Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B

Percentage Accurate: 72.1% → 99.8%

Time: 5.8s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x y)
  :precision binary64
  (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))

C

double code(double x, double y) {
	return 1.0 - log((1.0 - ((x - y) / (1.0 - y))));
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 - log((1.0d0 - ((x - y) / (1.0d0 - y))))
end function

Java

public static double code(double x, double y) {
	return 1.0 - Math.log((1.0 - ((x - y) / (1.0 - y))));
}

Python

def code(x, y):
	return 1.0 - math.log((1.0 - ((x - y) / (1.0 - y))))

Julia

function code(x, y)
	return Float64(1.0 - log(Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y)))))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - log((1.0 - ((x - y) / (1.0 - y))));
end

Wolfram

code[x_, y_] := N[(1.0 - N[Log[N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 72.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
  :precision binary64
  (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))

C

double code(double x, double y) {
	return 1.0 - log((1.0 - ((x - y) / (1.0 - y))));
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 - log((1.0d0 - ((x - y) / (1.0d0 - y))))
end function

Java

public static double code(double x, double y) {
	return 1.0 - Math.log((1.0 - ((x - y) / (1.0 - y))));
}

Python

def code(x, y):
	return 1.0 - math.log((1.0 - ((x - y) / (1.0 - y))))

Julia

function code(x, y)
	return Float64(1.0 - log(Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y)))))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - log((1.0 - ((x - y) / (1.0 - y))));
end

Wolfram

code[x_, y_] := N[(1.0 - N[Log[N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;1 - \log \left(1 - \frac{x - y}{1 - y}\right) \leq 18.5:\\ \;\;\;\;1 - \log \left(\frac{\left(y - x\right) - \left(y - 1\right)}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x - 1}{y}\right)\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (if (<= (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))) 18.5)
  (- 1.0 (log (/ (- (- y x) (- y 1.0)) (- 1.0 y))))
  (- 1.0 (log (/ (- x 1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if ((1.0 - log((1.0 - ((x - y) / (1.0 - y))))) <= 18.5) {
		tmp = 1.0 - log((((y - x) - (y - 1.0)) / (1.0 - y)));
	} else {
		tmp = 1.0 - log(((x - 1.0) / y));
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((1.0d0 - log((1.0d0 - ((x - y) / (1.0d0 - y))))) <= 18.5d0) then
        tmp = 1.0d0 - log((((y - x) - (y - 1.0d0)) / (1.0d0 - y)))
    else
        tmp = 1.0d0 - log(((x - 1.0d0) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((1.0 - Math.log((1.0 - ((x - y) / (1.0 - y))))) <= 18.5) {
		tmp = 1.0 - Math.log((((y - x) - (y - 1.0)) / (1.0 - y)));
	} else {
		tmp = 1.0 - Math.log(((x - 1.0) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (1.0 - math.log((1.0 - ((x - y) / (1.0 - y))))) <= 18.5:
		tmp = 1.0 - math.log((((y - x) - (y - 1.0)) / (1.0 - y)))
	else:
		tmp = 1.0 - math.log(((x - 1.0) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(1.0 - log(Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y))))) <= 18.5)
		tmp = Float64(1.0 - log(Float64(Float64(Float64(y - x) - Float64(y - 1.0)) / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(x - 1.0) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((1.0 - log((1.0 - ((x - y) / (1.0 - y))))) <= 18.5)
		tmp = 1.0 - log((((y - x) - (y - 1.0)) / (1.0 - y)));
	else
		tmp = 1.0 - log(((x - 1.0) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 - N[Log[N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 18.5], N[(1.0 - N[Log[N[(N[(N[(y - x), $MachinePrecision] - N[(y - 1.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))))) < 18.5

   1. Initial program 72.1%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-to-fraction N/A

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

      4. frac-2neg N/A

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

      5. distribute-neg-frac N/A

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

      6. distribute-frac-neg2 N/A

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

      7. sub-negate-rev N/A

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

      8. frac-2neg-rev N/A

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

      9. *-lft-identity N/A

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

      10. sub-negate-rev N/A

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

      11. *-lft-identity N/A

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

      12. *-lft-identity N/A

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

      13. sub-flip-reverse N/A

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

      14. distribute-neg-in N/A

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

      15. remove-double-neg N/A

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

      16. *-lft-identity N/A

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

      17. add-flip-rev N/A

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

      18. sub-negate-rev N/A

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

   3. Applied rewrites 72.4%

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

   if 18.5 < (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)))))

   1. Initial program 72.1%

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

   2. Taylor expanded in y around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 40.9%

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

   4. Applied rewrites 40.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lower--.f64 40.9%

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

   6. Applied rewrites 40.9%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := 1 - \log \left(1 - \frac{x - y}{1 - y}\right)\\ \mathbf{if}\;t\_0 \leq 18.5:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x - 1}{y}\right)\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y)))))))
  (if (<= t_0 18.5) t_0 (- 1.0 (log (/ (- x 1.0) y))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log((1.0 - ((x - y) / (1.0 - y))));
	double tmp;
	if (t_0 <= 18.5) {
		tmp = t_0;
	} else {
		tmp = 1.0 - log(((x - 1.0) / y));
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - log((1.0d0 - ((x - y) / (1.0d0 - y))))
    if (t_0 <= 18.5d0) then
        tmp = t_0
    else
        tmp = 1.0d0 - log(((x - 1.0d0) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - Math.log((1.0 - ((x - y) / (1.0 - y))));
	double tmp;
	if (t_0 <= 18.5) {
		tmp = t_0;
	} else {
		tmp = 1.0 - Math.log(((x - 1.0) / y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - math.log((1.0 - ((x - y) / (1.0 - y))))
	tmp = 0
	if t_0 <= 18.5:
		tmp = t_0
	else:
		tmp = 1.0 - math.log(((x - 1.0) / y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - log(Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y)))))
	tmp = 0.0
	if (t_0 <= 18.5)
		tmp = t_0;
	else
		tmp = Float64(1.0 - log(Float64(Float64(x - 1.0) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - log((1.0 - ((x - y) / (1.0 - y))));
	tmp = 0.0;
	if (t_0 <= 18.5)
		tmp = t_0;
	else
		tmp = 1.0 - log(((x - 1.0) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Log[N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 18.5], t$95$0, N[(1.0 - N[Log[N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))))) < 18.5

   1. Initial program 72.1%

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

   if 18.5 < (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)))))

   1. Initial program 72.1%

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

   2. Taylor expanded in y around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 40.9%

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

   4. Applied rewrites 40.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lower--.f64 40.9%

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

   6. Applied rewrites 40.9%

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

Alternative 3: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;1 - \log \left(1 - \frac{x - y}{1 - y}\right) \leq 2:\\ \;\;\;\;1 - \log \left(1 - \frac{x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x - 1}{y}\right)\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (if (<= (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))) 2.0)
  (- 1.0 (log (- 1.0 (/ x (- 1.0 y)))))
  (- 1.0 (log (/ (- x 1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if ((1.0 - log((1.0 - ((x - y) / (1.0 - y))))) <= 2.0) {
		tmp = 1.0 - log((1.0 - (x / (1.0 - y))));
	} else {
		tmp = 1.0 - log(((x - 1.0) / y));
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((1.0d0 - log((1.0d0 - ((x - y) / (1.0d0 - y))))) <= 2.0d0) then
        tmp = 1.0d0 - log((1.0d0 - (x / (1.0d0 - y))))
    else
        tmp = 1.0d0 - log(((x - 1.0d0) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((1.0 - Math.log((1.0 - ((x - y) / (1.0 - y))))) <= 2.0) {
		tmp = 1.0 - Math.log((1.0 - (x / (1.0 - y))));
	} else {
		tmp = 1.0 - Math.log(((x - 1.0) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (1.0 - math.log((1.0 - ((x - y) / (1.0 - y))))) <= 2.0:
		tmp = 1.0 - math.log((1.0 - (x / (1.0 - y))))
	else:
		tmp = 1.0 - math.log(((x - 1.0) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(1.0 - log(Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y))))) <= 2.0)
		tmp = Float64(1.0 - log(Float64(1.0 - Float64(x / Float64(1.0 - y)))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(x - 1.0) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((1.0 - log((1.0 - ((x - y) / (1.0 - y))))) <= 2.0)
		tmp = 1.0 - log((1.0 - (x / (1.0 - y))));
	else
		tmp = 1.0 - log(((x - 1.0) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 - N[Log[N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], N[(1.0 - N[Log[N[(1.0 - N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))))) < 2

   1. Initial program 72.1%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 72.8%

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

   4. Applied rewrites 72.8%

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

   if 2 < (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)))))

   1. Initial program 72.1%

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

   2. Taylor expanded in y around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 40.9%

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

   4. Applied rewrites 40.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lower--.f64 40.9%

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

   6. Applied rewrites 40.9%

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

Alternative 4: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := 1 - \log \left(\frac{x - 1}{y}\right)\\ \mathbf{if}\;y \leq -0.75:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 21500000:\\ \;\;\;\;1 - \left(y + \log \left(1 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- 1.0 (log (/ (- x 1.0) y)))))
  (if (<= y -0.75)
    t_0
    (if (<= y 21500000.0) (- 1.0 (+ y (log (- 1.0 x)))) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log(((x - 1.0) / y));
	double tmp;
	if (y <= -0.75) {
		tmp = t_0;
	} else if (y <= 21500000.0) {
		tmp = 1.0 - (y + log((1.0 - x)));
	} else {
		tmp = t_0;
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - log(((x - 1.0d0) / y))
    if (y <= (-0.75d0)) then
        tmp = t_0
    else if (y <= 21500000.0d0) then
        tmp = 1.0d0 - (y + log((1.0d0 - x)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - Math.log(((x - 1.0) / y));
	double tmp;
	if (y <= -0.75) {
		tmp = t_0;
	} else if (y <= 21500000.0) {
		tmp = 1.0 - (y + Math.log((1.0 - x)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - math.log(((x - 1.0) / y))
	tmp = 0
	if y <= -0.75:
		tmp = t_0
	elif y <= 21500000.0:
		tmp = 1.0 - (y + math.log((1.0 - x)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - log(Float64(Float64(x - 1.0) / y)))
	tmp = 0.0
	if (y <= -0.75)
		tmp = t_0;
	elseif (y <= 21500000.0)
		tmp = Float64(1.0 - Float64(y + log(Float64(1.0 - x))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - log(((x - 1.0) / y));
	tmp = 0.0;
	if (y <= -0.75)
		tmp = t_0;
	elseif (y <= 21500000.0)
		tmp = 1.0 - (y + log((1.0 - x)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Log[N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.75], t$95$0, If[LessEqual[y, 21500000.0], N[(1.0 - N[(y + N[Log[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.75 or 2.15e7 < y

   1. Initial program 72.1%

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

   2. Taylor expanded in y around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 40.9%

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

   4. Applied rewrites 40.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lower--.f64 40.9%

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

   6. Applied rewrites 40.9%

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

   if -0.75 < y < 2.15e7

   1. Initial program 72.1%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-to-fraction N/A

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

      4. frac-2neg N/A

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

      5. distribute-neg-frac N/A

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

      6. distribute-frac-neg2 N/A

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

      7. sub-negate-rev N/A

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

      8. frac-2neg-rev N/A

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

      9. *-lft-identity N/A

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

      10. sub-negate-rev N/A

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

      11. *-lft-identity N/A

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

      12. *-lft-identity N/A

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

      13. sub-flip-reverse N/A

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

      14. distribute-neg-in N/A

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

      15. remove-double-neg N/A

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

      16. *-lft-identity N/A

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

      17. add-flip-rev N/A

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

      18. sub-negate-rev N/A

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

   3. Applied rewrites 72.4%

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

   4. Taylor expanded in x around 0

      \[\leadsto 1 - \color{blue}{\left(\log \left(\frac{1}{1 - y}\right) + x \cdot \left(\frac{-1}{2} \cdot x - 1\right)\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 N/A

         \[\leadsto 1 - \left(\log \left(\frac{1}{1 - y}\right) + x \cdot \left(\frac{-1}{2} \cdot x - 1\right)\right) \]

      4. lower--.f64 N/A

         \[\leadsto 1 - \left(\log \left(\frac{1}{1 - y}\right) + x \cdot \left(\frac{-1}{2} \cdot x - 1\right)\right) \]

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 60.8%

         \[\leadsto 1 - \left(\log \left(\frac{1}{1 - y}\right) + x \cdot \left(-0.5 \cdot x - 1\right)\right) \]

   6. Applied rewrites 60.8%

      \[\leadsto 1 - \color{blue}{\left(\log \left(\frac{1}{1 - y}\right) + x \cdot \left(-0.5 \cdot x - 1\right)\right)} \]

   7. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower--.f64 60.5%

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

   9. Applied rewrites 60.5%

      \[\leadsto 1 - \color{blue}{\left(y + \log \left(1 - x\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq -540:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 9200:\\ \;\;\;\;1 - \left(y + \log \left(1 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (if (<= y -540.0)
  (- 1.0 (log (/ -1.0 y)))
  (if (<= y 9200.0)
    (- 1.0 (+ y (log (- 1.0 x))))
    (- 1.0 (log (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -540.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 9200.0) {
		tmp = 1.0 - (y + log((1.0 - x)));
	} else {
		tmp = 1.0 - log((x / y));
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-540.0d0)) then
        tmp = 1.0d0 - log(((-1.0d0) / y))
    else if (y <= 9200.0d0) then
        tmp = 1.0d0 - (y + log((1.0d0 - x)))
    else
        tmp = 1.0d0 - log((x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -540.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 9200.0) {
		tmp = 1.0 - (y + Math.log((1.0 - x)));
	} else {
		tmp = 1.0 - Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -540.0:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 9200.0:
		tmp = 1.0 - (y + math.log((1.0 - x)))
	else:
		tmp = 1.0 - math.log((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -540.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 9200.0)
		tmp = Float64(1.0 - Float64(y + log(Float64(1.0 - x))));
	else
		tmp = Float64(1.0 - log(Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -540.0)
		tmp = 1.0 - log((-1.0 / y));
	elseif (y <= 9200.0)
		tmp = 1.0 - (y + log((1.0 - x)));
	else
		tmp = 1.0 - log((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -540.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9200.0], N[(1.0 - N[(y + N[Log[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -540

   1. Initial program 72.1%

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

   2. Taylor expanded in y around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 40.9%

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

   4. Applied rewrites 40.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 22.9%

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

   7. Applied rewrites 22.9%

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

   if -540 < y < 9200

   1. Initial program 72.1%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-to-fraction N/A

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

      4. frac-2neg N/A

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

      5. distribute-neg-frac N/A

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

      6. distribute-frac-neg2 N/A

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

      7. sub-negate-rev N/A

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

      8. frac-2neg-rev N/A

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

      9. *-lft-identity N/A

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

      10. sub-negate-rev N/A

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

      11. *-lft-identity N/A

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

      12. *-lft-identity N/A

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

      13. sub-flip-reverse N/A

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

      14. distribute-neg-in N/A

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

      15. remove-double-neg N/A

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

      16. *-lft-identity N/A

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

      17. add-flip-rev N/A

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

      18. sub-negate-rev N/A

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

   3. Applied rewrites 72.4%

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

   4. Taylor expanded in x around 0

      \[\leadsto 1 - \color{blue}{\left(\log \left(\frac{1}{1 - y}\right) + x \cdot \left(\frac{-1}{2} \cdot x - 1\right)\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 N/A

         \[\leadsto 1 - \left(\log \left(\frac{1}{1 - y}\right) + x \cdot \left(\frac{-1}{2} \cdot x - 1\right)\right) \]

      4. lower--.f64 N/A

         \[\leadsto 1 - \left(\log \left(\frac{1}{1 - y}\right) + x \cdot \left(\frac{-1}{2} \cdot x - 1\right)\right) \]

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 60.8%

         \[\leadsto 1 - \left(\log \left(\frac{1}{1 - y}\right) + x \cdot \left(-0.5 \cdot x - 1\right)\right) \]

   6. Applied rewrites 60.8%

      \[\leadsto 1 - \color{blue}{\left(\log \left(\frac{1}{1 - y}\right) + x \cdot \left(-0.5 \cdot x - 1\right)\right)} \]

   7. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower--.f64 60.5%

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

   9. Applied rewrites 60.5%

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

   if 9200 < y

   1. Initial program 72.1%

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

   2. Taylor expanded in y around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 40.9%

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

   4. Applied rewrites 40.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 23.4%

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

   7. Applied rewrites 23.4%

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

Alternative 6: 80.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;1 - \log \left(1 - \frac{x - y}{1 - y}\right) \leq 2:\\ \;\;\;\;1 - \log \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (if (<= (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))) 2.0)
  (- 1.0 (log (- 1.0 x)))
  (- 1.0 (log (/ -1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if ((1.0 - log((1.0 - ((x - y) / (1.0 - y))))) <= 2.0) {
		tmp = 1.0 - log((1.0 - x));
	} else {
		tmp = 1.0 - log((-1.0 / y));
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((1.0d0 - log((1.0d0 - ((x - y) / (1.0d0 - y))))) <= 2.0d0) then
        tmp = 1.0d0 - log((1.0d0 - x))
    else
        tmp = 1.0d0 - log(((-1.0d0) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((1.0 - Math.log((1.0 - ((x - y) / (1.0 - y))))) <= 2.0) {
		tmp = 1.0 - Math.log((1.0 - x));
	} else {
		tmp = 1.0 - Math.log((-1.0 / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (1.0 - math.log((1.0 - ((x - y) / (1.0 - y))))) <= 2.0:
		tmp = 1.0 - math.log((1.0 - x))
	else:
		tmp = 1.0 - math.log((-1.0 / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(1.0 - log(Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y))))) <= 2.0)
		tmp = Float64(1.0 - log(Float64(1.0 - x)));
	else
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((1.0 - log((1.0 - ((x - y) / (1.0 - y))))) <= 2.0)
		tmp = 1.0 - log((1.0 - x));
	else
		tmp = 1.0 - log((-1.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 - N[Log[N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], N[(1.0 - N[Log[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))))) < 2

   1. Initial program 72.1%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 62.4%

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

   4. Applied rewrites 62.4%

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

   if 2 < (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)))))

   1. Initial program 72.1%

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

   2. Taylor expanded in y around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 40.9%

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

   4. Applied rewrites 40.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 22.9%

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

   7. Applied rewrites 22.9%

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

Alternative 7: 63.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := 1 - \log \left(1 - \frac{x - y}{1 - y}\right)\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;1 - \log \left(-x\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 - \left(y \cdot \left(1 + y \cdot \left(0.5 + y \cdot \left(0.3333333333333333 + 0.25 \cdot y\right)\right)\right) + x \cdot \left(-0.5 \cdot x - 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (let* ((t_0 (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y)))))))
  (if (<= t_0 -10.0)
    (- 1.0 (log (- x)))
    (if (<= t_0 2.0)
      (-
       1.0
       (+
        (*
         y
         (+
          1.0
          (* y (+ 0.5 (* y (+ 0.3333333333333333 (* 0.25 y)))))))
        (* x (- (* -0.5 x) 1.0))))
      1.0))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log((1.0 - ((x - y) / (1.0 - y))));
	double tmp;
	if (t_0 <= -10.0) {
		tmp = 1.0 - log(-x);
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - ((y * (1.0 + (y * (0.5 + (y * (0.3333333333333333 + (0.25 * y))))))) + (x * ((-0.5 * x) - 1.0)));
	} else {
		tmp = 1.0;
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - log((1.0d0 - ((x - y) / (1.0d0 - y))))
    if (t_0 <= (-10.0d0)) then
        tmp = 1.0d0 - log(-x)
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0 - ((y * (1.0d0 + (y * (0.5d0 + (y * (0.3333333333333333d0 + (0.25d0 * y))))))) + (x * (((-0.5d0) * x) - 1.0d0)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - Math.log((1.0 - ((x - y) / (1.0 - y))));
	double tmp;
	if (t_0 <= -10.0) {
		tmp = 1.0 - Math.log(-x);
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - ((y * (1.0 + (y * (0.5 + (y * (0.3333333333333333 + (0.25 * y))))))) + (x * ((-0.5 * x) - 1.0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - math.log((1.0 - ((x - y) / (1.0 - y))))
	tmp = 0
	if t_0 <= -10.0:
		tmp = 1.0 - math.log(-x)
	elif t_0 <= 2.0:
		tmp = 1.0 - ((y * (1.0 + (y * (0.5 + (y * (0.3333333333333333 + (0.25 * y))))))) + (x * ((-0.5 * x) - 1.0)))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - log(Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y)))))
	tmp = 0.0
	if (t_0 <= -10.0)
		tmp = Float64(1.0 - log(Float64(-x)));
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 - Float64(Float64(y * Float64(1.0 + Float64(y * Float64(0.5 + Float64(y * Float64(0.3333333333333333 + Float64(0.25 * y))))))) + Float64(x * Float64(Float64(-0.5 * x) - 1.0))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - log((1.0 - ((x - y) / (1.0 - y))));
	tmp = 0.0;
	if (t_0 <= -10.0)
		tmp = 1.0 - log(-x);
	elseif (t_0 <= 2.0)
		tmp = 1.0 - ((y * (1.0 + (y * (0.5 + (y * (0.3333333333333333 + (0.25 * y))))))) + (x * ((-0.5 * x) - 1.0)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Log[N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -10.0], N[(1.0 - N[Log[(-x)], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(1.0 - N[(N[(y * N[(1.0 + N[(y * N[(0.5 + N[(y * N[(0.3333333333333333 + N[(0.25 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(-0.5 * x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))))) < -10

   1. Initial program 72.1%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 62.4%

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

   4. Applied rewrites 62.4%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 24.7%

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

   7. Applied rewrites 24.7%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 24.7%

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

   9. Applied rewrites 24.7%

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

   if -10 < (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))))) < 2

   1. Initial program 72.1%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-to-fraction N/A

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

      4. frac-2neg N/A

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

      5. distribute-neg-frac N/A

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

      6. distribute-frac-neg2 N/A

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

      7. sub-negate-rev N/A

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

      8. frac-2neg-rev N/A

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

      9. *-lft-identity N/A

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

      10. sub-negate-rev N/A

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

      11. *-lft-identity N/A

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

      12. *-lft-identity N/A

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

      13. sub-flip-reverse N/A

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

      14. distribute-neg-in N/A

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

      15. remove-double-neg N/A

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

      16. *-lft-identity N/A

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

      17. add-flip-rev N/A

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

      18. sub-negate-rev N/A

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

   3. Applied rewrites 72.4%

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

   4. Taylor expanded in x around 0

      \[\leadsto 1 - \color{blue}{\left(\log \left(\frac{1}{1 - y}\right) + x \cdot \left(\frac{-1}{2} \cdot x - 1\right)\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 N/A

         \[\leadsto 1 - \left(\log \left(\frac{1}{1 - y}\right) + x \cdot \left(\frac{-1}{2} \cdot x - 1\right)\right) \]

      4. lower--.f64 N/A

         \[\leadsto 1 - \left(\log \left(\frac{1}{1 - y}\right) + x \cdot \left(\frac{-1}{2} \cdot x - 1\right)\right) \]

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 60.8%

         \[\leadsto 1 - \left(\log \left(\frac{1}{1 - y}\right) + x \cdot \left(-0.5 \cdot x - 1\right)\right) \]

   6. Applied rewrites 60.8%

      \[\leadsto 1 - \color{blue}{\left(\log \left(\frac{1}{1 - y}\right) + x \cdot \left(-0.5 \cdot x - 1\right)\right)} \]

   7. Taylor expanded in y around 0

      \[\leadsto 1 - \left(y \cdot \left(1 + y \cdot \left(\frac{1}{2} + y \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot y\right)\right)\right) + \color{blue}{x} \cdot \left(-0.5 \cdot x - 1\right)\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 1 - \left(y \cdot \left(1 + y \cdot \left(\frac{1}{2} + y \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot y\right)\right)\right) + x \cdot \left(\frac{-1}{2} \cdot x - 1\right)\right) \]

      2. lower-+.f64 N/A

         \[\leadsto 1 - \left(y \cdot \left(1 + y \cdot \left(\frac{1}{2} + y \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot y\right)\right)\right) + x \cdot \left(\frac{-1}{2} \cdot x - 1\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto 1 - \left(y \cdot \left(1 + y \cdot \left(\frac{1}{2} + y \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot y\right)\right)\right) + x \cdot \left(\frac{-1}{2} \cdot x - 1\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto 1 - \left(y \cdot \left(1 + y \cdot \left(\frac{1}{2} + y \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot y\right)\right)\right) + x \cdot \left(\frac{-1}{2} \cdot x - 1\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto 1 - \left(y \cdot \left(1 + y \cdot \left(\frac{1}{2} + y \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot y\right)\right)\right) + x \cdot \left(\frac{-1}{2} \cdot x - 1\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto 1 - \left(y \cdot \left(1 + y \cdot \left(\frac{1}{2} + y \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot y\right)\right)\right) + x \cdot \left(\frac{-1}{2} \cdot x - 1\right)\right) \]

      7. lower-*.f64 40.3%

         \[\leadsto 1 - \left(y \cdot \left(1 + y \cdot \left(0.5 + y \cdot \left(0.3333333333333333 + 0.25 \cdot y\right)\right)\right) + x \cdot \left(-0.5 \cdot x - 1\right)\right) \]

   9. Applied rewrites 40.3%

      \[\leadsto 1 - \left(y \cdot \left(1 + y \cdot \left(0.5 + y \cdot \left(0.3333333333333333 + 0.25 \cdot y\right)\right)\right) + \color{blue}{x} \cdot \left(-0.5 \cdot x - 1\right)\right) \]

   if 2 < (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)))))

   1. Initial program 72.1%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 42.1%

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

   4. Applied rewrites 42.1%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 42.1%

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

   7. Applied rewrites 42.1%

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

   8. Taylor expanded in y around 0

      \[\leadsto 1 + x \]
   9. Step-by-step derivation

      1. lower-+.f64 43.5%

         \[\leadsto 1 + x \]

   10. Applied rewrites 43.5%

       \[\leadsto 1 + x \]

   11. Taylor expanded in x around 0

       \[\leadsto 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 43.3%

       \[\leadsto 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 62.4% accurate, 1.2× speedup

Math

\[1 - \log \left(1 - x\right) \]

FPCore

(FPCore (x y)
  :precision binary64
  (- 1.0 (log (- 1.0 x))))

C

double code(double x, double y) {
	return 1.0 - log((1.0 - x));
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 - log((1.0d0 - x))
end function

Java

public static double code(double x, double y) {
	return 1.0 - Math.log((1.0 - x));
}

Python

def code(x, y):
	return 1.0 - math.log((1.0 - x))

Julia

function code(x, y)
	return Float64(1.0 - log(Float64(1.0 - x)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - log((1.0 - x));
end

Wolfram

code[x_, y_] := N[(1.0 - N[Log[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.1%

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

2. Taylor expanded in y around 0

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

   1. lower--.f64 62.4%

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

4. Applied rewrites 62.4%

   \[\leadsto 1 - \log \color{blue}{\left(1 - x\right)} \]
5. Add Preprocessing

Alternative 9: 45.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;1 - \log \left(1 - \frac{x - y}{1 - y}\right) \leq 2:\\ \;\;\;\;\left(1 + \frac{x}{\left(1 + \frac{y}{1 - y}\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + y \cdot \left(0.5 + y \cdot \left(0.3333333333333333 + 0.25 \cdot y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (if (<= (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))) 2.0)
  (-
   (+ 1.0 (/ x (* (+ 1.0 (/ y (- 1.0 y))) (- 1.0 y))))
   (*
    y
    (+ 1.0 (* y (+ 0.5 (* y (+ 0.3333333333333333 (* 0.25 y))))))))
  1.0))

C

double code(double x, double y) {
	double tmp;
	if ((1.0 - log((1.0 - ((x - y) / (1.0 - y))))) <= 2.0) {
		tmp = (1.0 + (x / ((1.0 + (y / (1.0 - y))) * (1.0 - y)))) - (y * (1.0 + (y * (0.5 + (y * (0.3333333333333333 + (0.25 * y)))))));
	} else {
		tmp = 1.0;
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((1.0d0 - log((1.0d0 - ((x - y) / (1.0d0 - y))))) <= 2.0d0) then
        tmp = (1.0d0 + (x / ((1.0d0 + (y / (1.0d0 - y))) * (1.0d0 - y)))) - (y * (1.0d0 + (y * (0.5d0 + (y * (0.3333333333333333d0 + (0.25d0 * y)))))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((1.0 - Math.log((1.0 - ((x - y) / (1.0 - y))))) <= 2.0) {
		tmp = (1.0 + (x / ((1.0 + (y / (1.0 - y))) * (1.0 - y)))) - (y * (1.0 + (y * (0.5 + (y * (0.3333333333333333 + (0.25 * y)))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (1.0 - math.log((1.0 - ((x - y) / (1.0 - y))))) <= 2.0:
		tmp = (1.0 + (x / ((1.0 + (y / (1.0 - y))) * (1.0 - y)))) - (y * (1.0 + (y * (0.5 + (y * (0.3333333333333333 + (0.25 * y)))))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(1.0 - log(Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y))))) <= 2.0)
		tmp = Float64(Float64(1.0 + Float64(x / Float64(Float64(1.0 + Float64(y / Float64(1.0 - y))) * Float64(1.0 - y)))) - Float64(y * Float64(1.0 + Float64(y * Float64(0.5 + Float64(y * Float64(0.3333333333333333 + Float64(0.25 * y))))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((1.0 - log((1.0 - ((x - y) / (1.0 - y))))) <= 2.0)
		tmp = (1.0 + (x / ((1.0 + (y / (1.0 - y))) * (1.0 - y)))) - (y * (1.0 + (y * (0.5 + (y * (0.3333333333333333 + (0.25 * y)))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 - N[Log[N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], N[(N[(1.0 + N[(x / N[(N[(1.0 + N[(y / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * N[(1.0 + N[(y * N[(0.5 + N[(y * N[(0.3333333333333333 + N[(0.25 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))))) < 2

   1. Initial program 72.1%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 42.1%

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

   4. Applied rewrites 42.1%

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

   5. Taylor expanded in y around 0

      \[\leadsto \left(1 + \frac{x}{\left(1 + \frac{y}{1 - y}\right) \cdot \left(1 - y\right)}\right) - y \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot y\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 41.2%

         \[\leadsto \left(1 + \frac{x}{\left(1 + \frac{y}{1 - y}\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + 0.5 \cdot y\right) \]

   7. Applied rewrites 41.2%

      \[\leadsto \left(1 + \frac{x}{\left(1 + \frac{y}{1 - y}\right) \cdot \left(1 - y\right)}\right) - y \cdot \color{blue}{\left(1 + 0.5 \cdot y\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \left(1 + \frac{x}{\left(1 + \frac{y}{1 - y}\right) \cdot \left(1 - y\right)}\right) - y \cdot \color{blue}{\left(1 + y \cdot \left(\frac{1}{2} + y \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot y\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{x}{\left(1 + \frac{y}{1 - y}\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + \color{blue}{y \cdot \left(\frac{1}{2} + y \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot y\right)\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(1 + \frac{x}{\left(1 + \frac{y}{1 - y}\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + y \cdot \color{blue}{\left(\frac{1}{2} + y \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot y\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{x}{\left(1 + \frac{y}{1 - y}\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + y \cdot \left(\frac{1}{2} + \color{blue}{y \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot y\right)}\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(1 + \frac{x}{\left(1 + \frac{y}{1 - y}\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + y \cdot \left(\frac{1}{2} + y \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot y\right)}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{x}{\left(1 + \frac{y}{1 - y}\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + y \cdot \left(\frac{1}{2} + y \cdot \left(\frac{1}{3} + \color{blue}{\frac{1}{4} \cdot y}\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(1 + \frac{x}{\left(1 + \frac{y}{1 - y}\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + y \cdot \left(\frac{1}{2} + y \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot \color{blue}{y}\right)\right)\right) \]

      7. lower-*.f64 41.3%

         \[\leadsto \left(1 + \frac{x}{\left(1 + \frac{y}{1 - y}\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + y \cdot \left(0.5 + y \cdot \left(0.3333333333333333 + 0.25 \cdot y\right)\right)\right) \]

   10. Applied rewrites 41.3%

       \[\leadsto \left(1 + \frac{x}{\left(1 + \frac{y}{1 - y}\right) \cdot \left(1 - y\right)}\right) - y \cdot \color{blue}{\left(1 + y \cdot \left(0.5 + y \cdot \left(0.3333333333333333 + 0.25 \cdot y\right)\right)\right)} \]

   if 2 < (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)))))

   1. Initial program 72.1%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 42.1%

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

   4. Applied rewrites 42.1%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 42.1%

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

   7. Applied rewrites 42.1%

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

   8. Taylor expanded in y around 0

      \[\leadsto 1 + x \]
   9. Step-by-step derivation

      1. lower-+.f64 43.5%

         \[\leadsto 1 + x \]

   10. Applied rewrites 43.5%

       \[\leadsto 1 + x \]

   11. Taylor expanded in x around 0

       \[\leadsto 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 43.3%

       \[\leadsto 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 45.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;1 - \log \left(1 - \frac{x - y}{1 - y}\right) \leq 2:\\ \;\;\;\;\left(1 + \frac{x}{\left(1 + y \cdot \left(1 + y \cdot \left(1 + y \cdot \left(1 + y\right)\right)\right)\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + 0.5 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (if (<= (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))) 2.0)
  (-
   (+
    1.0
    (/
     x
     (*
      (+ 1.0 (* y (+ 1.0 (* y (+ 1.0 (* y (+ 1.0 y)))))))
      (- 1.0 y))))
   (* y (+ 1.0 (* 0.5 y))))
  1.0))

C

double code(double x, double y) {
	double tmp;
	if ((1.0 - log((1.0 - ((x - y) / (1.0 - y))))) <= 2.0) {
		tmp = (1.0 + (x / ((1.0 + (y * (1.0 + (y * (1.0 + (y * (1.0 + y))))))) * (1.0 - y)))) - (y * (1.0 + (0.5 * y)));
	} else {
		tmp = 1.0;
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((1.0d0 - log((1.0d0 - ((x - y) / (1.0d0 - y))))) <= 2.0d0) then
        tmp = (1.0d0 + (x / ((1.0d0 + (y * (1.0d0 + (y * (1.0d0 + (y * (1.0d0 + y))))))) * (1.0d0 - y)))) - (y * (1.0d0 + (0.5d0 * y)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((1.0 - Math.log((1.0 - ((x - y) / (1.0 - y))))) <= 2.0) {
		tmp = (1.0 + (x / ((1.0 + (y * (1.0 + (y * (1.0 + (y * (1.0 + y))))))) * (1.0 - y)))) - (y * (1.0 + (0.5 * y)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (1.0 - math.log((1.0 - ((x - y) / (1.0 - y))))) <= 2.0:
		tmp = (1.0 + (x / ((1.0 + (y * (1.0 + (y * (1.0 + (y * (1.0 + y))))))) * (1.0 - y)))) - (y * (1.0 + (0.5 * y)))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(1.0 - log(Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y))))) <= 2.0)
		tmp = Float64(Float64(1.0 + Float64(x / Float64(Float64(1.0 + Float64(y * Float64(1.0 + Float64(y * Float64(1.0 + Float64(y * Float64(1.0 + y))))))) * Float64(1.0 - y)))) - Float64(y * Float64(1.0 + Float64(0.5 * y))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((1.0 - log((1.0 - ((x - y) / (1.0 - y))))) <= 2.0)
		tmp = (1.0 + (x / ((1.0 + (y * (1.0 + (y * (1.0 + (y * (1.0 + y))))))) * (1.0 - y)))) - (y * (1.0 + (0.5 * y)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 - N[Log[N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], N[(N[(1.0 + N[(x / N[(N[(1.0 + N[(y * N[(1.0 + N[(y * N[(1.0 + N[(y * N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * N[(1.0 + N[(0.5 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))))) < 2

   1. Initial program 72.1%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 42.1%

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

   4. Applied rewrites 42.1%

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

   5. Taylor expanded in y around 0

      \[\leadsto \left(1 + \frac{x}{\left(1 + \frac{y}{1 - y}\right) \cdot \left(1 - y\right)}\right) - y \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot y\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 41.2%

         \[\leadsto \left(1 + \frac{x}{\left(1 + \frac{y}{1 - y}\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + 0.5 \cdot y\right) \]

   7. Applied rewrites 41.2%

      \[\leadsto \left(1 + \frac{x}{\left(1 + \frac{y}{1 - y}\right) \cdot \left(1 - y\right)}\right) - y \cdot \color{blue}{\left(1 + 0.5 \cdot y\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \left(1 + \frac{x}{\left(1 + y \cdot \left(1 + y\right)\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + 0.5 \cdot y\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{x}{\left(1 + y \cdot \left(1 + y\right)\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + \frac{1}{2} \cdot y\right) \]

      2. lower-+.f64 41.3%

         \[\leadsto \left(1 + \frac{x}{\left(1 + y \cdot \left(1 + y\right)\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + 0.5 \cdot y\right) \]

   10. Applied rewrites 41.3%

       \[\leadsto \left(1 + \frac{x}{\left(1 + y \cdot \left(1 + y\right)\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + 0.5 \cdot y\right) \]

   11. Taylor expanded in y around 0

       \[\leadsto \left(1 + \frac{x}{\left(1 + y \cdot \left(1 + y \cdot \left(1 + y \cdot \left(1 + y\right)\right)\right)\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + 0.5 \cdot y\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(1 + \frac{x}{\left(1 + y \cdot \left(1 + y \cdot \left(1 + y \cdot \left(1 + y\right)\right)\right)\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + \frac{1}{2} \cdot y\right) \]

       2. lower-+.f64 N/A

          \[\leadsto \left(1 + \frac{x}{\left(1 + y \cdot \left(1 + y \cdot \left(1 + y \cdot \left(1 + y\right)\right)\right)\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + \frac{1}{2} \cdot y\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \left(1 + \frac{x}{\left(1 + y \cdot \left(1 + y \cdot \left(1 + y \cdot \left(1 + y\right)\right)\right)\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + \frac{1}{2} \cdot y\right) \]

       4. lower-+.f64 N/A

          \[\leadsto \left(1 + \frac{x}{\left(1 + y \cdot \left(1 + y \cdot \left(1 + y \cdot \left(1 + y\right)\right)\right)\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + \frac{1}{2} \cdot y\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \left(1 + \frac{x}{\left(1 + y \cdot \left(1 + y \cdot \left(1 + y \cdot \left(1 + y\right)\right)\right)\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + \frac{1}{2} \cdot y\right) \]

       6. lower-+.f64 41.3%

          \[\leadsto \left(1 + \frac{x}{\left(1 + y \cdot \left(1 + y \cdot \left(1 + y \cdot \left(1 + y\right)\right)\right)\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + 0.5 \cdot y\right) \]

   13. Applied rewrites 41.3%

       \[\leadsto \left(1 + \frac{x}{\left(1 + y \cdot \left(1 + y \cdot \left(1 + y \cdot \left(1 + y\right)\right)\right)\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + 0.5 \cdot y\right) \]

   if 2 < (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)))))

   1. Initial program 72.1%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 42.1%

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

   4. Applied rewrites 42.1%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 42.1%

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

   7. Applied rewrites 42.1%

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

   8. Taylor expanded in y around 0

      \[\leadsto 1 + x \]
   9. Step-by-step derivation

      1. lower-+.f64 43.5%

         \[\leadsto 1 + x \]

   10. Applied rewrites 43.5%

       \[\leadsto 1 + x \]

   11. Taylor expanded in x around 0

       \[\leadsto 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 43.3%

       \[\leadsto 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 45.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;1 - \log \left(1 - \frac{x - y}{1 - y}\right) \leq 2:\\ \;\;\;\;\left(1 + \frac{x}{\left(1 + y \cdot \left(1 + y\right)\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + 0.5 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (if (<= (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))) 2.0)
  (-
   (+ 1.0 (/ x (* (+ 1.0 (* y (+ 1.0 y))) (- 1.0 y))))
   (* y (+ 1.0 (* 0.5 y))))
  1.0))

C

double code(double x, double y) {
	double tmp;
	if ((1.0 - log((1.0 - ((x - y) / (1.0 - y))))) <= 2.0) {
		tmp = (1.0 + (x / ((1.0 + (y * (1.0 + y))) * (1.0 - y)))) - (y * (1.0 + (0.5 * y)));
	} else {
		tmp = 1.0;
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((1.0d0 - log((1.0d0 - ((x - y) / (1.0d0 - y))))) <= 2.0d0) then
        tmp = (1.0d0 + (x / ((1.0d0 + (y * (1.0d0 + y))) * (1.0d0 - y)))) - (y * (1.0d0 + (0.5d0 * y)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((1.0 - Math.log((1.0 - ((x - y) / (1.0 - y))))) <= 2.0) {
		tmp = (1.0 + (x / ((1.0 + (y * (1.0 + y))) * (1.0 - y)))) - (y * (1.0 + (0.5 * y)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (1.0 - math.log((1.0 - ((x - y) / (1.0 - y))))) <= 2.0:
		tmp = (1.0 + (x / ((1.0 + (y * (1.0 + y))) * (1.0 - y)))) - (y * (1.0 + (0.5 * y)))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(1.0 - log(Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y))))) <= 2.0)
		tmp = Float64(Float64(1.0 + Float64(x / Float64(Float64(1.0 + Float64(y * Float64(1.0 + y))) * Float64(1.0 - y)))) - Float64(y * Float64(1.0 + Float64(0.5 * y))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((1.0 - log((1.0 - ((x - y) / (1.0 - y))))) <= 2.0)
		tmp = (1.0 + (x / ((1.0 + (y * (1.0 + y))) * (1.0 - y)))) - (y * (1.0 + (0.5 * y)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 - N[Log[N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], N[(N[(1.0 + N[(x / N[(N[(1.0 + N[(y * N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * N[(1.0 + N[(0.5 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))))) < 2

   1. Initial program 72.1%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 42.1%

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

   4. Applied rewrites 42.1%

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

   5. Taylor expanded in y around 0

      \[\leadsto \left(1 + \frac{x}{\left(1 + \frac{y}{1 - y}\right) \cdot \left(1 - y\right)}\right) - y \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot y\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 41.2%

         \[\leadsto \left(1 + \frac{x}{\left(1 + \frac{y}{1 - y}\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + 0.5 \cdot y\right) \]

   7. Applied rewrites 41.2%

      \[\leadsto \left(1 + \frac{x}{\left(1 + \frac{y}{1 - y}\right) \cdot \left(1 - y\right)}\right) - y \cdot \color{blue}{\left(1 + 0.5 \cdot y\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \left(1 + \frac{x}{\left(1 + y \cdot \left(1 + y\right)\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + 0.5 \cdot y\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{x}{\left(1 + y \cdot \left(1 + y\right)\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + \frac{1}{2} \cdot y\right) \]

      2. lower-+.f64 41.3%

         \[\leadsto \left(1 + \frac{x}{\left(1 + y \cdot \left(1 + y\right)\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + 0.5 \cdot y\right) \]

   10. Applied rewrites 41.3%

       \[\leadsto \left(1 + \frac{x}{\left(1 + y \cdot \left(1 + y\right)\right) \cdot \left(1 - y\right)}\right) - y \cdot \left(1 + 0.5 \cdot y\right) \]

   if 2 < (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)))))

   1. Initial program 72.1%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 42.1%

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

   4. Applied rewrites 42.1%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 42.1%

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

   7. Applied rewrites 42.1%

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

   8. Taylor expanded in y around 0

      \[\leadsto 1 + x \]
   9. Step-by-step derivation

      1. lower-+.f64 43.5%

         \[\leadsto 1 + x \]

   10. Applied rewrites 43.5%

       \[\leadsto 1 + x \]

   11. Taylor expanded in x around 0

       \[\leadsto 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 43.3%

       \[\leadsto 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 45.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;1 - \log \left(1 - \frac{x - y}{1 - y}\right) \leq 2:\\ \;\;\;\;1 + \left(x + y \cdot \left(y \cdot \left(-0.3333333333333333 \cdot y - 0.5\right) - 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (if (<= (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))) 2.0)
  (+ 1.0 (+ x (* y (- (* y (- (* -0.3333333333333333 y) 0.5)) 1.0))))
  1.0))

C

double code(double x, double y) {
	double tmp;
	if ((1.0 - log((1.0 - ((x - y) / (1.0 - y))))) <= 2.0) {
		tmp = 1.0 + (x + (y * ((y * ((-0.3333333333333333 * y) - 0.5)) - 1.0)));
	} else {
		tmp = 1.0;
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((1.0d0 - log((1.0d0 - ((x - y) / (1.0d0 - y))))) <= 2.0d0) then
        tmp = 1.0d0 + (x + (y * ((y * (((-0.3333333333333333d0) * y) - 0.5d0)) - 1.0d0)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((1.0 - Math.log((1.0 - ((x - y) / (1.0 - y))))) <= 2.0) {
		tmp = 1.0 + (x + (y * ((y * ((-0.3333333333333333 * y) - 0.5)) - 1.0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (1.0 - math.log((1.0 - ((x - y) / (1.0 - y))))) <= 2.0:
		tmp = 1.0 + (x + (y * ((y * ((-0.3333333333333333 * y) - 0.5)) - 1.0)))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(1.0 - log(Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y))))) <= 2.0)
		tmp = Float64(1.0 + Float64(x + Float64(y * Float64(Float64(y * Float64(Float64(-0.3333333333333333 * y) - 0.5)) - 1.0))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((1.0 - log((1.0 - ((x - y) / (1.0 - y))))) <= 2.0)
		tmp = 1.0 + (x + (y * ((y * ((-0.3333333333333333 * y) - 0.5)) - 1.0)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 - N[Log[N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], N[(1.0 + N[(x + N[(y * N[(N[(y * N[(N[(-0.3333333333333333 * y), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))))) < 2

   1. Initial program 72.1%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 42.1%

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

   4. Applied rewrites 42.1%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 42.1%

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

   7. Applied rewrites 42.1%

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

   8. Taylor expanded in y around 0

      \[\leadsto 1 + x \]
   9. Step-by-step derivation

      1. lower-+.f64 43.5%

         \[\leadsto 1 + x \]

   10. Applied rewrites 43.5%

       \[\leadsto 1 + x \]

   11. Taylor expanded in y around 0

       \[\leadsto 1 + \color{blue}{\left(x + y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)} \]
   12. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto 1 + \left(x + \color{blue}{y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)}\right) \]

       2. lower-+.f64 N/A

          \[\leadsto 1 + \left(x + y \cdot \color{blue}{\left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto 1 + \left(x + y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - \color{blue}{1}\right)\right) \]

       4. lower--.f64 N/A

          \[\leadsto 1 + \left(x + y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right) \]

       5. lower-*.f64 N/A

          \[\leadsto 1 + \left(x + y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right) \]

       6. lower--.f64 N/A

          \[\leadsto 1 + \left(x + y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right) \]

       7. lower-*.f64 41.9%

          \[\leadsto 1 + \left(x + y \cdot \left(y \cdot \left(-0.3333333333333333 \cdot y - 0.5\right) - 1\right)\right) \]

   13. Applied rewrites 41.9%

       \[\leadsto 1 + \color{blue}{\left(x + y \cdot \left(y \cdot \left(-0.3333333333333333 \cdot y - 0.5\right) - 1\right)\right)} \]

   if 2 < (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)))))

   1. Initial program 72.1%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 42.1%

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

   4. Applied rewrites 42.1%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 42.1%

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

   7. Applied rewrites 42.1%

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

   8. Taylor expanded in y around 0

      \[\leadsto 1 + x \]
   9. Step-by-step derivation

      1. lower-+.f64 43.5%

         \[\leadsto 1 + x \]

   10. Applied rewrites 43.5%

       \[\leadsto 1 + x \]

   11. Taylor expanded in x around 0

       \[\leadsto 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 43.3%

       \[\leadsto 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 45.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;1 - \log \left(1 - \frac{x - y}{1 - y}\right) \leq 2:\\ \;\;\;\;1 + \left(x + y \cdot \left(-0.5 \cdot y - 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  (if (<= (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))) 2.0)
  (+ 1.0 (+ x (* y (- (* -0.5 y) 1.0))))
  1.0))

C

double code(double x, double y) {
	double tmp;
	if ((1.0 - log((1.0 - ((x - y) / (1.0 - y))))) <= 2.0) {
		tmp = 1.0 + (x + (y * ((-0.5 * y) - 1.0)));
	} else {
		tmp = 1.0;
	}
	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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((1.0d0 - log((1.0d0 - ((x - y) / (1.0d0 - y))))) <= 2.0d0) then
        tmp = 1.0d0 + (x + (y * (((-0.5d0) * y) - 1.0d0)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((1.0 - Math.log((1.0 - ((x - y) / (1.0 - y))))) <= 2.0) {
		tmp = 1.0 + (x + (y * ((-0.5 * y) - 1.0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (1.0 - math.log((1.0 - ((x - y) / (1.0 - y))))) <= 2.0:
		tmp = 1.0 + (x + (y * ((-0.5 * y) - 1.0)))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(1.0 - log(Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y))))) <= 2.0)
		tmp = Float64(1.0 + Float64(x + Float64(y * Float64(Float64(-0.5 * y) - 1.0))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((1.0 - log((1.0 - ((x - y) / (1.0 - y))))) <= 2.0)
		tmp = 1.0 + (x + (y * ((-0.5 * y) - 1.0)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 - N[Log[N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], N[(1.0 + N[(x + N[(y * N[(N[(-0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))))) < 2

   1. Initial program 72.1%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 42.1%

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

   4. Applied rewrites 42.1%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 42.1%

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

   7. Applied rewrites 42.1%

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

   8. Taylor expanded in y around 0

      \[\leadsto 1 + x \]
   9. Step-by-step derivation

      1. lower-+.f64 43.5%

         \[\leadsto 1 + x \]

   10. Applied rewrites 43.5%

       \[\leadsto 1 + x \]

   11. Taylor expanded in y around 0

       \[\leadsto 1 + \color{blue}{\left(x + y \cdot \left(\frac{-1}{2} \cdot y - 1\right)\right)} \]
   12. Step-by-step derivation

       1. lower-+.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower--.f64 N/A

          \[\leadsto 1 + \left(x + y \cdot \left(\frac{-1}{2} \cdot y - 1\right)\right) \]

       5. lower-*.f64 41.2%

          \[\leadsto 1 + \left(x + y \cdot \left(-0.5 \cdot y - 1\right)\right) \]

   13. Applied rewrites 41.2%

       \[\leadsto 1 + \color{blue}{\left(x + y \cdot \left(-0.5 \cdot y - 1\right)\right)} \]

   if 2 < (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)))))

   1. Initial program 72.1%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 42.1%

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

   4. Applied rewrites 42.1%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 42.1%

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

   7. Applied rewrites 42.1%

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

   8. Taylor expanded in y around 0

      \[\leadsto 1 + x \]
   9. Step-by-step derivation

      1. lower-+.f64 43.5%

         \[\leadsto 1 + x \]

   10. Applied rewrites 43.5%

       \[\leadsto 1 + x \]

   11. Taylor expanded in x around 0

       \[\leadsto 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 43.3%

       \[\leadsto 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 43.5% accurate, 31.0× speedup

Math

\[1 + x \]

FPCore

(FPCore (x y)
  :precision binary64
  (+ 1.0 x))

C

double code(double x, double y) {
	return 1.0 + x;
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + x
end function

Java

public static double code(double x, double y) {
	return 1.0 + x;
}

Python

def code(x, y):
	return 1.0 + x

Julia

function code(x, y)
	return Float64(1.0 + x)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + x;
end

Wolfram

code[x_, y_] := N[(1.0 + x), $MachinePrecision]

Derivation

1. Initial program 72.1%

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

2. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-log.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower--.f64 42.1%

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

4. Applied rewrites 42.1%

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

5. Taylor expanded in y around 0

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

   1. lower-+.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 42.1%

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

7. Applied rewrites 42.1%

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

8. Taylor expanded in y around 0

   \[\leadsto 1 + x \]
9. Step-by-step derivation

   1. lower-+.f64 43.5%

      \[\leadsto 1 + x \]

10. Applied rewrites 43.5%

    \[\leadsto 1 + x \]
11. Add Preprocessing

Alternative 15: 43.3% accurate, 124.0× speedup

Math

\[1 \]

FPCore

(FPCore (x y)
  :precision binary64
  1.0)

C

double code(double x, double y) {
	return 1.0;
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 72.1%

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

2. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-log.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower--.f64 42.1%

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

4. Applied rewrites 42.1%

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

5. Taylor expanded in y around 0

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

   1. lower-+.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 42.1%

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

7. Applied rewrites 42.1%

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

8. Taylor expanded in y around 0

   \[\leadsto 1 + x \]
9. Step-by-step derivation

   1. lower-+.f64 43.5%

      \[\leadsto 1 + x \]

10. Applied rewrites 43.5%

    \[\leadsto 1 + x \]

11. Taylor expanded in x around 0

    \[\leadsto 1 \]
12. Step-by-step derivation

13. Applied rewrites 43.3%

    \[\leadsto 1 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025258 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64
  (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))