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

Percentage Accurate: 72.9% → 99.9%

Time: 8.5s

Alternatives: 10

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \end{array} \]

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: 72.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \end{array} \]

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: 99.9% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y)))))))
   (if (<= t_0 2.0)
     t_0
     (-
      1.0
      (log
       (/
        (+ (+ -1.0 (/ (- (- x (/ (- (- 1.0 (/ (- x 1.0) y)) x) y)) 1.0) y)) x)
        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 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 - log((((-1.0 + (((x - (((1.0 - ((x - 1.0) / y)) - x) / y)) - 1.0) / y)) + 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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - log((1.0d0 - ((x - y) / (1.0d0 - y))))
    if (t_0 <= 2.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0 - log(((((-1.0d0) + (((x - (((1.0d0 - ((x - 1.0d0) / y)) - x) / y)) - 1.0d0) / y)) + x) / 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 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 - Math.log((((-1.0 + (((x - (((1.0 - ((x - 1.0) / y)) - x) / y)) - 1.0) / y)) + x) / 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 <= 2.0:
		tmp = t_0
	else:
		tmp = 1.0 - math.log((((-1.0 + (((x - (((1.0 - ((x - 1.0) / y)) - x) / y)) - 1.0) / y)) + x) / 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 <= 2.0)
		tmp = t_0;
	else
		tmp = Float64(1.0 - log(Float64(Float64(Float64(-1.0 + Float64(Float64(Float64(x - Float64(Float64(Float64(1.0 - Float64(Float64(x - 1.0) / y)) - x) / y)) - 1.0) / y)) + x) / 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 <= 2.0)
		tmp = t_0;
	else
		tmp = 1.0 - log((((-1.0 + (((x - (((1.0 - ((x - 1.0) / y)) - x) / y)) - 1.0) / y)) + x) / 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, 2.0], t$95$0, N[(1.0 - N[Log[N[(N[(N[(-1.0 + N[(N[(N[(x - N[(N[(N[(1.0 - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + x), $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 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   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 6.6%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(1 - \frac{x - y}{1 - y}\right)\\ \mathbf{if}\;t\_0 \leq 15:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\left(-1 + \frac{x - 1}{y}\right) + x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y)))))))
   (if (<= t_0 15.0) t_0 (- 1.0 (log (/ (+ (+ -1.0 (/ (- x 1.0) y)) x) 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 <= 15.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 - log((((-1.0 + ((x - 1.0) / y)) + 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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - log((1.0d0 - ((x - y) / (1.0d0 - y))))
    if (t_0 <= 15.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0 - log(((((-1.0d0) + ((x - 1.0d0) / y)) + x) / 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 <= 15.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 - Math.log((((-1.0 + ((x - 1.0) / y)) + x) / 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 <= 15.0:
		tmp = t_0
	else:
		tmp = 1.0 - math.log((((-1.0 + ((x - 1.0) / y)) + x) / 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 <= 15.0)
		tmp = t_0;
	else
		tmp = Float64(1.0 - log(Float64(Float64(Float64(-1.0 + Float64(Float64(x - 1.0) / y)) + x) / 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 <= 15.0)
		tmp = t_0;
	else
		tmp = 1.0 - log((((-1.0 + ((x - 1.0) / y)) + x) / 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, 15.0], t$95$0, N[(1.0 - N[Log[N[(N[(N[(-1.0 + N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + x), $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))))) < 15

   1. Initial program 99.9%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   if 15 < (-.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 5.4%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

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

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

      8. +-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \log \left(1 - \frac{x - y}{1 - y}\right) \leq 15:\\ \;\;\;\;1 - \log \left(1 - \frac{x - y}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\left(-1 + \frac{x - 1}{y}\right) + x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(1 - \frac{x - y}{1 - y}\right)\\ \mathbf{if}\;t\_0 \leq 15:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1 + x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y)))))))
   (if (<= t_0 15.0) t_0 (- 1.0 (log (/ (+ -1.0 x) 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 <= 15.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 - log(((-1.0 + 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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - log((1.0d0 - ((x - y) / (1.0d0 - y))))
    if (t_0 <= 15.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0 - log((((-1.0d0) + x) / 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 <= 15.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 - Math.log(((-1.0 + x) / 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 <= 15.0:
		tmp = t_0
	else:
		tmp = 1.0 - math.log(((-1.0 + x) / 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 <= 15.0)
		tmp = t_0;
	else
		tmp = Float64(1.0 - log(Float64(Float64(-1.0 + x) / 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 <= 15.0)
		tmp = t_0;
	else
		tmp = 1.0 - log(((-1.0 + x) / 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, 15.0], t$95$0, N[(1.0 - N[Log[N[(N[(-1.0 + x), $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))))) < 15

   1. Initial program 99.9%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   if 15 < (-.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 5.4%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-+.f64 99.7

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

   5. Applied rewrites 99.7%

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

Alternative 4: 80.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - \log \left(1 - \frac{x - y}{1 - y}\right) \leq 2:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(-\log \left(-y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))) 2.0)
   (- 1.0 (log1p (- x)))
   (- 1.0 (- (log (- 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 - log1p(-x);
	} else {
		tmp = 1.0 - -log(-y);
	}
	return tmp;
}

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.log1p(-x);
	} else {
		tmp = 1.0 - -Math.log(-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.log1p(-x)
	else:
		tmp = 1.0 - -math.log(-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 - log1p(Float64(-x)));
	else
		tmp = Float64(1.0 - Float64(-log(Float64(-y))));
	end
	return 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[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 - (-N[Log[(-y)], $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 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-lft-identity N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-log1p.f64 N/A

         \[\leadsto 1 - \mathsf{log1p}\left(-1 \cdot x\right) \]

      5. mul-1-neg N/A

         \[\leadsto 1 - \mathsf{log1p}\left(\mathsf{neg}\left(x\right)\right) \]

      6. lower-neg.f64 84.2

         \[\leadsto 1 - \mathsf{log1p}\left(-x\right) \]

   5. Applied rewrites 84.2%

      \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(-x\right)} \]

   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 6.6%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f64 N/A

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{y}{1 - y}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{y}{1 - y}\right) \]

      3. lower--.f64 5.0

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{y}{1 - y}\right) \]

   5. Applied rewrites 5.0%

      \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(\frac{y}{1 - y}\right)} \]

   6. Taylor expanded in y around -inf

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

      1. lower-log.f64 N/A

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

      2. lower-/.f64 61.9

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

   8. Applied rewrites 61.9%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. log-rec N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-neg.f64 61.9

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

   10. Applied rewrites 61.9%

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

Alternative 5: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.82 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \log \left(\frac{-1 + x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(y - -1\right) \cdot \left(1 - x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -0.82) (not (<= y 1.0)))
   (- 1.0 (log (/ (+ -1.0 x) y)))
   (- 1.0 (log (* (- y -1.0) (- 1.0 x))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -0.82) || !(y <= 1.0)) {
		tmp = 1.0 - log(((-1.0 + x) / y));
	} else {
		tmp = 1.0 - log(((y - -1.0) * (1.0 - x)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-0.82d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = 1.0d0 - log((((-1.0d0) + x) / y))
    else
        tmp = 1.0d0 - log(((y - (-1.0d0)) * (1.0d0 - x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -0.82) || !(y <= 1.0)) {
		tmp = 1.0 - Math.log(((-1.0 + x) / y));
	} else {
		tmp = 1.0 - Math.log(((y - -1.0) * (1.0 - x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -0.82) or not (y <= 1.0):
		tmp = 1.0 - math.log(((-1.0 + x) / y))
	else:
		tmp = 1.0 - math.log(((y - -1.0) * (1.0 - x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -0.82) || !(y <= 1.0))
		tmp = Float64(1.0 - log(Float64(Float64(-1.0 + x) / y)));
	else
		tmp = Float64(1.0 - log(Float64(Float64(y - -1.0) * Float64(1.0 - x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -0.82) || ~((y <= 1.0)))
		tmp = 1.0 - log(((-1.0 + x) / y));
	else
		tmp = 1.0 - log(((y - -1.0) * (1.0 - x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -0.82], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(1.0 - N[Log[N[(N[(-1.0 + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(y - -1.0), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.819999999999999951 or 1 < y

   1. Initial program 33.4%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-+.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if -0.819999999999999951 < y < 1

   1. Initial program 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower--.f64 99.4

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

   5. Applied rewrites 99.4%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.82 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \log \left(\frac{-1 + x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(y - -1\right) \cdot \left(1 - x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[y, -1.0], N[(1.0 - (-N[Log[(-y)], $MachinePrecision])), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[Log[N[(N[(y - -1.0), $MachinePrecision] * 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 < -1

   1. Initial program 24.3%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f64 N/A

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{y}{1 - y}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{y}{1 - y}\right) \]

      3. lower--.f64 4.7

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{y}{1 - y}\right) \]

   5. Applied rewrites 4.7%

      \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(\frac{y}{1 - y}\right)} \]

   6. Taylor expanded in y around -inf

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

      1. lower-log.f64 N/A

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

      2. lower-/.f64 67.8

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

   8. Applied rewrites 67.8%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. log-rec N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-neg.f64 67.8

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

   10. Applied rewrites 67.8%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower--.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 1 < y

   1. Initial program 50.5%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 95.7

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

   5. Applied rewrites 95.7%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 94.2

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

   8. Applied rewrites 94.2%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1 - \left(-\log \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \log \left(\left(y - -1\right) \cdot \left(1 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -225:\\ \;\;\;\;1 - \left(-\log \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -225.0)
   (- 1.0 (- (log (- y))))
   (if (<= y 1.0) (- 1.0 (log1p (- x))) (- 1.0 (log (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -225.0) {
		tmp = 1.0 - -log(-y);
	} else if (y <= 1.0) {
		tmp = 1.0 - log1p(-x);
	} else {
		tmp = 1.0 - log((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -225.0) {
		tmp = 1.0 - -Math.log(-y);
	} else if (y <= 1.0) {
		tmp = 1.0 - Math.log1p(-x);
	} else {
		tmp = 1.0 - Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -225.0:
		tmp = 1.0 - -math.log(-y)
	elif y <= 1.0:
		tmp = 1.0 - math.log1p(-x)
	else:
		tmp = 1.0 - math.log((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -225.0)
		tmp = Float64(1.0 - Float64(-log(Float64(-y))));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - log1p(Float64(-x)));
	else
		tmp = Float64(1.0 - log(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -225.0], N[(1.0 - (-N[Log[(-y)], $MachinePrecision])), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -225

   1. Initial program 24.3%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f64 N/A

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{y}{1 - y}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{y}{1 - y}\right) \]

      3. lower--.f64 4.7

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{y}{1 - y}\right) \]

   5. Applied rewrites 4.7%

      \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(\frac{y}{1 - y}\right)} \]

   6. Taylor expanded in y around -inf

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

      1. lower-log.f64 N/A

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

      2. lower-/.f64 67.8

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

   8. Applied rewrites 67.8%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. log-rec N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-neg.f64 67.8

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

   10. Applied rewrites 67.8%

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

   if -225 < y < 1

   1. Initial program 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-lft-identity N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-log1p.f64 N/A

         \[\leadsto 1 - \mathsf{log1p}\left(-1 \cdot x\right) \]

      5. mul-1-neg N/A

         \[\leadsto 1 - \mathsf{log1p}\left(\mathsf{neg}\left(x\right)\right) \]

      6. lower-neg.f64 98.4

         \[\leadsto 1 - \mathsf{log1p}\left(-x\right) \]

   5. Applied rewrites 98.4%

      \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(-x\right)} \]

   if 1 < y

   1. Initial program 50.5%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 95.7

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

   5. Applied rewrites 95.7%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 94.2

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

   8. Applied rewrites 94.2%

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

Alternative 8: 63.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 1 - \mathsf{log1p}\left(-x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 (log1p (- x))))

C

double code(double x, double y) {
	return 1.0 - log1p(-x);
}

Java

public static double code(double x, double y) {
	return 1.0 - Math.log1p(-x);
}

Python

def code(x, y):
	return 1.0 - math.log1p(-x)

Julia

function code(x, y)
	return Float64(1.0 - log1p(Float64(-x)))
end

Wolfram

code[x_, y_] := N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.7%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. *-lft-identity N/A

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

   2. metadata-eval N/A

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

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

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

   4. lower-log1p.f64 N/A

      \[\leadsto 1 - \mathsf{log1p}\left(-1 \cdot x\right) \]

   5. mul-1-neg N/A

      \[\leadsto 1 - \mathsf{log1p}\left(\mathsf{neg}\left(x\right)\right) \]

   6. lower-neg.f64 63.0

      \[\leadsto 1 - \mathsf{log1p}\left(-x\right) \]

5. Applied rewrites 63.0%

   \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(-x\right)} \]
6. Add Preprocessing

Alternative 9: 43.7% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ 1 - \left(-x\right) \end{array} \]

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 - Float64(-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 73.7%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. *-lft-identity N/A

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

   2. metadata-eval N/A

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

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

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

   4. lower-log1p.f64 N/A

      \[\leadsto 1 - \mathsf{log1p}\left(-1 \cdot x\right) \]

   5. mul-1-neg N/A

      \[\leadsto 1 - \mathsf{log1p}\left(\mathsf{neg}\left(x\right)\right) \]

   6. lower-neg.f64 63.0

      \[\leadsto 1 - \mathsf{log1p}\left(-x\right) \]

5. Applied rewrites 63.0%

   \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(-x\right)} \]

6. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 38.5

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

8. Applied rewrites 38.5%

   \[\leadsto 1 - \left(-x\right) \]
9. Add Preprocessing

Alternative 10: 41.2% accurate, 31.0× speedup

Math

\[\begin{array}{l} \\ 1 - y \end{array} \]

FPCore

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

C

double code(double x, double y) {
	return 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 - y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.7%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. lower-log1p.f64 N/A

      \[\leadsto 1 - \mathsf{log1p}\left(\frac{y}{1 - y}\right) \]

   2. lower-/.f64 N/A

      \[\leadsto 1 - \mathsf{log1p}\left(\frac{y}{1 - y}\right) \]

   3. lower--.f64 35.8

      \[\leadsto 1 - \mathsf{log1p}\left(\frac{y}{1 - y}\right) \]

5. Applied rewrites 35.8%

   \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(\frac{y}{1 - y}\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto 1 - y \]
7. Step-by-step derivation

8. Applied rewrites 35.7%

   \[\leadsto 1 - y \]
9. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{if}\;y < -81284752.61947241:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 3.0094271212461764 \cdot 10^{+25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y)))))))
   (if (< y -81284752.61947241)
     t_0
     (if (< y 3.0094271212461764e+25)
       (log (/ (exp 1.0) (- 1.0 (/ (- x y) (- 1.0 y)))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log(((x / (y * y)) - ((1.0 / y) - (x / y))));
	double tmp;
	if (y < -81284752.61947241) {
		tmp = t_0;
	} else if (y < 3.0094271212461764e+25) {
		tmp = log((exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))));
	} 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 / (y * y)) - ((1.0d0 / y) - (x / y))))
    if (y < (-81284752.61947241d0)) then
        tmp = t_0
    else if (y < 3.0094271212461764d+25) then
        tmp = log((exp(1.0d0) / (1.0d0 - ((x - y) / (1.0d0 - y)))))
    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 / (y * y)) - ((1.0 / y) - (x / y))));
	double tmp;
	if (y < -81284752.61947241) {
		tmp = t_0;
	} else if (y < 3.0094271212461764e+25) {
		tmp = Math.log((Math.exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - math.log(((x / (y * y)) - ((1.0 / y) - (x / y))))
	tmp = 0
	if y < -81284752.61947241:
		tmp = t_0
	elif y < 3.0094271212461764e+25:
		tmp = math.log((math.exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - log(Float64(Float64(x / Float64(y * y)) - Float64(Float64(1.0 / y) - Float64(x / y)))))
	tmp = 0.0
	if (y < -81284752.61947241)
		tmp = t_0;
	elseif (y < 3.0094271212461764e+25)
		tmp = log(Float64(exp(1.0) / Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - log(((x / (y * y)) - ((1.0 / y) - (x / y))));
	tmp = 0.0;
	if (y < -81284752.61947241)
		tmp = t_0;
	elseif (y < 3.0094271212461764e+25)
		tmp = log((exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))));
	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 / N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / y), $MachinePrecision] - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Less[y, -81284752.61947241], t$95$0, If[Less[y, 3.0094271212461764e+25], N[Log[N[(N[Exp[1.0], $MachinePrecision] / N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2025026 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -8128475261947241/100000000) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))) (if (< y 30094271212461764000000000) (log (/ (exp 1) (- 1 (/ (- x y) (- 1 y))))) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))))))

  (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))