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

Percentage Accurate: 77.5% → 99.5%

Time: 4.6s

Alternatives: 7

Speedup: 0.6×

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: 77.5% 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: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-311}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4e-311)
   (- (* 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 <= -4e-311) {
		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 <= (-4d-311)) 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 <= -4e-311) {
		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 <= -4e-311:
		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 <= -4e-311)
		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 <= -4e-311)
		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, -4e-311], 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 < -3.99999999999979e-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 -3.99999999999979e-311 < y

   1. Initial program 77.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 2: 90.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-94}:\\ \;\;\;\;\mathsf{fma}\left(-x, \log \left(\frac{y}{x}\right), -z\right)\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-304}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.9e-94)
   (fma (- x) (log (/ y x)) (- z))
   (if (<= x -1e-304) (- z) (- (* x (- (log x) (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.9e-94) {
		tmp = fma(-x, log((y / x)), -z);
	} else if (x <= -1e-304) {
		tmp = -z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.9e-94)
		tmp = fma(Float64(-x), log(Float64(y / x)), Float64(-z));
	elseif (x <= -1e-304)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.9e-94], N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision] + (-z)), $MachinePrecision], If[LessEqual[x, -1e-304], (-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 < -3.9000000000000002e-94

   1. Initial program 81.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 82.5%

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

   4. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift-/.f64 82.5

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

   6. Applied rewrites 82.5%

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

   if -3.9000000000000002e-94 < x < -9.99999999999999971e-305

   1. Initial program 69.6%

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

   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 81.4

         \[\leadsto -z \]

   4. Applied rewrites 81.4%

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

   if -9.99999999999999971e-305 < x

   1. Initial program 77.6%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. 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 98.8%

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

Alternative 3: 87.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+299}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* x (log (/ x y))) z)))
   (if (<= t_0 (- INFINITY)) (- z) (if (<= t_0 5e+299) 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+299) {
		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+299) {
		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+299:
		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+299)
		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+299)
		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+299], 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 5.0000000000000003e299 < (-.f64 (*.f64 x (log.f64 (/.f64 x y))) z)

   1. Initial program 9.2%

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

   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 48.5

         \[\leadsto -z \]

   4. Applied rewrites 48.5%

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

   if -inf.0 < (-.f64 (*.f64 x (log.f64 (/.f64 x y))) z) < 5.0000000000000003e299

   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: 66.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-55}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+50}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.2e-55)
   (* (- x) (log (/ y x)))
   (if (<= x 1.4e+50) (- z) (* (log (/ x y)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.2e-55) {
		tmp = -x * log((y / x));
	} else if (x <= 1.4e+50) {
		tmp = -z;
	} else {
		tmp = log((x / y)) * x;
	}
	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 <= (-5.2d-55)) then
        tmp = -x * log((y / x))
    else if (x <= 1.4d+50) then
        tmp = -z
    else
        tmp = log((x / y)) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.2e-55) {
		tmp = -x * Math.log((y / x));
	} else if (x <= 1.4e+50) {
		tmp = -z;
	} else {
		tmp = Math.log((x / y)) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.2e-55:
		tmp = -x * math.log((y / x))
	elif x <= 1.4e+50:
		tmp = -z
	else:
		tmp = math.log((x / y)) * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.2e-55)
		tmp = Float64(Float64(-x) * log(Float64(y / x)));
	elseif (x <= 1.4e+50)
		tmp = Float64(-z);
	else
		tmp = Float64(log(Float64(x / y)) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5.2e-55)
		tmp = -x * log((y / x));
	elseif (x <= 1.4e+50)
		tmp = -z;
	else
		tmp = log((x / y)) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.2e-55], N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.4e+50], (-z), N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.1999999999999998e-55

   1. Initial program 80.3%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 81.9%

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

   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. lower-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 55.9

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

   6. Applied rewrites 55.9%

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

   if -5.1999999999999998e-55 < x < 1.3999999999999999e50

   1. Initial program 77.6%

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

   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 74.0

         \[\leadsto -z \]

   4. Applied rewrites 74.0%

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

   if 1.3999999999999999e50 < x

   1. Initial program 73.4%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. log-div N/A

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

      4. metadata-eval N/A

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

      5. associate-+r- N/A

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

      6. mul-1-neg N/A

         \[\leadsto \left(\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right) + 0\right) - \log y\right) \cdot x \]

      7. log-rec N/A

         \[\leadsto \left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right) + 0\right) - \log y\right) \cdot x \]

      8. remove-double-neg N/A

         \[\leadsto \left(\left(\log x + 0\right) - \log y\right) \cdot x \]

      9. metadata-eval N/A

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

      10. sum-log N/A

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

      11. log-div N/A

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

      12. associate-*r/ N/A

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

      13. *-commutative N/A

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

      14. sum-log N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. distribute-rgt-in N/A

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

   4. Applied rewrites 57.1%

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

Alternative 5: 65.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+21}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-81}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.35e+21) (- z) (if (<= z 6e-81) (* (log (/ x y)) x) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.35e+21) {
		tmp = -z;
	} else if (z <= 6e-81) {
		tmp = log((x / 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 <= (-1.35d+21)) then
        tmp = -z
    else if (z <= 6d-81) then
        tmp = log((x / y)) * x
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.35e+21) {
		tmp = -z;
	} else if (z <= 6e-81) {
		tmp = Math.log((x / y)) * x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.35e+21:
		tmp = -z
	elif z <= 6e-81:
		tmp = math.log((x / y)) * x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.35e+21)
		tmp = Float64(-z);
	elseif (z <= 6e-81)
		tmp = Float64(log(Float64(x / y)) * x);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.35e+21)
		tmp = -z;
	elseif (z <= 6e-81)
		tmp = log((x / y)) * x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.35e+21], (-z), If[LessEqual[z, 6e-81], N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -1.35e21 or 5.9999999999999998e-81 < z

   1. Initial program 77.8%

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

   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 70.5

         \[\leadsto -z \]

   4. Applied rewrites 70.5%

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

   if -1.35e21 < z < 5.9999999999999998e-81

   1. Initial program 77.2%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. log-div N/A

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

      4. metadata-eval N/A

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

      5. associate-+r- N/A

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

      6. mul-1-neg N/A

         \[\leadsto \left(\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right) + 0\right) - \log y\right) \cdot x \]

      7. log-rec N/A

         \[\leadsto \left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right) + 0\right) - \log y\right) \cdot x \]

      8. remove-double-neg N/A

         \[\leadsto \left(\left(\log x + 0\right) - \log y\right) \cdot x \]

      9. metadata-eval N/A

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

      10. sum-log N/A

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

      11. log-div N/A

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

      12. associate-*r/ N/A

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

      13. *-commutative N/A

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

      14. sum-log N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. distribute-rgt-in N/A

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

   4. Applied rewrites 61.9%

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

Alternative 6: 49.6% accurate, 8.0× 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 77.5%

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

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 49.6

      \[\leadsto -z \]

4. Applied rewrites 49.6%

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

Alternative 7: 2.2% accurate, 15.7× 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 z
end

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 77.5%

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

3. Applied rewrites 16.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z \cdot z, z, {\left(\log \left(\frac{x}{y}\right) \cdot x\right)}^{3}\right)}{\mathsf{fma}\left(\log \left(\frac{x}{y}\right) \cdot x, \log \left(\frac{x}{y}\right) \cdot x - \left(-z\right), z \cdot z\right)}} \]

4. Taylor expanded in x around 0

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

6. Applied rewrites 2.2%

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

Reproduce

herbie shell --seed 2025119 
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64
  (- (* x (log (/ x y))) z))