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

Percentage Accurate: 72.6% → 99.9%

Time: 4.6s

Alternatives: 13

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: 72.6% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;1 - \log \left(1 - t\_0\right) \leq 10:\\ \;\;\;\;1 - \mathsf{log1p}\left(-t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(-\frac{\left(\left(-\frac{x - 1}{y}\right) + 1\right) - x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (<= (- 1.0 (log (- 1.0 t_0))) 10.0)
     (- 1.0 (log1p (- t_0)))
     (- 1.0 (log (- (/ (- (+ (- (/ (- x 1.0) y)) 1.0) x) y)))))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if ((1.0 - log((1.0 - t_0))) <= 10.0) {
		tmp = 1.0 - log1p(-t_0);
	} else {
		tmp = 1.0 - log(-(((-((x - 1.0) / y) + 1.0) - x) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if ((1.0 - Math.log((1.0 - t_0))) <= 10.0) {
		tmp = 1.0 - Math.log1p(-t_0);
	} else {
		tmp = 1.0 - Math.log(-(((-((x - 1.0) / y) + 1.0) - x) / y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (1.0 - y)
	tmp = 0
	if (1.0 - math.log((1.0 - t_0))) <= 10.0:
		tmp = 1.0 - math.log1p(-t_0)
	else:
		tmp = 1.0 - math.log(-(((-((x - 1.0) / y) + 1.0) - x) / y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(1.0 - y))
	tmp = 0.0
	if (Float64(1.0 - log(Float64(1.0 - t_0))) <= 10.0)
		tmp = Float64(1.0 - log1p(Float64(-t_0)));
	else
		tmp = Float64(1.0 - log(Float64(-Float64(Float64(Float64(Float64(-Float64(Float64(x - 1.0) / y)) + 1.0) - x) / y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 - N[Log[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 10.0], N[(1.0 - N[Log[1 + (-t$95$0)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[(-N[(N[(N[((-N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]) + 1.0), $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))))) < 10

   1. Initial program 72.6%

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

      1. lift-log.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sub-flip N/A

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

      7. lower-log1p.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 72.6

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

   3. Applied rewrites 72.6%

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

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

   1. Initial program 72.6%

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

   2. Taylor expanded in y around -inf

      \[\leadsto 1 - \log \color{blue}{\left(-1 \cdot \frac{\left(1 + -1 \cdot \frac{x - 1}{y}\right) - x}{y}\right)} \]
   3. 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. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 39.7

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

   4. Applied rewrites 39.7%

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

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;1 - \log \left(1 - t\_0\right) \leq 10:\\ \;\;\;\;1 - \mathsf{log1p}\left(-t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x - 1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (<= (- 1.0 (log (- 1.0 t_0))) 10.0)
     (- 1.0 (log1p (- t_0)))
     (- 1.0 (log (/ (- x 1.0) y))))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if ((1.0 - log((1.0 - t_0))) <= 10.0) {
		tmp = 1.0 - log1p(-t_0);
	} else {
		tmp = 1.0 - log(((x - 1.0) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if ((1.0 - Math.log((1.0 - t_0))) <= 10.0) {
		tmp = 1.0 - Math.log1p(-t_0);
	} else {
		tmp = 1.0 - Math.log(((x - 1.0) / y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (1.0 - y)
	tmp = 0
	if (1.0 - math.log((1.0 - t_0))) <= 10.0:
		tmp = 1.0 - math.log1p(-t_0)
	else:
		tmp = 1.0 - math.log(((x - 1.0) / y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(1.0 - y))
	tmp = 0.0
	if (Float64(1.0 - log(Float64(1.0 - t_0))) <= 10.0)
		tmp = Float64(1.0 - log1p(Float64(-t_0)));
	else
		tmp = Float64(1.0 - log(Float64(Float64(x - 1.0) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 - N[Log[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 10.0], N[(1.0 - N[Log[1 + (-t$95$0)], $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))))) < 10

   1. Initial program 72.6%

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

      1. lift-log.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sub-flip N/A

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

      7. lower-log1p.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 72.6

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

   3. Applied rewrites 72.6%

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

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

   1. Initial program 72.6%

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

   2. Taylor expanded in y around inf

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

      1. sum-log N/A

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

      2. mul-1-neg N/A

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

      3. sub-flip N/A

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

      4. sub-negate-rev N/A

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

      5. mult-flip N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 40.8

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

   4. Applied rewrites 40.8%

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

Alternative 3: 99.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y)))))))
   (if (<= t_0 10.0) 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 <= 10.0) {
		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 <= 10.0d0) 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 <= 10.0) {
		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 <= 10.0:
		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 <= 10.0)
		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 <= 10.0)
		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, 10.0], 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))))) < 10

   1. Initial program 72.6%

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

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

   1. Initial program 72.6%

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

   2. Taylor expanded in y around inf

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

      1. sum-log N/A

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

      2. mul-1-neg N/A

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

      3. sub-flip N/A

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

      4. sub-negate-rev N/A

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

      5. mult-flip N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 40.8

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

   4. Applied rewrites 40.8%

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

Alternative 4: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(1 - \frac{x - y}{1 - y}\right)\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;1 - \log \left(\frac{-x}{1 - y}\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 - \log \left(1 - \frac{x - y}{1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x - 1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y)))))))
   (if (<= t_0 -5.0)
     (- 1.0 (log (/ (- x) (- 1.0 y))))
     (if (<= t_0 2.0)
       (- 1.0 (log (- 1.0 (/ (- x y) 1.0))))
       (- 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 <= -5.0) {
		tmp = 1.0 - log((-x / (1.0 - y)));
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - log((1.0 - ((x - y) / 1.0)));
	} 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 <= (-5.0d0)) then
        tmp = 1.0d0 - log((-x / (1.0d0 - y)))
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0 - log((1.0d0 - ((x - y) / 1.0d0)))
    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 <= -5.0) {
		tmp = 1.0 - Math.log((-x / (1.0 - y)));
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - Math.log((1.0 - ((x - y) / 1.0)));
	} 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 <= -5.0:
		tmp = 1.0 - math.log((-x / (1.0 - y)))
	elif t_0 <= 2.0:
		tmp = 1.0 - math.log((1.0 - ((x - y) / 1.0)))
	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 <= -5.0)
		tmp = Float64(1.0 - log(Float64(Float64(-x) / Float64(1.0 - y))));
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 - log(Float64(1.0 - Float64(Float64(x - y) / 1.0))));
	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 <= -5.0)
		tmp = 1.0 - log((-x / (1.0 - y)));
	elseif (t_0 <= 2.0)
		tmp = 1.0 - log((1.0 - ((x - y) / 1.0)));
	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, -5.0], N[(1.0 - N[Log[N[((-x) / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(1.0 - N[Log[N[(1.0 - N[(N[(x - y), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

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))))) < -5

   1. Initial program 72.6%

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

   2. Taylor expanded in x around inf

      \[\leadsto 1 - \log \color{blue}{\left(-1 \cdot \frac{x}{1 - y}\right)} \]
   3. 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. lift--.f64 44.4

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

   4. Applied rewrites 44.4%

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

   if -5 < (-.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 72.6%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 60.1%

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

   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 72.6%

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

   2. Taylor expanded in y around inf

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

      1. sum-log N/A

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

      2. mul-1-neg N/A

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

      3. sub-flip N/A

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

      4. sub-negate-rev N/A

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

      5. mult-flip N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 40.8

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

   4. Applied rewrites 40.8%

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

Alternative 5: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(1 - \frac{x - y}{1 - y}\right)\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;1 - \log \left(\frac{-x}{1 - y}\right)\\ \mathbf{elif}\;t\_0 \leq 10:\\ \;\;\;\;1 - \log \left(\frac{y}{1 - y} - -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x - 1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y)))))))
   (if (<= t_0 -5.0)
     (- 1.0 (log (/ (- x) (- 1.0 y))))
     (if (<= t_0 10.0)
       (- 1.0 (log (- (/ y (- 1.0 y)) -1.0)))
       (- 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 <= -5.0) {
		tmp = 1.0 - log((-x / (1.0 - y)));
	} else if (t_0 <= 10.0) {
		tmp = 1.0 - log(((y / (1.0 - y)) - -1.0));
	} 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 <= (-5.0d0)) then
        tmp = 1.0d0 - log((-x / (1.0d0 - y)))
    else if (t_0 <= 10.0d0) then
        tmp = 1.0d0 - log(((y / (1.0d0 - y)) - (-1.0d0)))
    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 <= -5.0) {
		tmp = 1.0 - Math.log((-x / (1.0 - y)));
	} else if (t_0 <= 10.0) {
		tmp = 1.0 - Math.log(((y / (1.0 - y)) - -1.0));
	} 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 <= -5.0:
		tmp = 1.0 - math.log((-x / (1.0 - y)))
	elif t_0 <= 10.0:
		tmp = 1.0 - math.log(((y / (1.0 - y)) - -1.0))
	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 <= -5.0)
		tmp = Float64(1.0 - log(Float64(Float64(-x) / Float64(1.0 - y))));
	elseif (t_0 <= 10.0)
		tmp = Float64(1.0 - log(Float64(Float64(y / Float64(1.0 - y)) - -1.0)));
	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 <= -5.0)
		tmp = 1.0 - log((-x / (1.0 - y)));
	elseif (t_0 <= 10.0)
		tmp = 1.0 - log(((y / (1.0 - y)) - -1.0));
	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, -5.0], N[(1.0 - N[Log[N[((-x) / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 10.0], N[(1.0 - N[Log[N[(N[(y / N[(1.0 - y), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

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))))) < -5

   1. Initial program 72.6%

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

   2. Taylor expanded in x around inf

      \[\leadsto 1 - \log \color{blue}{\left(-1 \cdot \frac{x}{1 - y}\right)} \]
   3. 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. lift--.f64 44.4

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

   4. Applied rewrites 44.4%

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

   if -5 < (-.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 72.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 40.5

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

   4. Applied rewrites 40.5%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. add-flip N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 40.5

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

   6. Applied rewrites 40.5%

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

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

   1. Initial program 72.6%

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

   2. Taylor expanded in y around inf

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

      1. sum-log N/A

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

      2. mul-1-neg N/A

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

      3. sub-flip N/A

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

      4. sub-negate-rev N/A

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

      5. mult-flip N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 40.8

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

   4. Applied rewrites 40.8%

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

Alternative 6: 98.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(1 - \frac{x - y}{1 - y}\right)\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;1 - \log \left(\frac{-x}{1 - y}\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x - 1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y)))))))
   (if (<= t_0 -5.0)
     (- 1.0 (log (/ (- x) (- 1.0 y))))
     (if (<= t_0 2.0) (- 1.0 (log1p (- x))) (- 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 <= -5.0) {
		tmp = 1.0 - log((-x / (1.0 - y)));
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - log1p(-x);
	} else {
		tmp = 1.0 - log(((x - 1.0) / y));
	}
	return tmp;
}

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 <= -5.0) {
		tmp = 1.0 - Math.log((-x / (1.0 - y)));
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - Math.log1p(-x);
	} 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 <= -5.0:
		tmp = 1.0 - math.log((-x / (1.0 - y)))
	elif t_0 <= 2.0:
		tmp = 1.0 - math.log1p(-x)
	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 <= -5.0)
		tmp = Float64(1.0 - log(Float64(Float64(-x) / Float64(1.0 - y))));
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 - log1p(Float64(-x)));
	else
		tmp = Float64(1.0 - log(Float64(Float64(x - 1.0) / y)));
	end
	return 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, -5.0], N[(1.0 - N[Log[N[((-x) / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

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))))) < -5

   1. Initial program 72.6%

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

   2. Taylor expanded in x around inf

      \[\leadsto 1 - \log \color{blue}{\left(-1 \cdot \frac{x}{1 - y}\right)} \]
   3. 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. lift--.f64 44.4

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

   4. Applied rewrites 44.4%

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

   if -5 < (-.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 72.6%

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

      1. lift-log.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sub-flip N/A

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

      7. lower-log1p.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 72.6

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

   3. Applied rewrites 72.6%

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

   4. Taylor expanded in x around inf

      \[\leadsto 1 - \mathsf{log1p}\left(-\color{blue}{\frac{x}{1 - y}}\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lift--.f64 73.4

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

   6. Applied rewrites 73.4%

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

   7. Taylor expanded in y around 0

      \[\leadsto 1 - \mathsf{log1p}\left(-\left(x + \color{blue}{x \cdot y}\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 60.4

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

   9. Applied rewrites 60.4%

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

   10. Taylor expanded in y around 0

       \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{-1 \cdot x}\right) \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. lower-neg.f64 62.6

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

   12. Applied rewrites 62.6%

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

   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 72.6%

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

   2. Taylor expanded in y around inf

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

      1. sum-log N/A

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

      2. mul-1-neg N/A

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

      3. sub-flip N/A

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

      4. sub-negate-rev N/A

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

      5. mult-flip N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 40.8

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

   4. Applied rewrites 40.8%

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

Alternative 7: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(\frac{x - 1}{y}\right)\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (/ (- x 1.0) y)))))
   (if (<= y -1.0) t_0 (if (<= y 1.0) (- 1.0 (log1p (- x))) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log(((x - 1.0) / y));
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = 1.0 - log1p(-x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - Math.log(((x - 1.0) / y));
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = 1.0 - Math.log1p(-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 <= -1.0:
		tmp = t_0
	elif y <= 1.0:
		tmp = 1.0 - math.log1p(-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 <= -1.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(1.0 - log1p(Float64(-x)));
	else
		tmp = t_0;
	end
	return 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, -1.0], t$95$0, If[LessEqual[y, 1.0], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 72.6%

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

   2. Taylor expanded in y around inf

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

      1. sum-log N/A

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

      2. mul-1-neg N/A

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

      3. sub-flip N/A

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

      4. sub-negate-rev N/A

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

      5. mult-flip N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 40.8

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

   4. Applied rewrites 40.8%

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

   if -1 < y < 1

   1. Initial program 72.6%

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

      1. lift-log.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sub-flip N/A

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

      7. lower-log1p.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 72.6

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

   3. Applied rewrites 72.6%

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

   4. Taylor expanded in x around inf

      \[\leadsto 1 - \mathsf{log1p}\left(-\color{blue}{\frac{x}{1 - y}}\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lift--.f64 73.4

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

   6. Applied rewrites 73.4%

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

   7. Taylor expanded in y around 0

      \[\leadsto 1 - \mathsf{log1p}\left(-\left(x + \color{blue}{x \cdot y}\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 60.4

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

   9. Applied rewrites 60.4%

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

   10. Taylor expanded in y around 0

       \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{-1 \cdot x}\right) \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. lower-neg.f64 62.6

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

   12. Applied rewrites 62.6%

       \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{-x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 98.0% accurate, 0.5× speedup

Math

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

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 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 - log1p(-(x / (1.0 - y)));
	} else {
		tmp = 1.0 - log(((x - 1.0) / 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 / (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.log1p(-(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 - log1p(Float64(-Float64(x / Float64(1.0 - y)))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(x - 1.0) / 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 + (-N[(x / 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))))) < 2

   1. Initial program 72.6%

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

      1. lift-log.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sub-flip N/A

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

      7. lower-log1p.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 72.6

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

   3. Applied rewrites 72.6%

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

   4. Taylor expanded in x around inf

      \[\leadsto 1 - \mathsf{log1p}\left(-\color{blue}{\frac{x}{1 - y}}\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lift--.f64 73.4

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

   6. Applied rewrites 73.4%

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

   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 72.6%

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

   2. Taylor expanded in y around inf

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

      1. sum-log N/A

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

      2. mul-1-neg N/A

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

      3. sub-flip N/A

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

      4. sub-negate-rev N/A

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

      5. mult-flip N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 40.8

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

   4. Applied rewrites 40.8%

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

Alternative 9: 89.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.8)
   (- 1.0 (log (/ -1.0 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 <= -4.8) {
		tmp = 1.0 - log((-1.0 / 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 <= -4.8) {
		tmp = 1.0 - Math.log((-1.0 / 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 <= -4.8:
		tmp = 1.0 - math.log((-1.0 / 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 <= -4.8)
		tmp = Float64(1.0 - log(Float64(-1.0 / 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, -4.8], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $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 < -4.79999999999999982

   1. Initial program 72.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 40.5

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

   4. Applied rewrites 40.5%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. add-flip N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 40.5

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

   6. Applied rewrites 40.5%

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

   7. Taylor expanded in y around inf

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

      1. lower-/.f64 22.2

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

   9. Applied rewrites 22.2%

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

   if -4.79999999999999982 < y < 1

   1. Initial program 72.6%

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

      1. lift-log.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. sub-flip N/A

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

      7. lower-log1p.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 72.6

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

   3. Applied rewrites 72.6%

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

   4. Taylor expanded in x around inf

      \[\leadsto 1 - \mathsf{log1p}\left(-\color{blue}{\frac{x}{1 - y}}\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lift--.f64 73.4

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

   6. Applied rewrites 73.4%

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

   7. Taylor expanded in y around 0

      \[\leadsto 1 - \mathsf{log1p}\left(-\left(x + \color{blue}{x \cdot y}\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 60.4

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

   9. Applied rewrites 60.4%

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

   10. Taylor expanded in y around 0

       \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{-1 \cdot x}\right) \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. lower-neg.f64 62.6

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

   12. Applied rewrites 62.6%

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

   if 1 < y

   1. Initial program 72.6%

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

   2. Taylor expanded in x around inf

      \[\leadsto 1 - \log \color{blue}{\left(-1 \cdot \frac{x}{1 - y}\right)} \]
   3. 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. lift--.f64 44.4

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

   4. Applied rewrites 44.4%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lift-neg.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-neg.f64 23.7

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

   7. Applied rewrites 23.7%

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

   8. Taylor expanded in y around inf

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

      1. lower-/.f64 23.8

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

   10. Applied rewrites 23.8%

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

Alternative 10: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \log \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.8)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 1.0) (- 1.0 (log (- 1.0 x))) (- 1.0 (log (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.8) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - 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 <= (-4.8d0)) then
        tmp = 1.0d0 - log(((-1.0d0) / y))
    else if (y <= 1.0d0) then
        tmp = 1.0d0 - 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 <= -4.8) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - Math.log((1.0 - x));
	} else {
		tmp = 1.0 - Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.8:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 1.0:
		tmp = 1.0 - 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 <= -4.8)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - 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 <= -4.8)
		tmp = 1.0 - log((-1.0 / y));
	elseif (y <= 1.0)
		tmp = 1.0 - log((1.0 - x));
	else
		tmp = 1.0 - log((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.8], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[Log[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.79999999999999982

   1. Initial program 72.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 40.5

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

   4. Applied rewrites 40.5%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. add-flip N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 40.5

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

   6. Applied rewrites 40.5%

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

   7. Taylor expanded in y around inf

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

      1. lower-/.f64 22.2

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

   9. Applied rewrites 22.2%

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

   if -4.79999999999999982 < y < 1

   1. Initial program 72.6%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 62.6

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

   4. Applied rewrites 62.6%

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

   if 1 < y

   1. Initial program 72.6%

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

   2. Taylor expanded in x around inf

      \[\leadsto 1 - \log \color{blue}{\left(-1 \cdot \frac{x}{1 - y}\right)} \]
   3. 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. lift--.f64 44.4

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

   4. Applied rewrites 44.4%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lift-neg.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-neg.f64 23.7

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

   7. Applied rewrites 23.7%

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

   8. Taylor expanded in y around inf

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

      1. lower-/.f64 23.8

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

   10. Applied rewrites 23.8%

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

Alternative 11: 79.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (log((1.0 - ((x - y) / (1.0 - y)))) <= -1.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - log((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 (log((1.0d0 - ((x - y) / (1.0d0 - y)))) <= (-1.0d0)) then
        tmp = 1.0d0 - log(((-1.0d0) / y))
    else
        tmp = 1.0d0 - log((1.0d0 - x))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 72.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 40.5

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

   4. Applied rewrites 40.5%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. add-flip N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift--.f64 40.5

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

   6. Applied rewrites 40.5%

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

   7. Taylor expanded in y around inf

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

      1. lower-/.f64 22.2

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

   9. Applied rewrites 22.2%

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

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

   1. Initial program 72.6%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 62.6

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

   4. Applied rewrites 62.6%

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

Alternative 12: 62.6% accurate, 1.7× speedup

Math

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

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 72.6%

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

2. Taylor expanded in y around 0

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

   1. lower--.f64 62.6

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

4. Applied rewrites 62.6%

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

Alternative 13: 42.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return 1.0 - log(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 - log(1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.6%

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

2. Taylor expanded in y around 0

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

   1. lower--.f64 62.6

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

4. Applied rewrites 62.6%

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

5. Taylor expanded in x around 0

   \[\leadsto 1 - \log 1 \]
6. Step-by-step derivation

7. Applied rewrites 42.5%

   \[\leadsto 1 - \log 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025136 
(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))))))