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

Percentage Accurate: 76.8% → 99.4%

Time: 8.9s

Alternatives: 8

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: 76.8% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\log \left(-x\right) \cdot x - \log \left(-y\right) \cdot x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4e-310)
   (- (- (* (log (- x)) x) (* (log (- y)) x)) z)
   (- (* (- (log x) (log y)) x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e-310) {
		tmp = ((log(-x) * 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 <= (-4d-310)) then
        tmp = ((log(-x) * 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 <= -4e-310) {
		tmp = ((Math.log(-x) * 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 <= -4e-310:
		tmp = ((math.log(-x) * 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 <= -4e-310)
		tmp = Float64(Float64(Float64(log(Float64(-x)) * x) - Float64(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 <= -4e-310)
		tmp = ((log(-x) * 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, -4e-310], N[(N[(N[(N[Log[(-x)], $MachinePrecision] * x), $MachinePrecision] - N[(N[Log[(-y)], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.999999999999988e-310

   1. Initial program 79.7%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 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

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. unpow2 N/A

         \[\leadsto x \cdot \frac{\color{blue}{{\log x}^{2}} - \log y \cdot \log y}{\log x + \log y} - z \]

      5. lift-pow.f64 N/A

         \[\leadsto x \cdot \frac{\color{blue}{{\log x}^{2}} - \log y \cdot \log y}{\log x + \log y} - z \]

      6. unpow2 N/A

         \[\leadsto x \cdot \frac{{\log x}^{2} - \color{blue}{{\log y}^{2}}}{\log x + \log y} - z \]

      7. lift-pow.f64 N/A

         \[\leadsto x \cdot \frac{{\log x}^{2} - \color{blue}{{\log y}^{2}}}{\log x + \log y} - z \]

      8. lift--.f64 N/A

         \[\leadsto x \cdot \frac{\color{blue}{{\log x}^{2} - {\log y}^{2}}}{\log x + \log y} - z \]

      9. lift-log.f64 N/A

         \[\leadsto x \cdot \frac{{\log x}^{2} - {\log y}^{2}}{\color{blue}{\log x} + \log y} - z \]

      10. lift-log.f64 N/A

          \[\leadsto x \cdot \frac{{\log x}^{2} - {\log y}^{2}}{\log x + \color{blue}{\log y}} - z \]

      11. sum-log N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. clear-num N/A

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

      16. un-div-inv N/A

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

      17. lower-/.f64 N/A

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

      18. clear-num N/A

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

   6. Applied rewrites 79.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. /-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. remove-double-div N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. inv-pow N/A

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

      10. lower-pow.f64 79.7

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

   8. Applied rewrites 79.7%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow-1 N/A

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

      4. associate-/r/ N/A

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

      5. /-rgt-identity N/A

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

      6. *-commutative N/A

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

      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. frac-2neg N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. diff-log N/A

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

      13. lift-log.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. sub-neg N/A

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

      16. distribute-rgt-in N/A

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

      17. lower-+.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 99.6

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

   10. Applied rewrites 99.6%

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

   if -3.999999999999988e-310 < y

   1. Initial program 77.3%

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

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\log \left(-x\right) \cdot x - \log \left(-y\right) \cdot x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right) \cdot x\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t\_0 \leq 10^{+308}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log (/ x y)) x)))
   (if (<= t_0 (- INFINITY)) (- z) (if (<= t_0 1e+308) (- 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+308) {
		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+308) {
		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+308:
		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+308)
		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+308)
		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+308], 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 1e308 < (*.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 50.9

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

   5. Applied rewrites 50.9%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1e308

   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 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(\frac{x}{y}\right) \cdot x \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;\log \left(\frac{x}{y}\right) \cdot x \leq 10^{+308}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+197}:\\ \;\;\;\;\left(\log \left(-x\right) - \log \left(-y\right)\right) \cdot x\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-167}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.8e+197)
   (* (- (log (- x)) (log (- y))) x)
   (if (<= x -2.7e-167)
     (- (* (log (/ x y)) x) z)
     (if (<= x -2e-309) (- z) (- (* (- (log x) (log y)) x) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.8e+197) {
		tmp = (log(-x) - log(-y)) * x;
	} else if (x <= -2.7e-167) {
		tmp = (log((x / y)) * x) - z;
	} else if (x <= -2e-309) {
		tmp = -z;
	} else {
		tmp = ((log(x) - log(y)) * x) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.8d+197)) then
        tmp = (log(-x) - log(-y)) * x
    else if (x <= (-2.7d-167)) then
        tmp = (log((x / y)) * x) - z
    else if (x <= (-2d-309)) then
        tmp = -z
    else
        tmp = ((log(x) - log(y)) * x) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.8e+197) {
		tmp = (Math.log(-x) - Math.log(-y)) * x;
	} else if (x <= -2.7e-167) {
		tmp = (Math.log((x / y)) * x) - z;
	} else if (x <= -2e-309) {
		tmp = -z;
	} else {
		tmp = ((Math.log(x) - Math.log(y)) * x) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.8e+197:
		tmp = (math.log(-x) - math.log(-y)) * x
	elif x <= -2.7e-167:
		tmp = (math.log((x / y)) * x) - z
	elif x <= -2e-309:
		tmp = -z
	else:
		tmp = ((math.log(x) - math.log(y)) * x) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.8e+197)
		tmp = Float64(Float64(log(Float64(-x)) - log(Float64(-y))) * x);
	elseif (x <= -2.7e-167)
		tmp = Float64(Float64(log(Float64(x / y)) * x) - z);
	elseif (x <= -2e-309)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(Float64(log(x) - log(y)) * x) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.8e+197)
		tmp = (log(-x) - log(-y)) * x;
	elseif (x <= -2.7e-167)
		tmp = (log((x / y)) * x) - z;
	elseif (x <= -2e-309)
		tmp = -z;
	else
		tmp = ((log(x) - log(y)) * x) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.8e+197], N[(N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, -2.7e-167], N[(N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -2e-309], (-z), N[(N[(N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.79999999999999991e197

   1. Initial program 59.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 59.8

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

   5. Applied rewrites 59.8%

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

   7. Applied rewrites 89.5%

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

   if -1.79999999999999991e197 < x < -2.7000000000000001e-167

   1. Initial program 88.7%

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

   if -2.7000000000000001e-167 < x < -1.9999999999999988e-309

   1. Initial program 74.5%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 97.5

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

   5. Applied rewrites 97.5%

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

   if -1.9999999999999988e-309 < x

   1. Initial program 77.3%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+197}:\\ \;\;\;\;\left(\log \left(-x\right) - \log \left(-y\right)\right) \cdot x\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-167}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(-x\right), x, \log \left(-y\right) \cdot \left(-x\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4e-310)
   (- (fma (log (- x)) x (* (log (- y)) (- x))) z)
   (- (* (- (log x) (log y)) x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e-310) {
		tmp = fma(log(-x), x, (log(-y) * -x)) - z;
	} else {
		tmp = ((log(x) - log(y)) * x) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4e-310)
		tmp = Float64(fma(log(Float64(-x)), x, Float64(log(Float64(-y)) * Float64(-x))) - z);
	else
		tmp = Float64(Float64(Float64(log(x) - log(y)) * x) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4e-310], N[(N[(N[Log[(-x)], $MachinePrecision] * x + N[(N[Log[(-y)], $MachinePrecision] * (-x)), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.999999999999988e-310

   1. Initial program 79.7%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 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

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. unpow2 N/A

         \[\leadsto x \cdot \frac{\color{blue}{{\log x}^{2}} - \log y \cdot \log y}{\log x + \log y} - z \]

      5. lift-pow.f64 N/A

         \[\leadsto x \cdot \frac{\color{blue}{{\log x}^{2}} - \log y \cdot \log y}{\log x + \log y} - z \]

      6. unpow2 N/A

         \[\leadsto x \cdot \frac{{\log x}^{2} - \color{blue}{{\log y}^{2}}}{\log x + \log y} - z \]

      7. lift-pow.f64 N/A

         \[\leadsto x \cdot \frac{{\log x}^{2} - \color{blue}{{\log y}^{2}}}{\log x + \log y} - z \]

      8. lift--.f64 N/A

         \[\leadsto x \cdot \frac{\color{blue}{{\log x}^{2} - {\log y}^{2}}}{\log x + \log y} - z \]

      9. lift-log.f64 N/A

         \[\leadsto x \cdot \frac{{\log x}^{2} - {\log y}^{2}}{\color{blue}{\log x} + \log y} - z \]

      10. lift-log.f64 N/A

          \[\leadsto x \cdot \frac{{\log x}^{2} - {\log y}^{2}}{\log x + \color{blue}{\log y}} - z \]

      11. sum-log N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. clear-num N/A

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

      16. un-div-inv N/A

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

      17. lower-/.f64 N/A

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

      18. clear-num N/A

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

   6. Applied rewrites 79.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. /-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. remove-double-div N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. inv-pow N/A

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

      10. lower-pow.f64 79.7

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

   8. Applied rewrites 79.7%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow-1 N/A

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

      4. associate-/r/ N/A

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

      5. /-rgt-identity N/A

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

      6. *-commutative N/A

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

      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. frac-2neg N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. diff-log N/A

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

      13. lift-log.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. sub-neg N/A

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

      16. distribute-rgt-in N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 99.6

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

   10. Applied rewrites 99.6%

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

   if -3.999999999999988e-310 < y

   1. Initial program 77.3%

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

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(-x\right), x, \log \left(-y\right) \cdot \left(-x\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-167}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.7e-167)
   (- (* (log (/ x y)) x) z)
   (if (<= x -2e-309) (- z) (- (* (- (log x) (log y)) x) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.7e-167) {
		tmp = (log((x / y)) * x) - z;
	} else if (x <= -2e-309) {
		tmp = -z;
	} else {
		tmp = ((log(x) - log(y)) * x) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-2.7d-167)) then
        tmp = (log((x / y)) * x) - z
    else if (x <= (-2d-309)) then
        tmp = -z
    else
        tmp = ((log(x) - log(y)) * x) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.7e-167) {
		tmp = (Math.log((x / y)) * x) - z;
	} else if (x <= -2e-309) {
		tmp = -z;
	} else {
		tmp = ((Math.log(x) - Math.log(y)) * x) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.7e-167:
		tmp = (math.log((x / y)) * x) - z
	elif x <= -2e-309:
		tmp = -z
	else:
		tmp = ((math.log(x) - math.log(y)) * x) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.7e-167)
		tmp = Float64(Float64(log(Float64(x / y)) * x) - z);
	elseif (x <= -2e-309)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(Float64(log(x) - log(y)) * x) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.7e-167)
		tmp = (log((x / y)) * x) - z;
	elseif (x <= -2e-309)
		tmp = -z;
	else
		tmp = ((log(x) - log(y)) * x) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.7e-167], N[(N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -2e-309], (-z), N[(N[(N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.7000000000000001e-167

   1. Initial program 82.0%

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

   if -2.7000000000000001e-167 < x < -1.9999999999999988e-309

   1. Initial program 74.5%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 97.5

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

   5. Applied rewrites 97.5%

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

   if -1.9999999999999988e-309 < x

   1. Initial program 77.3%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 99.6

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

   4. Applied rewrites 99.6%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-167}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\log \left(-x\right) - \log \left(-y\right)\right) \cdot x - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4e-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 <= -4e-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 <= (-4d-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 <= -4e-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 <= -4e-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 <= -4e-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 <= -4e-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, -4e-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 < -3.999999999999988e-310

   1. Initial program 79.7%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-neg.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if -3.999999999999988e-310 < y

   1. Initial program 77.3%

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

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left(\log \left(-x\right) - \log \left(-y\right)\right) \cdot x - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right) \cdot x\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{-67}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+42}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log (/ x y)) x)))
   (if (<= x -3.2e-67) t_0 (if (<= x 3.3e+42) (- z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y)) * x;
	double tmp;
	if (x <= -3.2e-67) {
		tmp = t_0;
	} else if (x <= 3.3e+42) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log((x / y)) * x
    if (x <= (-3.2d-67)) then
        tmp = t_0
    else if (x <= 3.3d+42) then
        tmp = -z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.log((x / y)) * x;
	double tmp;
	if (x <= -3.2e-67) {
		tmp = t_0;
	} else if (x <= 3.3e+42) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.log((x / y)) * x
	tmp = 0
	if x <= -3.2e-67:
		tmp = t_0
	elif x <= 3.3e+42:
		tmp = -z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(log(Float64(x / y)) * x)
	tmp = 0.0
	if (x <= -3.2e-67)
		tmp = t_0;
	elseif (x <= 3.3e+42)
		tmp = Float64(-z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = log((x / y)) * x;
	tmp = 0.0;
	if (x <= -3.2e-67)
		tmp = t_0;
	elseif (x <= 3.3e+42)
		tmp = -z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -3.2e-67], t$95$0, If[LessEqual[x, 3.3e+42], (-z), t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.20000000000000021e-67 or 3.2999999999999999e42 < x

   1. Initial program 75.5%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 60.5

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

   5. Applied rewrites 60.5%

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

   if -3.20000000000000021e-67 < x < 3.2999999999999999e42

   1. Initial program 80.9%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 76.5

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

   5. Applied rewrites 76.5%

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

Alternative 8: 49.7% 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 78.5%

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

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 52.6

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

5. Applied rewrites 52.6%

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

Developer Target 1: 88.4% 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 2024294 
(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))