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

Percentage Accurate: 77.6% → 99.4%

Time: 12.2s

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: 77.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-310)
   (fma (+ (log (/ -1.0 y)) (log (- x))) (/ 1.0 (/ 1.0 x)) (- z))
   (- (- (* x (log x)) (* x (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-310) {
		tmp = fma((log((-1.0 / y)) + log(-x)), (1.0 / (1.0 / x)), -z);
	} else {
		tmp = ((x * log(x)) - (x * log(y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1e-310)
		tmp = fma(Float64(log(Float64(-1.0 / y)) + log(Float64(-x))), Float64(1.0 / Float64(1.0 / x)), Float64(-z));
	else
		tmp = Float64(Float64(Float64(x * log(x)) - Float64(x * log(y))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1e-310], N[(N[(N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision] + N[Log[(-x)], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision] + (-z)), $MachinePrecision], N[(N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.999999999999969e-311

   1. Initial program 75.5%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. unpow1 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow-prod-down N/A

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

      7. metadata-eval N/A

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

      8. pow-pow N/A

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

      9. inv-pow N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. unpow-1 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 75.5

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

   4. Applied rewrites 75.5%

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

   5. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 99.5

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

   7. Applied rewrites 99.5%

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

   if -9.999999999999969e-311 < y

   1. Initial program 79.9%

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

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

      3. lift-/.f64 N/A

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

      4. log-div N/A

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

      5. sub-neg N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-log.f64 99.6

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

   4. Applied rewrites 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 87.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+296}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY)) (- z) (if (<= t_0 2e+296) (- t_0 z) (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_0 <= 2e+296) {
		tmp = t_0 - z;
	} else {
		tmp = -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) {
		tmp = -z;
	} else if (t_0 <= 2e+296) {
		tmp = t_0 - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -z
	elif t_0 <= 2e+296:
		tmp = t_0 - z
	else:
		tmp = -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))
		tmp = Float64(-z);
	elseif (t_0 <= 2e+296)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(-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)
		tmp = -z;
	elseif (t_0 <= 2e+296)
		tmp = t_0 - z;
	else
		tmp = -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[LessEqual[t$95$0, (-Infinity)], (-z), If[LessEqual[t$95$0, 2e+296], 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 1.99999999999999996e296 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 7.2%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 53.7

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

   5. Applied rewrites 53.7%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1.99999999999999996e296

   1. Initial program 99.7%

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

Alternative 3: 96.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(-x\right) - \log \left(-y\right)\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{+144}:\\ \;\;\;\;x \cdot t\_0\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(t\_0, \frac{x}{z}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x \cdot \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (log (- x)) (log (- y)))))
   (if (<= x -7.2e+144)
     (* x t_0)
     (if (<= x -5e-310)
       (* z (fma t_0 (/ x z) -1.0))
       (- (- (* x (log x)) (* x (log y))) z)))))

C

double code(double x, double y, double z) {
	double t_0 = log(-x) - log(-y);
	double tmp;
	if (x <= -7.2e+144) {
		tmp = x * t_0;
	} else if (x <= -5e-310) {
		tmp = z * fma(t_0, (x / z), -1.0);
	} else {
		tmp = ((x * log(x)) - (x * log(y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(log(Float64(-x)) - log(Float64(-y)))
	tmp = 0.0
	if (x <= -7.2e+144)
		tmp = Float64(x * t_0);
	elseif (x <= -5e-310)
		tmp = Float64(z * fma(t_0, Float64(x / z), -1.0));
	else
		tmp = Float64(Float64(Float64(x * log(x)) - Float64(x * log(y))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.2e+144], N[(x * t$95$0), $MachinePrecision], If[LessEqual[x, -5e-310], N[(z * N[(t$95$0 * N[(x / z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.1999999999999995e144

   1. Initial program 74.5%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. remove-double-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. log-rec N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. mul-1-neg N/A

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

      12. log-rec N/A

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

      13. metadata-eval N/A

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

      14. associate-/r* N/A

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

      15. remove-double-div N/A

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

      16. lower-log.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 90.0

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

   5. Applied rewrites 90.0%

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

   if -7.1999999999999995e144 < x < -4.999999999999985e-310

   1. Initial program 75.8%

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

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 75.7

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 75.7

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

   4. Applied rewrites 75.7%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 71.9

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

   7. Applied rewrites 71.9%

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

   9. Applied rewrites 94.9%

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

   if -4.999999999999985e-310 < x

   1. Initial program 79.9%

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

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

      3. lift-/.f64 N/A

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

      4. log-div N/A

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

      5. sub-neg N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-log.f64 99.6

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

   4. Applied rewrites 99.6%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+144}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(\log \left(-x\right) - \log \left(-y\right), \frac{x}{z}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x \cdot \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+89}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{-107}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x \cdot \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -8.2e+89)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -1.65e-107)
     (- (* x (log (/ x y))) z)
     (if (<= x -2e-309) (- z) (- (- (* x (log x)) (* x (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -8.2e+89) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -1.65e-107) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -2e-309) {
		tmp = -z;
	} else {
		tmp = ((x * log(x)) - (x * log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-8.2d+89)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-1.65d-107)) then
        tmp = (x * log((x / y))) - z
    else if (x <= (-2d-309)) then
        tmp = -z
    else
        tmp = ((x * log(x)) - (x * log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -8.2e+89) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -1.65e-107) {
		tmp = (x * Math.log((x / y))) - z;
	} else if (x <= -2e-309) {
		tmp = -z;
	} else {
		tmp = ((x * Math.log(x)) - (x * Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -8.2e+89:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -1.65e-107:
		tmp = (x * math.log((x / y))) - z
	elif x <= -2e-309:
		tmp = -z
	else:
		tmp = ((x * math.log(x)) - (x * math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -8.2e+89)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -1.65e-107)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -2e-309)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(Float64(x * log(x)) - Float64(x * log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -8.2e+89)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -1.65e-107)
		tmp = (x * log((x / y))) - z;
	elseif (x <= -2e-309)
		tmp = -z;
	else
		tmp = ((x * log(x)) - (x * log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -8.2e+89], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.65e-107], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -2e-309], (-z), N[(N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.1999999999999997e89

   1. Initial program 69.7%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. remove-double-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. log-rec N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. mul-1-neg N/A

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

      12. log-rec N/A

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

      13. metadata-eval N/A

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

      14. associate-/r* N/A

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

      15. remove-double-div N/A

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

      16. lower-log.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 81.9

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

   5. Applied rewrites 81.9%

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

   if -8.1999999999999997e89 < x < -1.65000000000000002e-107

   1. Initial program 91.1%

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

   if -1.65000000000000002e-107 < x < -1.9999999999999988e-309

   1. Initial program 66.1%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 88.4

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

   5. Applied rewrites 88.4%

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

   if -1.9999999999999988e-309 < x

   1. Initial program 79.9%

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

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

      3. lift-/.f64 N/A

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

      4. log-div N/A

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

      5. sub-neg N/A

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

      6. distribute-rgt-in N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-log.f64 99.6

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

   4. Applied rewrites 99.6%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+89}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{-107}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x \cdot \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+89}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{-107}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -8.2e+89)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -1.65e-107)
     (- (* x (log (/ x y))) z)
     (if (<= x -2e-309) (- z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -8.2e+89) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -1.65e-107) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -2e-309) {
		tmp = -z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-8.2d+89)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-1.65d-107)) then
        tmp = (x * log((x / y))) - z
    else if (x <= (-2d-309)) then
        tmp = -z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -8.2e+89) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -1.65e-107) {
		tmp = (x * Math.log((x / y))) - z;
	} else if (x <= -2e-309) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -8.2e+89:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -1.65e-107:
		tmp = (x * math.log((x / y))) - z
	elif x <= -2e-309:
		tmp = -z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -8.2e+89)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -1.65e-107)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -2e-309)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -8.2e+89)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -1.65e-107)
		tmp = (x * log((x / y))) - z;
	elseif (x <= -2e-309)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -8.2e+89], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.65e-107], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -2e-309], (-z), N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.1999999999999997e89

   1. Initial program 69.7%

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

   3. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. remove-double-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. log-rec N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. mul-1-neg N/A

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

      12. log-rec N/A

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

      13. metadata-eval N/A

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

      14. associate-/r* N/A

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

      15. remove-double-div N/A

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

      16. lower-log.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 81.9

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

   5. Applied rewrites 81.9%

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

   if -8.1999999999999997e89 < x < -1.65000000000000002e-107

   1. Initial program 91.1%

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

   if -1.65000000000000002e-107 < x < -1.9999999999999988e-309

   1. Initial program 66.1%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 88.4

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

   5. Applied rewrites 88.4%

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

   if -1.9999999999999988e-309 < x

   1. Initial program 79.9%

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

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

   4. Applied rewrites 99.5%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+89}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{-107}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-107}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.65e-107)
   (- (* x (log (/ x y))) z)
   (if (<= x -2e-309) (- z) (- (* x (- (log x) (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.65e-107) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -2e-309) {
		tmp = -z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.65d-107)) then
        tmp = (x * log((x / y))) - z
    else if (x <= (-2d-309)) then
        tmp = -z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.65e-107) {
		tmp = (x * Math.log((x / y))) - z;
	} else if (x <= -2e-309) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.65e-107:
		tmp = (x * math.log((x / y))) - z
	elif x <= -2e-309:
		tmp = -z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.65e-107)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -2e-309)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.65e-107)
		tmp = (x * log((x / y))) - z;
	elseif (x <= -2e-309)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.65e-107], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -2e-309], (-z), N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.65000000000000002e-107

   1. Initial program 81.3%

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

   if -1.65000000000000002e-107 < x < -1.9999999999999988e-309

   1. Initial program 66.1%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 88.4

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

   5. Applied rewrites 88.4%

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

   if -1.9999999999999988e-309 < x

   1. Initial program 79.9%

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

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

   4. Applied rewrites 99.5%

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

Alternative 7: 66.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-124}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-46}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.1e-124) (- z) (if (<= z 8e-46) (* (- x) (log (/ y x))) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.1e-124) {
		tmp = -z;
	} else if (z <= 8e-46) {
		tmp = -x * log((y / x));
	} else {
		tmp = -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 (z <= (-2.1d-124)) then
        tmp = -z
    else if (z <= 8d-46) then
        tmp = -x * log((y / x))
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.1e-124) {
		tmp = -z;
	} else if (z <= 8e-46) {
		tmp = -x * Math.log((y / x));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.1e-124:
		tmp = -z
	elif z <= 8e-46:
		tmp = -x * math.log((y / x))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.1e-124)
		tmp = Float64(-z);
	elseif (z <= 8e-46)
		tmp = Float64(Float64(-x) * log(Float64(y / x)));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.1e-124)
		tmp = -z;
	elseif (z <= 8e-46)
		tmp = -x * log((y / x));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.1e-124], (-z), If[LessEqual[z, 8e-46], N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1000000000000001e-124 or 8.00000000000000018e-46 < z

   1. Initial program 76.9%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 78.8

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

   5. Applied rewrites 78.8%

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

   if -2.1000000000000001e-124 < z < 8.00000000000000018e-46

   1. Initial program 79.0%

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

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

      5. lower-neg.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 80.5

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

   4. Applied rewrites 80.5%

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

   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. 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 65.5

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

   7. Applied rewrites 65.5%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-124}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-46}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-124}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-46}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.1e-124) (- z) (if (<= z 8e-46) (* x (log (/ x y))) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.1e-124) {
		tmp = -z;
	} else if (z <= 8e-46) {
		tmp = x * log((x / y));
	} else {
		tmp = -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 (z <= (-2.1d-124)) then
        tmp = -z
    else if (z <= 8d-46) then
        tmp = x * log((x / y))
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.1e-124) {
		tmp = -z;
	} else if (z <= 8e-46) {
		tmp = x * Math.log((x / y));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.1e-124:
		tmp = -z
	elif z <= 8e-46:
		tmp = x * math.log((x / y))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.1e-124)
		tmp = Float64(-z);
	elseif (z <= 8e-46)
		tmp = Float64(x * log(Float64(x / y)));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.1e-124)
		tmp = -z;
	elseif (z <= 8e-46)
		tmp = x * log((x / y));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.1e-124], (-z), If[LessEqual[z, 8e-46], N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1000000000000001e-124 or 8.00000000000000018e-46 < z

   1. Initial program 76.9%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 78.8

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

   5. Applied rewrites 78.8%

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

   if -2.1000000000000001e-124 < z < 8.00000000000000018e-46

   1. Initial program 79.0%

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

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 64.1

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

   5. Applied rewrites 64.1%

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

Alternative 9: 50.3% 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 77.7%

   \[x \cdot \log \left(\frac{x}{y}\right) - z \]
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 58.4

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

5. Applied rewrites 58.4%

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

Alternative 10: 2.7% accurate, 120.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 0.0)

C

double code(double x, double y, double z) {
	return 0.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 0.0;
}

Python

def code(x, y, z):
	return 0.0

Julia

function code(x, y, z)
	return 0.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.0;
end

Wolfram

code[x_, y_, z_] := 0.0

Derivation

1. Initial program 77.7%

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

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

   4. inv-pow N/A

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

   5. sqr-pow N/A

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

   6. log-prod N/A

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

   7. lower-+.f64 N/A

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

   8. lower-log.f64 N/A

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

   9. clear-num N/A

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

   10. lift-/.f64 N/A

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

   11. inv-pow N/A

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

   12. pow-pow N/A

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

   13. lower-pow.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower-log.f64 N/A

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

   17. clear-num N/A

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

   18. lift-/.f64 N/A

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

   19. inv-pow N/A

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

   20. pow-pow N/A

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

   21. lower-pow.f64 N/A

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

   22. metadata-eval N/A

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

   23. metadata-eval 77.7

       \[\leadsto x \cdot \left(\log \left({\left(\frac{x}{y}\right)}^{0.5}\right) + \log \left({\left(\frac{x}{y}\right)}^{\color{blue}{0.5}}\right)\right) - z \]

4. Applied rewrites 77.7%

   \[\leadsto x \cdot \color{blue}{\left(\log \left({\left(\frac{x}{y}\right)}^{0.5}\right) + \log \left({\left(\frac{x}{y}\right)}^{0.5}\right)\right)} - z \]

5. Taylor expanded in z around 0

   \[\leadsto \color{blue}{2 \cdot \left(x \cdot \log \left(\sqrt{\frac{x}{y}}\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot \log \left(\sqrt{\frac{x}{y}}\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\log \left(\sqrt{\frac{x}{y}}\right) \cdot \left(2 \cdot x\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\log \left(\sqrt{\frac{x}{y}}\right) \cdot \left(2 \cdot x\right)} \]

   4. lower-log.f64 N/A

      \[\leadsto \color{blue}{\log \left(\sqrt{\frac{x}{y}}\right)} \cdot \left(2 \cdot x\right) \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \log \color{blue}{\left(\sqrt{\frac{x}{y}}\right)} \cdot \left(2 \cdot x\right) \]

   6. lower-/.f64 N/A

      \[\leadsto \log \left(\sqrt{\color{blue}{\frac{x}{y}}}\right) \cdot \left(2 \cdot x\right) \]

   7. *-commutative N/A

      \[\leadsto \log \left(\sqrt{\frac{x}{y}}\right) \cdot \color{blue}{\left(x \cdot 2\right)} \]

   8. lower-*.f64 33.4

      \[\leadsto \log \left(\sqrt{\frac{x}{y}}\right) \cdot \color{blue}{\left(x \cdot 2\right)} \]

7. Applied rewrites 33.4%

   \[\leadsto \color{blue}{\log \left(\sqrt{\frac{x}{y}}\right) \cdot \left(x \cdot 2\right)} \]
8. Step-by-step derivation

9. Applied rewrites 2.7%

   \[\leadsto \color{blue}{0} \]
10. 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 77.7%

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

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. lower-/.f64 77.5

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

   9. lift--.f64 N/A

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

   10. sub-neg N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-neg.f64 77.5

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

4. Applied rewrites 77.5%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. remove-double-div 77.7

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

   4. lift-fma.f64 N/A

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

   5. lift-neg.f64 N/A

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

6. Applied rewrites 2.1%

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

Developer Target 1: 88.2% 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 2024233 
(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))