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

Percentage Accurate: 77.4% → 99.4%

Time: 4.8s

Alternatives: 9

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.4% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -9.999999999999969e-311

   1. Initial program 77.3%

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

      1. lift-/.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. 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. mul-1-neg N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-log.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-neg.f64 99.5

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

   3. Applied rewrites 99.5%

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

   if -9.999999999999969e-311 < y

   1. Initial program 77.5%

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

   3. Applied rewrites 99.2%

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

Alternative 2: 86.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+231}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \left(\log y - \log x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY))
     (- z)
     (if (<= t_0 5e+231) (- t_0 z) (* (- x) (- (log y) (log x)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_0 <= 5e+231) {
		tmp = t_0 - z;
	} else {
		tmp = -x * (log(y) - log(x));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = -z;
	} else if (t_0 <= 5e+231) {
		tmp = t_0 - z;
	} else {
		tmp = -x * (Math.log(y) - Math.log(x));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -z
	elif t_0 <= 5e+231:
		tmp = t_0 - z
	else:
		tmp = -x * (math.log(y) - math.log(x))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(-z);
	elseif (t_0 <= 5e+231)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(Float64(-x) * Float64(log(y) - log(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = -z;
	elseif (t_0 <= 5e+231)
		tmp = t_0 - z;
	else
		tmp = -x * (log(y) - log(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], (-z), If[LessEqual[t$95$0, 5e+231], N[(t$95$0 - z), $MachinePrecision], N[((-x) * N[(N[Log[y], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.3%

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

   2. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 46.9

         \[\leadsto -z \]

   4. Applied rewrites 46.9%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 5.00000000000000028e231

   1. Initial program 99.7%

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

   if 5.00000000000000028e231 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 26.6%

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

      1. lift-/.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. 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. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. frac-2neg N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*r/ N/A

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

      12. log-div N/A

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

      13. sum-log N/A

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

      14. remove-double-neg N/A

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

      15. log-rec N/A

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

      16. mul-1-neg N/A

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

      17. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(-1 \cdot \log \left(\frac{1}{x}\right) + \color{blue}{0}\right) - \log y\right) - z \]

      18. associate-+r- N/A

          \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(0 - \log y\right)\right)} - z \]

      19. metadata-eval N/A

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

      20. log-div N/A

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

      21. +-commutative N/A

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

      22. log-div N/A

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

      23. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(\color{blue}{0} - \log y\right) + -1 \cdot \log \left(\frac{1}{x}\right)\right) - z \]

      24. associate-+l- N/A

          \[\leadsto x \cdot \color{blue}{\left(0 - \left(\log y - -1 \cdot \log \left(\frac{1}{x}\right)\right)\right)} - z \]

      25. lower--.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(0 - \left(\log y - -1 \cdot \log \left(\frac{1}{x}\right)\right)\right)} - z \]

   3. Applied rewrites 29.4%

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

   4. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-/.f64 26.6

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

   6. Applied rewrites 26.6%

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

      1. lift-/.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. log-div N/A

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

      4. lower--.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 54.1

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

   8. Applied rewrites 54.1%

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

Alternative 3: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+273}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* x (log (/ x y))) z)))
   (if (<= t_0 (- INFINITY)) (- z) (if (<= t_0 5e+273) t_0 (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = (x * log((x / y))) - z;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_0 <= 5e+273) {
		tmp = t_0;
	} else {
		tmp = -z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = (x * Math.log((x / y))) - z;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = -z;
	} else if (t_0 <= 5e+273) {
		tmp = t_0;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * math.log((x / y))) - z
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -z
	elif t_0 <= 5e+273:
		tmp = t_0
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * log(Float64(x / y))) - z)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(-z);
	elseif (t_0 <= 5e+273)
		tmp = t_0;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * log((x / y))) - z;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = -z;
	elseif (t_0 <= 5e+273)
		tmp = t_0;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], (-z), If[LessEqual[t$95$0, 5e+273], t$95$0, (-z)]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x (log.f64 (/.f64 x y))) z) < -inf.0 or 4.99999999999999961e273 < (-.f64 (*.f64 x (log.f64 (/.f64 x y))) z)

   1. Initial program 17.6%

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

   2. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 47.3

         \[\leadsto -z \]

   4. Applied rewrites 47.3%

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

   if -inf.0 < (-.f64 (*.f64 x (log.f64 (/.f64 x y))) z) < 4.99999999999999961e273

   1. Initial program 99.7%

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

Alternative 4: 93.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+183}:\\ \;\;\;\;\left(-x\right) \cdot \left(\log \left(-y\right) - \log \left(-x\right)\right)\\ \mathbf{elif}\;x \leq -1.08 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-307}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.6e+183)
   (* (- x) (- (log (- y)) (log (- x))))
   (if (<= x -1.08e-110)
     (- (* x (log (/ x y))) z)
     (if (<= x -2e-307) (- z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.6e+183) {
		tmp = -x * (log(-y) - log(-x));
	} else if (x <= -1.08e-110) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -2e-307) {
		tmp = -z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-3.6d+183)) then
        tmp = -x * (log(-y) - log(-x))
    else if (x <= (-1.08d-110)) then
        tmp = (x * log((x / y))) - z
    else if (x <= (-2d-307)) then
        tmp = -z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.6e+183) {
		tmp = -x * (Math.log(-y) - Math.log(-x));
	} else if (x <= -1.08e-110) {
		tmp = (x * Math.log((x / y))) - z;
	} else if (x <= -2e-307) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.6e+183:
		tmp = -x * (math.log(-y) - math.log(-x))
	elif x <= -1.08e-110:
		tmp = (x * math.log((x / y))) - z
	elif x <= -2e-307:
		tmp = -z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.6e+183)
		tmp = Float64(Float64(-x) * Float64(log(Float64(-y)) - log(Float64(-x))));
	elseif (x <= -1.08e-110)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -2e-307)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.6e+183)
		tmp = -x * (log(-y) - log(-x));
	elseif (x <= -1.08e-110)
		tmp = (x * log((x / y))) - z;
	elseif (x <= -2e-307)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.6e+183], N[((-x) * N[(N[Log[(-y)], $MachinePrecision] - N[Log[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.08e-110], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -2e-307], (-z), N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.60000000000000023e183

   1. Initial program 64.2%

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

      1. lift-/.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. 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. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. frac-2neg N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*r/ N/A

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

      12. log-div N/A

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

      13. sum-log N/A

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

      14. remove-double-neg N/A

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

      15. log-rec N/A

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

      16. mul-1-neg N/A

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

      17. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(-1 \cdot \log \left(\frac{1}{x}\right) + \color{blue}{0}\right) - \log y\right) - z \]

      18. associate-+r- N/A

          \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(0 - \log y\right)\right)} - z \]

      19. metadata-eval N/A

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

      20. log-div N/A

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

      21. +-commutative N/A

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

      22. log-div N/A

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

      23. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(\color{blue}{0} - \log y\right) + -1 \cdot \log \left(\frac{1}{x}\right)\right) - z \]

      24. associate-+l- N/A

          \[\leadsto x \cdot \color{blue}{\left(0 - \left(\log y - -1 \cdot \log \left(\frac{1}{x}\right)\right)\right)} - z \]

      25. lower--.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(0 - \left(\log y - -1 \cdot \log \left(\frac{1}{x}\right)\right)\right)} - z \]

   3. Applied rewrites 65.9%

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

   4. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-/.f64 59.4

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

   6. Applied rewrites 59.4%

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

      1. lift-/.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. log-div N/A

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

      6. mul-1-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. mul-1-neg N/A

         \[\leadsto \left(-x\right) \cdot \left(\log \left(-1 \cdot y\right) - \log \left(-1 \cdot x\right)\right) \]

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-log.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lift-neg.f64 N/A

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

      15. lower-log.f64 89.5

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

   8. Applied rewrites 89.5%

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

   if -3.60000000000000023e183 < x < -1.07999999999999995e-110

   1. Initial program 87.6%

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

   if -1.07999999999999995e-110 < x < -1.99999999999999982e-307

   1. Initial program 71.2%

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

   2. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 84.8

         \[\leadsto -z \]

   4. Applied rewrites 84.8%

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

   if -1.99999999999999982e-307 < x

   1. Initial program 77.3%

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

      1. lift-/.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. 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. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. frac-2neg N/A

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

      9. *-commutative N/A

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

      10. sum-log N/A

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

      11. +-commutative N/A

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

      12. flip-+ N/A

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

      13. log-rec N/A

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

      14. mul-1-neg N/A

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

      15. log-rec N/A

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

      16. mul-1-neg N/A

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

      17. log-rec N/A

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

      18. mul-1-neg N/A

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

      19. flip-+ N/A

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

      20. fp-cancel-sign-sub-inv N/A

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

   3. Applied rewrites 99.3%

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

Alternative 5: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \left(0 - \log \left(\frac{y}{x}\right)\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-307}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.7e-110)
   (- (* x (- 0.0 (log (/ y x)))) z)
   (if (<= x -2e-307) (- z) (- (* x (- (log x) (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.7e-110) {
		tmp = (x * (0.0 - log((y / x)))) - z;
	} else if (x <= -2e-307) {
		tmp = -z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.7d-110)) then
        tmp = (x * (0.0d0 - log((y / x)))) - z
    else if (x <= (-2d-307)) then
        tmp = -z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.7e-110) {
		tmp = (x * (0.0 - Math.log((y / x)))) - z;
	} else if (x <= -2e-307) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.7e-110:
		tmp = (x * (0.0 - math.log((y / x)))) - z
	elif x <= -2e-307:
		tmp = -z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.7e-110)
		tmp = Float64(Float64(x * Float64(0.0 - log(Float64(y / x)))) - z);
	elseif (x <= -2e-307)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.7e-110)
		tmp = (x * (0.0 - log((y / x)))) - z;
	elseif (x <= -2e-307)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.7e-110], N[(N[(x * N[(0.0 - N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -2e-307], (-z), N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7000000000000001e-110

   1. Initial program 80.5%

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

      1. lift-/.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. 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. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. frac-2neg N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*r/ N/A

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

      12. log-div N/A

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

      13. sum-log N/A

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

      14. remove-double-neg N/A

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

      15. log-rec N/A

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

      16. mul-1-neg N/A

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

      17. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(-1 \cdot \log \left(\frac{1}{x}\right) + \color{blue}{0}\right) - \log y\right) - z \]

      18. associate-+r- N/A

          \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(0 - \log y\right)\right)} - z \]

      19. metadata-eval N/A

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

      20. log-div N/A

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

      21. +-commutative N/A

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

      22. log-div N/A

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

      23. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(\color{blue}{0} - \log y\right) + -1 \cdot \log \left(\frac{1}{x}\right)\right) - z \]

      24. associate-+l- N/A

          \[\leadsto x \cdot \color{blue}{\left(0 - \left(\log y - -1 \cdot \log \left(\frac{1}{x}\right)\right)\right)} - z \]

      25. lower--.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(0 - \left(\log y - -1 \cdot \log \left(\frac{1}{x}\right)\right)\right)} - z \]

   3. Applied rewrites 81.5%

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

   if -1.7000000000000001e-110 < x < -1.99999999999999982e-307

   1. Initial program 71.2%

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

   2. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 84.8

         \[\leadsto -z \]

   4. Applied rewrites 84.8%

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

   if -1.99999999999999982e-307 < x

   1. Initial program 77.3%

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

      1. lift-/.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. 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. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. frac-2neg N/A

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

      9. *-commutative N/A

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

      10. sum-log N/A

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

      11. +-commutative N/A

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

      12. flip-+ N/A

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

      13. log-rec N/A

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

      14. mul-1-neg N/A

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

      15. log-rec N/A

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

      16. mul-1-neg N/A

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

      17. log-rec N/A

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

      18. mul-1-neg N/A

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

      19. flip-+ N/A

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

      20. fp-cancel-sign-sub-inv N/A

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

   3. Applied rewrites 99.3%

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

Alternative 6: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1d-310)) then
        tmp = (x * (log(-x) - log(-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 <= -1e-310) {
		tmp = (x * (Math.log(-x) - Math.log(-y))) - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1e-310:
		tmp = (x * (math.log(-x) - math.log(-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 <= -1e-310)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-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 <= -1e-310)
		tmp = (x * (log(-x) - log(-y))) - z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -9.999999999999969e-311

   1. Initial program 77.3%

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

      1. lift-/.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. 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. mul-1-neg N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-log.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-neg.f64 99.5

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

   3. Applied rewrites 99.5%

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

   if -9.999999999999969e-311 < y

   1. Initial program 77.5%

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

      1. lift-/.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. 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. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. frac-2neg N/A

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

      9. *-commutative N/A

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

      10. sum-log N/A

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

      11. +-commutative N/A

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

      12. flip-+ N/A

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

      13. log-rec N/A

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

      14. mul-1-neg N/A

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

      15. log-rec N/A

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

      16. mul-1-neg N/A

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

      17. log-rec N/A

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

      18. mul-1-neg N/A

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

      19. flip-+ N/A

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

      20. fp-cancel-sign-sub-inv N/A

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

   3. Applied rewrites 99.5%

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

Alternative 7: 64.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{+119}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 70000000000:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.4e+119)
   (- z)
   (if (<= z 70000000000.0) (* (- x) (log (/ y x))) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.4e+119) {
		tmp = -z;
	} else if (z <= 70000000000.0) {
		tmp = -x * log((y / x));
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-6.4d+119)) then
        tmp = -z
    else if (z <= 70000000000.0d0) then
        tmp = -x * log((y / x))
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.4e+119) {
		tmp = -z;
	} else if (z <= 70000000000.0) {
		tmp = -x * Math.log((y / x));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -6.4e+119:
		tmp = -z
	elif z <= 70000000000.0:
		tmp = -x * math.log((y / x))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6.4e+119)
		tmp = Float64(-z);
	elseif (z <= 70000000000.0)
		tmp = Float64(Float64(-x) * log(Float64(y / x)));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -6.4e+119)
		tmp = -z;
	elseif (z <= 70000000000.0)
		tmp = -x * log((y / x));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6.4e+119], (-z), If[LessEqual[z, 70000000000.0], N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -6.39999999999999979e119 or 7e10 < z

   1. Initial program 77.5%

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

   2. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 77.5

         \[\leadsto -z \]

   4. Applied rewrites 77.5%

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

   if -6.39999999999999979e119 < z < 7e10

   1. Initial program 77.3%

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

      1. lift-/.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. 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. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. frac-2neg N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*r/ N/A

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

      12. log-div N/A

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

      13. sum-log N/A

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

      14. remove-double-neg N/A

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

      15. log-rec N/A

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

      16. mul-1-neg N/A

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

      17. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(-1 \cdot \log \left(\frac{1}{x}\right) + \color{blue}{0}\right) - \log y\right) - z \]

      18. associate-+r- N/A

          \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(0 - \log y\right)\right)} - z \]

      19. metadata-eval N/A

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

      20. log-div N/A

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

      21. +-commutative N/A

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

      22. log-div N/A

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

      23. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(\color{blue}{0} - \log y\right) + -1 \cdot \log \left(\frac{1}{x}\right)\right) - z \]

      24. associate-+l- N/A

          \[\leadsto x \cdot \color{blue}{\left(0 - \left(\log y - -1 \cdot \log \left(\frac{1}{x}\right)\right)\right)} - z \]

      25. lower--.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(0 - \left(\log y - -1 \cdot \log \left(\frac{1}{x}\right)\right)\right)} - z \]

   3. Applied rewrites 76.9%

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

   4. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-/.f64 55.6

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

   6. Applied rewrites 55.6%

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

Alternative 8: 52.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-307}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2e-307) (- z) (fma (log x) x (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2e-307) {
		tmp = -z;
	} else {
		tmp = fma(log(x), x, -z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2e-307)
		tmp = Float64(-z);
	else
		tmp = fma(log(x), x, Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2e-307], (-z), N[(N[Log[x], $MachinePrecision] * x + (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.99999999999999982e-307

   1. Initial program 77.5%

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

   2. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 50.6

         \[\leadsto -z \]

   4. Applied rewrites 50.6%

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

   if -1.99999999999999982e-307 < x

   1. Initial program 77.3%

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

   3. Applied rewrites 99.1%

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

   4. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 54.3

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

   6. Applied rewrites 54.3%

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

Alternative 9: 49.4% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

2. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 49.4

      \[\leadsto -z \]

4. Applied rewrites 49.4%

   \[\leadsto \color{blue}{-z} \]
5. 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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 2025095 
(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))