y)) * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.5e-82) || !(z <= 7.2e-9))
		tmp = Float64(-z);
	else
		tmp = Float64(log(Float64(x / y)) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -5.5e-82) || ~((z <= 7.2e-9)))
		tmp = -z;
	else
		tmp = log((x / y)) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.5e-82], N[Not[LessEqual[z, 7.2e-9]], $MachinePrecision]], (-z), N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4999999999999998e-82 or 7.2e-9 < z

   1. Initial program 76.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 74.4

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

   5. Applied rewrites 74.4%

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

   if -5.4999999999999998e-82 < z < 7.2e-9

   1. Initial program 86.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 73.1

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

   5. Applied rewrites 73.1%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-82} \lor \neg \left(z \leq 7.2 \cdot 10^{-9}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.6% 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 80.5%

   \[x \cdot \log \left(\frac{x}{y}\right) - z \]
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 52.6

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

5. Applied rewrites 52.6%

   \[\leadsto \color{blue}{-z} \]
6. 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 2024322 
(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))