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

Percentage Accurate: 72.8% → 99.7%

Time: 9.4s

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: 72.8% 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: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double t_0 = 1.0 - log((1.0 - ((x - y) / (1.0 - y))));
	double tmp;
	if (t_0 <= 10.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 - log(((-1.0 + x) / 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 = 1.0d0 - log((1.0d0 - ((x - y) / (1.0d0 - y))))
    if (t_0 <= 10.0d0) then
        tmp = t_0
    else
        tmp = 1.0d0 - log((((-1.0d0) + x) / y))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - log((1.0 - ((x - y) / (1.0 - y))));
	tmp = 0.0;
	if (t_0 <= 10.0)
		tmp = t_0;
	else
		tmp = 1.0 - log(((-1.0 + x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   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 4.1%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 2: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(1 - \frac{x - y}{1 - y}\right)\\ \mathbf{if}\;t\_0 \leq -28:\\ \;\;\;\;1 - \log \left(\frac{x}{-1 + y}\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 - \log \left(\left(y + 1\right) \cdot \left(1 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1 + x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y)))))))
   (if (<= t_0 -28.0)
     (- 1.0 (log (/ x (+ -1.0 y))))
     (if (<= t_0 2.0)
       (- 1.0 (log (* (+ y 1.0) (- 1.0 x))))
       (- 1.0 (log (/ (+ -1.0 x) y)))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log((1.0 - ((x - y) / (1.0 - y))));
	double tmp;
	if (t_0 <= -28.0) {
		tmp = 1.0 - log((x / (-1.0 + y)));
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - log(((y + 1.0) * (1.0 - x)));
	} else {
		tmp = 1.0 - log(((-1.0 + x) / 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 = 1.0d0 - log((1.0d0 - ((x - y) / (1.0d0 - y))))
    if (t_0 <= (-28.0d0)) then
        tmp = 1.0d0 - log((x / ((-1.0d0) + y)))
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0 - log(((y + 1.0d0) * (1.0d0 - x)))
    else
        tmp = 1.0d0 - log((((-1.0d0) + x) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - Math.log((1.0 - ((x - y) / (1.0 - y))));
	double tmp;
	if (t_0 <= -28.0) {
		tmp = 1.0 - Math.log((x / (-1.0 + y)));
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - Math.log(((y + 1.0) * (1.0 - x)));
	} else {
		tmp = 1.0 - Math.log(((-1.0 + x) / y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - math.log((1.0 - ((x - y) / (1.0 - y))))
	tmp = 0
	if t_0 <= -28.0:
		tmp = 1.0 - math.log((x / (-1.0 + y)))
	elif t_0 <= 2.0:
		tmp = 1.0 - math.log(((y + 1.0) * (1.0 - x)))
	else:
		tmp = 1.0 - math.log(((-1.0 + x) / y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - log(Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y)))))
	tmp = 0.0
	if (t_0 <= -28.0)
		tmp = Float64(1.0 - log(Float64(x / Float64(-1.0 + y))));
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 - log(Float64(Float64(y + 1.0) * Float64(1.0 - x))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(-1.0 + x) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - log((1.0 - ((x - y) / (1.0 - y))));
	tmp = 0.0;
	if (t_0 <= -28.0)
		tmp = 1.0 - log((x / (-1.0 + y)));
	elseif (t_0 <= 2.0)
		tmp = 1.0 - log(((y + 1.0) * (1.0 - x)));
	else
		tmp = 1.0 - log(((-1.0 + x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Log[N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -28.0], 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[(N[(y + 1.0), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(-1.0 + x), $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))))) < -28

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -28 < (-.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 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower--.f64 98.7

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

   5. Applied rewrites 98.7%

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

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

   1. Initial program 6.7%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-+.f64 99.1

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

   5. Applied rewrites 99.1%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \log \left(1 - \frac{x - y}{1 - y}\right) \leq -28:\\ \;\;\;\;1 - \log \left(\frac{x}{-1 + y}\right)\\ \mathbf{elif}\;1 - \log \left(1 - \frac{x - y}{1 - y}\right) \leq 2:\\ \;\;\;\;1 - \log \left(\left(y + 1\right) \cdot \left(1 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1 + x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.6% accurate, 0.5× speedup

Math

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

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 (log1p (- 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 - log1p(-x);
	}
	return tmp;
}

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.log1p(-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.log1p(-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 - log1p(Float64(-x)));
	end
	return 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[1 + (-x)], $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 6.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 4.4

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

   5. Applied rewrites 4.4%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 65.3%

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

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

   1. Initial program 99.9%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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. *-lft-identity N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-log1p.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 85.0

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

   5. Applied rewrites 85.0%

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

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -0.819999999999999951 or 1 < y

   1. Initial program 30.9%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if -0.819999999999999951 < y < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower--.f64 99.1

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

   5. Applied rewrites 99.1%

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

4. Final simplification 98.7%

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

Alternative 5: 88.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 18.9%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 4.1

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

   5. Applied rewrites 4.1%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 67.3%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower--.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if 1 < y

   1. Initial program 61.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 96.5

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

   5. Applied rewrites 96.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 93.9%

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

Alternative 6: 88.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1800:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1800.0)
   (- 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 <= -1800.0) {
		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 <= -1800.0) {
		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 <= -1800.0:
		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 <= -1800.0)
		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, -1800.0], 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 < -1800

   1. Initial program 16.6%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 2.9

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

   5. Applied rewrites 2.9%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 68.7%

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

   if -1800 < y < 1

   1. Initial program 99.9%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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. *-lft-identity N/A

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

      2. metadata-eval N/A

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

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

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

      4. lower-log1p.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 97.1

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

   5. Applied rewrites 97.1%

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

   if 1 < y

   1. Initial program 61.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 96.5

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

   5. Applied rewrites 96.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 93.9%

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

Alternative 7: 62.8% 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 73.0%

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

3. Taylor expanded in y around 0

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

   1. *-lft-identity N/A

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

   2. metadata-eval N/A

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

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

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

   4. lower-log1p.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 63.2

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

5. Applied rewrites 63.2%

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

Alternative 8: 43.4% 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 73.0%

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

3. Taylor expanded in y around 0

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

   1. *-lft-identity N/A

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

   2. metadata-eval N/A

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

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

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

   4. lower-log1p.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 63.2

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

5. Applied rewrites 63.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 44.8%

   \[\leadsto 1 - \left(-x\right) \]
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 2024338 
(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))))))