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

Percentage Accurate: 77.0% → 99.5%

Time: 9.0s

Alternatives: 10

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 75.0%

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

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 76.2

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

   4. Applied rewrites 76.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-rgt-neg-out N/A

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

      6. distribute-neg-out N/A

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

      7. remove-double-neg N/A

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

      8. distribute-neg-out N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lift-log.f64 N/A

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

      12. neg-log N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. lift-/.f64 N/A

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

      16. lift-log.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. sub-neg N/A

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

   6. Applied rewrites 76.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-neg-in N/A

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-log.f64 N/A

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

      8. log-rec N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. frac-2neg N/A

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

      12. lift-neg.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg N/A

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

      16. lift-/.f64 N/A

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

      17. log-prod N/A

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

      18. +-commutative N/A

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

      19. distribute-rgt-in N/A

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

   8. Applied rewrites 99.6%

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

   if -4.999999999999985e-310 < y

   1. Initial program 79.6%

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

4. Final simplification 99.6%

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

Alternative 2: 86.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right) \cdot x\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t\_0 \leq 10^{+296}:\\ \;\;\;\;\log \left(\frac{-1}{-y} \cdot x\right) \cdot x - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log (/ x y)) x)))
   (if (<= t_0 (- INFINITY))
     (- z)
     (if (<= t_0 1e+296) (- (* (log (* (/ -1.0 (- y)) x)) x) z) (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y)) * x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_0 <= 1e+296) {
		tmp = (log(((-1.0 / -y) * x)) * x) - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.log((x / y)) * x;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = -z;
	} else if (t_0 <= 1e+296) {
		tmp = (Math.log(((-1.0 / -y) * x)) * x) - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.log((x / y)) * x
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -z
	elif t_0 <= 1e+296:
		tmp = (math.log(((-1.0 / -y) * x)) * x) - z
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(log(Float64(x / y)) * x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(-z);
	elseif (t_0 <= 1e+296)
		tmp = Float64(Float64(log(Float64(Float64(-1.0 / Float64(-y)) * x)) * x) - z);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = log((x / y)) * x;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = -z;
	elseif (t_0 <= 1e+296)
		tmp = (log(((-1.0 / -y) * x)) * x) - z;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], (-z), If[LessEqual[t$95$0, 1e+296], N[(N[(N[Log[N[(N[(-1.0 / (-y)), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] - z), $MachinePrecision], (-z)]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 9.99999999999999981e295 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 4.3%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 46.4

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

   5. Applied rewrites 46.4%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 9.99999999999999981e295

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(\frac{x}{y}\right) \cdot x \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;\log \left(\frac{x}{y}\right) \cdot x \leq 10^{+296}:\\ \;\;\;\;\log \left(\frac{-1}{-y} \cdot x\right) \cdot x - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.0% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log (/ x y)) x)))
   (if (<= t_0 (- INFINITY)) (- z) (if (<= t_0 1e+296) (- t_0 z) (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y)) * x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_0 <= 1e+296) {
		tmp = t_0 - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.log((x / y)) * x;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = -z;
	} else if (t_0 <= 1e+296) {
		tmp = t_0 - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.log((x / y)) * x
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -z
	elif t_0 <= 1e+296:
		tmp = t_0 - z
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(log(Float64(x / y)) * x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(-z);
	elseif (t_0 <= 1e+296)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = log((x / y)) * x;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = -z;
	elseif (t_0 <= 1e+296)
		tmp = t_0 - z;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], (-z), If[LessEqual[t$95$0, 1e+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 9.99999999999999981e295 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 4.3%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 46.4

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

   5. Applied rewrites 46.4%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 9.99999999999999981e295

   1. Initial program 99.8%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(\frac{x}{y}\right) \cdot x \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;\log \left(\frac{x}{y}\right) \cdot x \leq 10^{+296}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.7% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log (/ x y)) x)))
   (if (<= t_0 (- INFINITY))
     (- z)
     (if (<= t_0 1e+296) (- (fma (log (/ y x)) x z)) (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y)) * x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_0 <= 1e+296) {
		tmp = -fma(log((y / x)), x, z);
	} else {
		tmp = -z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(log(Float64(x / y)) * x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(-z);
	elseif (t_0 <= 1e+296)
		tmp = Float64(-fma(log(Float64(y / x)), x, z));
	else
		tmp = Float64(-z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], (-z), If[LessEqual[t$95$0, 1e+296], (-N[(N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision] * x + z), $MachinePrecision]), (-z)]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 9.99999999999999981e295 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 4.3%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 46.4

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

   5. Applied rewrites 46.4%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 9.99999999999999981e295

   1. Initial program 99.8%

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

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-rgt-neg-out N/A

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

      6. distribute-neg-out N/A

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

      7. remove-double-neg N/A

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

      8. distribute-neg-out N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lift-log.f64 N/A

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

      12. neg-log N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. lift-/.f64 N/A

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

      16. lift-log.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. sub-neg N/A

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

   6. Applied rewrites 99.4%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(\frac{x}{y}\right) \cdot x \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;\log \left(\frac{x}{y}\right) \cdot x \leq 10^{+296}:\\ \;\;\;\;-\mathsf{fma}\left(\log \left(\frac{y}{x}\right), x, z\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+170}:\\ \;\;\;\;-\left(\log \left(-y\right) - \log \left(-x\right)\right) \cdot x\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-114}:\\ \;\;\;\;-\mathsf{fma}\left(\log \left(\frac{y}{x}\right), x, z\right)\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-305}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.45e+170)
   (- (* (- (log (- y)) (log (- x))) x))
   (if (<= x -2.8e-114)
     (- (fma (log (/ y x)) x z))
     (if (<= x -4e-305) (- z) (- (* (- (log x) (log y)) x) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.45e+170) {
		tmp = -((log(-y) - log(-x)) * x);
	} else if (x <= -2.8e-114) {
		tmp = -fma(log((y / x)), x, z);
	} else if (x <= -4e-305) {
		tmp = -z;
	} else {
		tmp = ((log(x) - log(y)) * x) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.45e+170)
		tmp = Float64(-Float64(Float64(log(Float64(-y)) - log(Float64(-x))) * x));
	elseif (x <= -2.8e-114)
		tmp = Float64(-fma(log(Float64(y / x)), x, z));
	elseif (x <= -4e-305)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(Float64(log(x) - log(y)) * x) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.45e+170], (-N[(N[(N[Log[(-y)], $MachinePrecision] - N[Log[(-x)], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), If[LessEqual[x, -2.8e-114], (-N[(N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision] * x + z), $MachinePrecision]), If[LessEqual[x, -4e-305], (-z), N[(N[(N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.45e170

   1. Initial program 51.8%

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

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 61.5

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

   4. Applied rewrites 61.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-rgt-neg-out N/A

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

      6. distribute-neg-out N/A

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

      7. remove-double-neg N/A

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

      8. distribute-neg-out N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lift-log.f64 N/A

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

      12. neg-log N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. lift-/.f64 N/A

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

      16. lift-log.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. sub-neg N/A

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

   6. Applied rewrites 61.5%

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

   7. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-/.f64 49.0

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

   9. Applied rewrites 49.0%

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

   11. Applied rewrites 85.6%

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

   if -1.45e170 < x < -2.8000000000000001e-114

   1. Initial program 90.6%

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

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 90.6

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

   4. Applied rewrites 90.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-rgt-neg-out N/A

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

      6. distribute-neg-out N/A

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

      7. remove-double-neg N/A

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

      8. distribute-neg-out N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lift-log.f64 N/A

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

      12. neg-log N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. lift-/.f64 N/A

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

      16. lift-log.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. sub-neg N/A

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

   6. Applied rewrites 90.6%

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

   if -2.8000000000000001e-114 < x < -3.99999999999999999e-305

   1. Initial program 62.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 90.7

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

   5. Applied rewrites 90.7%

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

   if -3.99999999999999999e-305 < x

   1. Initial program 79.6%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+170}:\\ \;\;\;\;-\left(\log \left(-y\right) - \log \left(-x\right)\right) \cdot x\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-114}:\\ \;\;\;\;-\mathsf{fma}\left(\log \left(\frac{y}{x}\right), x, z\right)\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-305}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-114}:\\ \;\;\;\;-\mathsf{fma}\left(\log \left(\frac{y}{x}\right), x, z\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-184}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+182}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.8e-114)
   (- (fma (log (/ y x)) x z))
   (if (<= x 5e-184)
     (- z)
     (if (<= x 6.2e+182)
       (- (* (log (/ x y)) x) z)
       (* (- (log x) (log y)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.8e-114) {
		tmp = -fma(log((y / x)), x, z);
	} else if (x <= 5e-184) {
		tmp = -z;
	} else if (x <= 6.2e+182) {
		tmp = (log((x / y)) * x) - z;
	} else {
		tmp = (log(x) - log(y)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.8e-114)
		tmp = Float64(-fma(log(Float64(y / x)), x, z));
	elseif (x <= 5e-184)
		tmp = Float64(-z);
	elseif (x <= 6.2e+182)
		tmp = Float64(Float64(log(Float64(x / y)) * x) - z);
	else
		tmp = Float64(Float64(log(x) - log(y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.8e-114], (-N[(N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision] * x + z), $MachinePrecision]), If[LessEqual[x, 5e-184], (-z), If[LessEqual[x, 6.2e+182], N[(N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.8000000000000001e-114

   1. Initial program 79.6%

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

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 82.3

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

   4. Applied rewrites 82.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-rgt-neg-out N/A

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

      6. distribute-neg-out N/A

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

      7. remove-double-neg N/A

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

      8. distribute-neg-out N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lift-log.f64 N/A

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

      12. neg-log N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. lift-/.f64 N/A

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

      16. lift-log.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. sub-neg N/A

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

   6. Applied rewrites 82.3%

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

   if -2.8000000000000001e-114 < x < 5.00000000000000003e-184

   1. Initial program 58.2%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 91.6

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

   5. Applied rewrites 91.6%

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

   if 5.00000000000000003e-184 < x < 6.19999999999999993e182

   1. Initial program 96.3%

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

   if 6.19999999999999993e182 < x

   1. Initial program 53.5%

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

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. log-rec N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-log.f64 88.1

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

   5. Applied rewrites 88.1%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-114}:\\ \;\;\;\;-\mathsf{fma}\left(\log \left(\frac{y}{x}\right), x, z\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-184}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+182}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-114}:\\ \;\;\;\;-\mathsf{fma}\left(\log \left(\frac{y}{x}\right), x, z\right)\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-305}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.8e-114)
   (- (fma (log (/ y x)) x z))
   (if (<= x -4e-305) (- z) (- (* (- (log x) (log y)) x) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.8e-114) {
		tmp = -fma(log((y / x)), x, z);
	} else if (x <= -4e-305) {
		tmp = -z;
	} else {
		tmp = ((log(x) - log(y)) * x) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.8e-114)
		tmp = Float64(-fma(log(Float64(y / x)), x, z));
	elseif (x <= -4e-305)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(Float64(log(x) - log(y)) * x) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.8e-114], (-N[(N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision] * x + z), $MachinePrecision]), If[LessEqual[x, -4e-305], (-z), N[(N[(N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.8000000000000001e-114

   1. Initial program 79.6%

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

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 82.3

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

   4. Applied rewrites 82.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-rgt-neg-out N/A

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

      6. distribute-neg-out N/A

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

      7. remove-double-neg N/A

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

      8. distribute-neg-out N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lift-log.f64 N/A

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

      12. neg-log N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. lift-/.f64 N/A

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

      16. lift-log.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. sub-neg N/A

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

   6. Applied rewrites 82.3%

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

   if -2.8000000000000001e-114 < x < -3.99999999999999999e-305

   1. Initial program 62.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 90.7

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

   5. Applied rewrites 90.7%

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

   if -3.99999999999999999e-305 < x

   1. Initial program 79.6%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-114}:\\ \;\;\;\;-\mathsf{fma}\left(\log \left(\frac{y}{x}\right), x, z\right)\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-305}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-5d-310)) then
        tmp = ((log(-x) - log(-y)) * x) - z
    else
        tmp = ((log(x) - log(y)) * x) - z
    end if
    code = tmp
end function

Java

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

Python

def code(x, y, z):
	tmp = 0
	if y <= -5e-310:
		tmp = ((math.log(-x) - math.log(-y)) * x) - z
	else:
		tmp = ((math.log(x) - math.log(y)) * x) - z
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5e-310)
		tmp = ((log(-x) - log(-y)) * x) - z;
	else
		tmp = ((log(x) - log(y)) * x) - z;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 75.0%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. 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 79.6%

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

4. Final simplification 99.6%

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

Alternative 9: 67.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-75}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-67}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.2e-75) (- z) (if (<= z 2.4e-67) (* (log (/ x y)) x) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.2e-75) {
		tmp = -z;
	} else if (z <= 2.4e-67) {
		tmp = log((x / 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 <= (-3.2d-75)) then
        tmp = -z
    else if (z <= 2.4d-67) then
        tmp = log((x / 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 <= -3.2e-75) {
		tmp = -z;
	} else if (z <= 2.4e-67) {
		tmp = Math.log((x / y)) * x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.2e-75:
		tmp = -z
	elif z <= 2.4e-67:
		tmp = math.log((x / y)) * x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.2e-75)
		tmp = Float64(-z);
	elseif (z <= 2.4e-67)
		tmp = Float64(log(Float64(x / y)) * x);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3.2e-75)
		tmp = -z;
	elseif (z <= 2.4e-67)
		tmp = log((x / y)) * x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.2e-75], (-z), If[LessEqual[z, 2.4e-67], N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -3.19999999999999977e-75 or 2.4e-67 < z

   1. Initial program 77.9%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 74.5

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

   5. Applied rewrites 74.5%

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

   if -3.19999999999999977e-75 < z < 2.4e-67

   1. Initial program 76.5%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 65.3

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

   5. Applied rewrites 65.3%

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

Alternative 10: 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 77.4%

   \[x \cdot \log \left(\frac{x}{y}\right) - z \]
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 55.1

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

5. Applied rewrites 55.1%

   \[\leadsto \color{blue}{-z} \]
6. 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 2024296 
(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))