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

Percentage Accurate: 76.4% → 99.4%

Time: 10.9s

Alternatives: 11

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 78.2%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. 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 0.0

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

   4. Applied rewrites 0.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift-log.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. diff-log N/A

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

      10. lift-/.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lower-/.f64 78.1

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

   6. Applied rewrites 78.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. remove-double-div N/A

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

      5. lift-log.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. diff-log N/A

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

      11. lift-log.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. sub-neg N/A

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

      14. lift-neg.f64 N/A

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

      15. distribute-rgt-in N/A

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

      16. +-commutative N/A

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

   8. Applied rewrites 99.4%

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

   if -4.999999999999985e-310 < y

   1. Initial program 79.1%

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

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

      8. lower-log.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-log.f64 99.5

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

   4. Applied rewrites 99.5%

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

Alternative 2: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right) \cdot x\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+303}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log (/ x y)) x)))
   (if (<= t_0 (- INFINITY))
     (- z)
     (if (<= t_0 2e+303) (- t_0 z) (* (- (log x) (log y)) x)))))

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 <= 2e+303) {
		tmp = t_0 - z;
	} else {
		tmp = (log(x) - log(y)) * x;
	}
	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 <= 2e+303) {
		tmp = t_0 - z;
	} else {
		tmp = (Math.log(x) - Math.log(y)) * x;
	}
	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 <= 2e+303:
		tmp = t_0 - z
	else:
		tmp = (math.log(x) - math.log(y)) * x
	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 <= 2e+303)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(Float64(log(x) - log(y)) * x);
	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 <= 2e+303)
		tmp = t_0 - z;
	else
		tmp = (log(x) - log(y)) * x;
	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, 2e+303], N[(t$95$0 - z), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 4.4%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 51.1

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

   5. Applied rewrites 51.1%

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

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

   1. Initial program 99.5%

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

   if 2e303 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 13.6%

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

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      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 54.2

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

   5. Applied rewrites 54.2%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(\frac{x}{y}\right) \cdot x \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;\log \left(\frac{x}{y}\right) \cdot x \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right) \cdot x\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t\_0 \leq 10^{+306}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log (/ x y)) x)))
   (if (<= t_0 (- INFINITY)) (- z) (if (<= t_0 1e+306) (- 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+306) {
		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+306) {
		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+306:
		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+306)
		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+306)
		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+306], 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 1.00000000000000002e306 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 6.1%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 49.0

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

   5. Applied rewrites 49.0%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1.00000000000000002e306

   1. Initial program 99.5%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(\frac{x}{y}\right) \cdot x \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;\log \left(\frac{x}{y}\right) \cdot x \leq 10^{+306}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+116}:\\ \;\;\;\;\left(\log \left(-x\right) - \log \left(-y\right)\right) \cdot x\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-169}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-301}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x, -\mathsf{fma}\left(\log y, x, z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -8e+116)
   (* (- (log (- x)) (log (- y))) x)
   (if (<= x -2.1e-169)
     (- (* (log (/ x y)) x) z)
     (if (<= x -5e-301) (- z) (fma (log x) x (- (fma (log y) x z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -8e+116) {
		tmp = (log(-x) - log(-y)) * x;
	} else if (x <= -2.1e-169) {
		tmp = (log((x / y)) * x) - z;
	} else if (x <= -5e-301) {
		tmp = -z;
	} else {
		tmp = fma(log(x), x, -fma(log(y), x, z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -8e+116)
		tmp = Float64(Float64(log(Float64(-x)) - log(Float64(-y))) * x);
	elseif (x <= -2.1e-169)
		tmp = Float64(Float64(log(Float64(x / y)) * x) - z);
	elseif (x <= -5e-301)
		tmp = Float64(-z);
	else
		tmp = fma(log(x), x, Float64(-fma(log(y), x, z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -8e+116], N[(N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, -2.1e-169], N[(N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -5e-301], (-z), N[(N[Log[x], $MachinePrecision] * x + (-N[(N[Log[y], $MachinePrecision] * x + z), $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.00000000000000012e116

   1. Initial program 58.6%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-neg.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-neg.f64 83.3

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

   7. Applied rewrites 83.3%

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

   if -8.00000000000000012e116 < x < -2.1000000000000001e-169

   1. Initial program 93.3%

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

   if -2.1000000000000001e-169 < x < -5.00000000000000013e-301

   1. Initial program 71.7%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.6

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

   5. Applied rewrites 95.6%

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

   if -5.00000000000000013e-301 < x

   1. Initial program 79.1%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift-log.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. diff-log N/A

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

      10. lift-/.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lower-/.f64 79.1

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

   6. Applied rewrites 79.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 79.1

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

      5. lift-/.f64 N/A

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

      6. inv-pow N/A

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

      7. lower-pow.f64 79.1

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

   8. Applied rewrites 79.1%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-1 N/A

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

      6. remove-double-div N/A

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

      7. lift-log.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. log-div N/A

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

      10. lift-log.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. sub-neg N/A

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

      13. lift-neg.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. lift-*.f64 N/A

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

      16. +-commutative N/A

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

      17. +-commutative N/A

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

      18. associate--l+ N/A

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

      19. lower-fma.f64 N/A

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

   10. Applied rewrites 99.5%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+116}:\\ \;\;\;\;\left(\log \left(-x\right) - \log \left(-y\right)\right) \cdot x\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-169}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-301}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x, -\mathsf{fma}\left(\log y, x, z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+116}:\\ \;\;\;\;\left(\log \left(-x\right) - \log \left(-y\right)\right) \cdot x\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-169}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-301}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -8e+116)
   (* (- (log (- x)) (log (- y))) x)
   (if (<= x -2.1e-169)
     (- (* (log (/ x y)) x) z)
     (if (<= x -5e-301) (- z) (- (* (- (log x) (log y)) x) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -8e+116) {
		tmp = (log(-x) - log(-y)) * x;
	} else if (x <= -2.1e-169) {
		tmp = (log((x / y)) * x) - z;
	} else if (x <= -5e-301) {
		tmp = -z;
	} else {
		tmp = ((log(x) - log(y)) * x) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-8d+116)) then
        tmp = (log(-x) - log(-y)) * x
    else if (x <= (-2.1d-169)) then
        tmp = (log((x / y)) * x) - z
    else if (x <= (-5d-301)) then
        tmp = -z
    else
        tmp = ((log(x) - log(y)) * x) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -8e+116) {
		tmp = (Math.log(-x) - Math.log(-y)) * x;
	} else if (x <= -2.1e-169) {
		tmp = (Math.log((x / y)) * x) - z;
	} else if (x <= -5e-301) {
		tmp = -z;
	} else {
		tmp = ((Math.log(x) - Math.log(y)) * x) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -8e+116:
		tmp = (math.log(-x) - math.log(-y)) * x
	elif x <= -2.1e-169:
		tmp = (math.log((x / y)) * x) - z
	elif x <= -5e-301:
		tmp = -z
	else:
		tmp = ((math.log(x) - math.log(y)) * x) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -8e+116)
		tmp = Float64(Float64(log(Float64(-x)) - log(Float64(-y))) * x);
	elseif (x <= -2.1e-169)
		tmp = Float64(Float64(log(Float64(x / y)) * x) - z);
	elseif (x <= -5e-301)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(Float64(log(x) - log(y)) * x) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -8e+116)
		tmp = (log(-x) - log(-y)) * x;
	elseif (x <= -2.1e-169)
		tmp = (log((x / y)) * x) - z;
	elseif (x <= -5e-301)
		tmp = -z;
	else
		tmp = ((log(x) - log(y)) * x) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -8e+116], N[(N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, -2.1e-169], N[(N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -5e-301], (-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 < -8.00000000000000012e116

   1. Initial program 58.6%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-neg.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-neg.f64 83.3

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

   7. Applied rewrites 83.3%

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

   if -8.00000000000000012e116 < x < -2.1000000000000001e-169

   1. Initial program 93.3%

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

   if -2.1000000000000001e-169 < x < -5.00000000000000013e-301

   1. Initial program 71.7%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.6

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

   5. Applied rewrites 95.6%

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

   if -5.00000000000000013e-301 < x

   1. Initial program 79.1%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 99.4

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

   4. Applied rewrites 99.4%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+116}:\\ \;\;\;\;\left(\log \left(-x\right) - \log \left(-y\right)\right) \cdot x\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-169}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-301}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-169}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-301}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.1e-169)
   (- (* (log (/ x y)) x) z)
   (if (<= x -5e-301) (- z) (- (* (- (log x) (log y)) x) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.1e-169) {
		tmp = (log((x / y)) * x) - z;
	} else if (x <= -5e-301) {
		tmp = -z;
	} else {
		tmp = ((log(x) - log(y)) * x) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-2.1d-169)) then
        tmp = (log((x / y)) * x) - z
    else if (x <= (-5d-301)) then
        tmp = -z
    else
        tmp = ((log(x) - log(y)) * x) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.1e-169) {
		tmp = (Math.log((x / y)) * x) - z;
	} else if (x <= -5e-301) {
		tmp = -z;
	} else {
		tmp = ((Math.log(x) - Math.log(y)) * x) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.1e-169:
		tmp = (math.log((x / y)) * x) - z
	elif x <= -5e-301:
		tmp = -z
	else:
		tmp = ((math.log(x) - math.log(y)) * x) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.1e-169)
		tmp = Float64(Float64(log(Float64(x / y)) * x) - z);
	elseif (x <= -5e-301)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(Float64(log(x) - log(y)) * x) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.1e-169)
		tmp = (log((x / y)) * x) - z;
	elseif (x <= -5e-301)
		tmp = -z;
	else
		tmp = ((log(x) - log(y)) * x) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.1e-169], N[(N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -5e-301], (-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 < -2.1000000000000001e-169

   1. Initial program 79.7%

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

   if -2.1000000000000001e-169 < x < -5.00000000000000013e-301

   1. Initial program 71.7%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.6

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

   5. Applied rewrites 95.6%

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

   if -5.00000000000000013e-301 < x

   1. Initial program 79.1%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 99.4

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

   4. Applied rewrites 99.4%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-169}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-301}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 78.2%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-neg.f64 99.4

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

   4. Applied rewrites 99.4%

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

   if -4.999999999999985e-310 < y

   1. Initial program 79.1%

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

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

      8. lower-log.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-log.f64 99.5

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

   4. Applied rewrites 99.5%

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

4. Final simplification 99.5%

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

Alternative 8: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 78.2%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-neg.f64 99.4

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

   4. Applied rewrites 99.4%

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

   if -4.999999999999985e-310 < y

   1. Initial program 79.1%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift-log.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. diff-log N/A

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

      10. lift-/.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lower-/.f64 79.1

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

   6. Applied rewrites 79.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 79.1

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

      5. lift-/.f64 N/A

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

      6. inv-pow N/A

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

      7. lower-pow.f64 79.1

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

   8. Applied rewrites 79.1%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow-1 N/A

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

      6. remove-double-div N/A

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

      7. lift-log.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. log-div N/A

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

      10. lift-log.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. sub-neg N/A

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

      13. lift-neg.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. lift-*.f64 N/A

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

      16. +-commutative N/A

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

      17. +-commutative N/A

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

      18. associate--l+ N/A

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

      19. lower-fma.f64 N/A

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

   10. Applied rewrites 99.5%

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

4. Final simplification 99.4%

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

Alternative 9: 67.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9500000000:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -9500000000.0) (- z) (if (<= z 1.0) (* (log (/ x y)) x) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -9500000000.0) {
		tmp = -z;
	} else if (z <= 1.0) {
		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 <= (-9500000000.0d0)) then
        tmp = -z
    else if (z <= 1.0d0) 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 <= -9500000000.0) {
		tmp = -z;
	} else if (z <= 1.0) {
		tmp = Math.log((x / y)) * x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -9500000000.0:
		tmp = -z
	elif z <= 1.0:
		tmp = math.log((x / y)) * x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -9500000000.0)
		tmp = Float64(-z);
	elseif (z <= 1.0)
		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 <= -9500000000.0)
		tmp = -z;
	elseif (z <= 1.0)
		tmp = log((x / y)) * x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -9500000000.0], (-z), If[LessEqual[z, 1.0], N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -9.5e9 or 1 < z

   1. Initial program 75.5%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 76.9

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

   5. Applied rewrites 76.9%

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

   if -9.5e9 < z < 1

   1. Initial program 81.7%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 71.1

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

   5. Applied rewrites 71.1%

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

Alternative 10: 50.1% accurate, 40.0× speedup

Math

\[\begin{array}{l} \\ -z \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.7%

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

3. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 44.4

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

5. Applied rewrites 44.4%

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

Alternative 11: 2.2% 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 78.7%

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

3. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 44.4

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

5. Applied rewrites 44.4%

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

7. Applied rewrites 1.2%

   \[\leadsto \frac{0 + {z}^{3}}{\color{blue}{0 + \left(z \cdot z - 0 \cdot \left(-z\right)\right)}} \]
8. Step-by-step derivation

9. Applied rewrites 2.3%

   \[\leadsto z \]
10. Add Preprocessing

Developer Target 1: 87.6% 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 2024270 
(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))