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

Percentage Accurate: 72.4% → 99.9%

Time: 8.8s

Alternatives: 8

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

real(8) function code(x, y)
    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: 72.4% 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

real(8) function code(x, y)
    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, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t\_0 \leq 0.999:\\ \;\;\;\;1 - \log \left(1 - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\left(x - \frac{1 - \left(x - \frac{1 - x}{y}\right)}{y}\right) - 1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (<= t_0 0.999)
     (- 1.0 (log (- 1.0 t_0)))
     (- 1.0 (log (/ (- (- x (/ (- 1.0 (- x (/ (- 1.0 x) y))) y)) 1.0) y))))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if (t_0 <= 0.999) {
		tmp = 1.0 - log((1.0 - t_0));
	} else {
		tmp = 1.0 - log((((x - ((1.0 - (x - ((1.0 - x) / y))) / y)) - 1.0) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / (1.0d0 - y)
    if (t_0 <= 0.999d0) then
        tmp = 1.0d0 - log((1.0d0 - t_0))
    else
        tmp = 1.0d0 - log((((x - ((1.0d0 - (x - ((1.0d0 - x) / y))) / y)) - 1.0d0) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if (t_0 <= 0.999) {
		tmp = 1.0 - Math.log((1.0 - t_0));
	} else {
		tmp = 1.0 - Math.log((((x - ((1.0 - (x - ((1.0 - x) / y))) / y)) - 1.0) / y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (1.0 - y)
	tmp = 0
	if t_0 <= 0.999:
		tmp = 1.0 - math.log((1.0 - t_0))
	else:
		tmp = 1.0 - math.log((((x - ((1.0 - (x - ((1.0 - x) / y))) / y)) - 1.0) / y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(1.0 - y))
	tmp = 0.0
	if (t_0 <= 0.999)
		tmp = Float64(1.0 - log(Float64(1.0 - t_0)));
	else
		tmp = Float64(1.0 - log(Float64(Float64(Float64(x - Float64(Float64(1.0 - Float64(x - Float64(Float64(1.0 - x) / y))) / y)) - 1.0) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / (1.0 - y);
	tmp = 0.0;
	if (t_0 <= 0.999)
		tmp = 1.0 - log((1.0 - t_0));
	else
		tmp = 1.0 - log((((x - ((1.0 - (x - ((1.0 - x) / y))) / y)) - 1.0) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.999], N[(1.0 - N[Log[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(N[(x - N[(N[(1.0 - N[(x - N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.998999999999999999

   1. Initial program 100.0%

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

   if 0.998999999999999999 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 6.1%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 100.0%

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

Alternative 2: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t\_0 \leq 0.999:\\ \;\;\;\;1 - \log \left(1 - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\left(x - \frac{1 - x}{y}\right) - 1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (<= t_0 0.999)
     (- 1.0 (log (- 1.0 t_0)))
     (- 1.0 (log (/ (- (- x (/ (- 1.0 x) y)) 1.0) y))))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if (t_0 <= 0.999) {
		tmp = 1.0 - log((1.0 - t_0));
	} else {
		tmp = 1.0 - log((((x - ((1.0 - x) / y)) - 1.0) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / (1.0d0 - y)
    if (t_0 <= 0.999d0) then
        tmp = 1.0d0 - log((1.0d0 - t_0))
    else
        tmp = 1.0d0 - log((((x - ((1.0d0 - x) / y)) - 1.0d0) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if (t_0 <= 0.999) {
		tmp = 1.0 - Math.log((1.0 - t_0));
	} else {
		tmp = 1.0 - Math.log((((x - ((1.0 - x) / y)) - 1.0) / y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (1.0 - y)
	tmp = 0
	if t_0 <= 0.999:
		tmp = 1.0 - math.log((1.0 - t_0))
	else:
		tmp = 1.0 - math.log((((x - ((1.0 - x) / y)) - 1.0) / y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(1.0 - y))
	tmp = 0.0
	if (t_0 <= 0.999)
		tmp = Float64(1.0 - log(Float64(1.0 - t_0)));
	else
		tmp = Float64(1.0 - log(Float64(Float64(Float64(x - Float64(Float64(1.0 - x) / y)) - 1.0) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / (1.0 - y);
	tmp = 0.0;
	if (t_0 <= 0.999)
		tmp = 1.0 - log((1.0 - t_0));
	else
		tmp = 1.0 - log((((x - ((1.0 - x) / y)) - 1.0) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.999], N[(1.0 - N[Log[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(N[(x - N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.998999999999999999

   1. Initial program 100.0%

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

   if 0.998999999999999999 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 6.1%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 100.0%

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

Alternative 3: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t\_0 \leq 0.99999998:\\ \;\;\;\;1 - \log \left(1 - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x - 1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (<= t_0 0.99999998)
     (- 1.0 (log (- 1.0 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 (t_0 <= 0.99999998) {
		tmp = 1.0 - log((1.0 - t_0));
	} else {
		tmp = 1.0 - log(((x - 1.0) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / (1.0d0 - y)
    if (t_0 <= 0.99999998d0) then
        tmp = 1.0d0 - log((1.0d0 - 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 = (x - y) / (1.0 - y);
	double tmp;
	if (t_0 <= 0.99999998) {
		tmp = 1.0 - Math.log((1.0 - 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 t_0 <= 0.99999998:
		tmp = 1.0 - math.log((1.0 - 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 (t_0 <= 0.99999998)
		tmp = Float64(1.0 - log(Float64(1.0 - 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 = (x - y) / (1.0 - y);
	tmp = 0.0;
	if (t_0 <= 0.99999998)
		tmp = 1.0 - log((1.0 - 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[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.99999998], N[(1.0 - N[Log[N[(1.0 - t$95$0), $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 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.999999980000000011

   1. Initial program 99.8%

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

   if 0.999999980000000011 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 5.2%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. neg-mul-1 N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. sub-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 99.5

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

   5. Applied rewrites 99.5%

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

Alternative 4: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - \frac{x - y}{1 - y} \leq 0.1:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- 1.0 (/ (- x y) (- 1.0 y))) 0.1)
   (- 1.0 (log (/ -1.0 y)))
   (- 1.0 (log (- 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if ((1.0 - ((x - y) / (1.0 - y))) <= 0.1) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - log((1.0 - x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((1.0d0 - ((x - y) / (1.0d0 - y))) <= 0.1d0) 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 ((1.0 - ((x - y) / (1.0 - y))) <= 0.1) {
		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 (1.0 - ((x - y) / (1.0 - y))) <= 0.1:
		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 (Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y))) <= 0.1)
		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 ((1.0 - ((x - y) / (1.0 - y))) <= 0.1)
		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[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.1], 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 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))) < 0.10000000000000001

   1. Initial program 8.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 8.6

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

   5. Applied rewrites 8.6%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 70.6%

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

   if 0.10000000000000001 < (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 87.2

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

   5. Applied rewrites 87.2%

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

Alternative 5: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.45 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \log \left(\frac{x - 1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{fma}\left(\mathsf{fma}\left(0.5, y, 1\right), y, \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.45) (not (<= y 1.0)))
   (- 1.0 (log (/ (- x 1.0) y)))
   (- 1.0 (fma (fma 0.5 y 1.0) y (log1p (- x))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.45) || !(y <= 1.0)) {
		tmp = 1.0 - log(((x - 1.0) / y));
	} else {
		tmp = 1.0 - fma(fma(0.5, y, 1.0), y, log1p(-x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.45) || !(y <= 1.0))
		tmp = Float64(1.0 - log(Float64(Float64(x - 1.0) / y)));
	else
		tmp = Float64(1.0 - fma(fma(0.5, y, 1.0), y, log1p(Float64(-x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.45], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(1.0 - N[Log[N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(0.5 * y + 1.0), $MachinePrecision] * y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.44999999999999996 or 1 < y

   1. Initial program 31.2%

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

   3. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. neg-mul-1 N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. sub-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 97.8

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

   5. Applied rewrites 97.8%

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

   if -1.44999999999999996 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 99.5%

      \[\leadsto 1 - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, 1\right), y, \mathsf{log1p}\left(-x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \log \left(\frac{x - 1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{fma}\left(\mathsf{fma}\left(0.5, y, 1\right), y, \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.5% accurate, 1.2× 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

real(8) function code(x, y)
    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 73.9%

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

3. Taylor expanded in y around 0

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

   1. lower--.f64 65.3

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

5. Applied rewrites 65.3%

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

Alternative 7: 63.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

3. Taylor expanded in y around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 65.3

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

5. Applied rewrites 65.3%

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

Alternative 8: 43.9% accurate, 20.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

3. Taylor expanded in y around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 65.3

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

5. Applied rewrites 65.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 44.9%

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

Developer Target 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - log(((x / (y * y)) - ((1.0d0 / y) - (x / y))))
    if (y < (-81284752.61947241d0)) then
        tmp = t_0
    else if (y < 3.0094271212461764d+25) then
        tmp = log((exp(1.0d0) / (1.0d0 - ((x - y) / (1.0d0 - y)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - log(((x / (y * y)) - ((1.0 / y) - (x / y))));
	tmp = 0.0;
	if (y < -81284752.61947241)
		tmp = t_0;
	elseif (y < 3.0094271212461764e+25)
		tmp = log((exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Log[N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / y), $MachinePrecision] - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Less[y, -81284752.61947241], t$95$0, If[Less[y, 3.0094271212461764e+25], N[Log[N[(N[Exp[1.0], $MachinePrecision] / N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024318 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

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

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