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

Percentage Accurate: 71.8% → 100.0%

Time: 11.0s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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: 71.8% 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: 100.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.998:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + \left(-1 + \frac{-1 + \left(x + \frac{x + -1}{y}\right)}{y}\right)}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.998)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (- 1.0 (log (/ (+ x (+ -1.0 (/ (+ -1.0 (+ x (/ (+ x -1.0) y))) y))) y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.998) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - log(((x + (-1.0 + ((-1.0 + (x + ((x + -1.0) / y))) / y))) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.998) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - Math.log(((x + (-1.0 + ((-1.0 + (x + ((x + -1.0) / y))) / y))) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.998:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 - math.log(((x + (-1.0 + ((-1.0 + (x + ((x + -1.0) / y))) / y))) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.998)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(x + Float64(-1.0 + Float64(Float64(-1.0 + Float64(x + Float64(Float64(x + -1.0) / y))) / y))) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.998], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(x + N[(-1.0 + N[(N[(-1.0 + N[(x + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.998

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. log1p-define 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. neg-sub0 99.9%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{0 - \left(1 - y\right)}}\right) \]

      5. associate--r- 99.9%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{\left(0 - 1\right) + y}}\right) \]

      6. metadata-eval 99.9%

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

      7. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   if 0.998 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 6.9%

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

   3. Taylor expanded in y around -inf 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.998:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + \left(-1 + \frac{-1 + \left(x + \frac{x + -1}{y}\right)}{y}\right)}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.998:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\left(x + -1\right) + \frac{x + -1}{y}}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.998)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (- 1.0 (log (/ (+ (+ x -1.0) (/ (+ x -1.0) y)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.998) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - log((((x + -1.0) + ((x + -1.0) / y)) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.998) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - Math.log((((x + -1.0) + ((x + -1.0) / y)) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.998:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 - math.log((((x + -1.0) + ((x + -1.0) / y)) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.998)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(Float64(x + -1.0) + Float64(Float64(x + -1.0) / y)) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.998], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(N[(x + -1.0), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.998

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. log1p-define 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. neg-sub0 99.9%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{0 - \left(1 - y\right)}}\right) \]

      5. associate--r- 99.9%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{\left(0 - 1\right) + y}}\right) \]

      6. metadata-eval 99.9%

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

      7. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   if 0.998 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 6.9%

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

   3. Taylor expanded in y around -inf 100.0%

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

      1. associate-*r/ 100.0%

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

   5. Simplified 100.0%

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

Alternative 3: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.998:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.998)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (- 1.0 (log (/ (+ x -1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.998) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - log(((x + -1.0) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.998) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - Math.log(((x + -1.0) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.998:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 - math.log(((x + -1.0) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.998)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(x + -1.0) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.998], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -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 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.998

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. log1p-define 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. neg-sub0 99.9%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{0 - \left(1 - y\right)}}\right) \]

      5. associate--r- 99.9%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{\left(0 - 1\right) + y}}\right) \]

      6. metadata-eval 99.9%

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

      7. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   if 0.998 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 6.9%

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

   3. Taylor expanded in y around inf 99.2%

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

      1. associate-*r/ 99.2%

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

      2. neg-mul-1 99.2%

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

      3. +-commutative 99.2%

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

      4. distribute-neg-in 99.2%

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

      5. mul-1-neg 99.2%

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

      6. remove-double-neg 99.2%

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

      7. metadata-eval 99.2%

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

   5. Simplified 99.2%

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

Alternative 4: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.75 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.75) (not (<= y 1.0)))
   (- 1.0 (log (/ (+ x -1.0) y)))
   (- (- 1.0 y) (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.75) || !(y <= 1.0)) {
		tmp = 1.0 - log(((x + -1.0) / y));
	} else {
		tmp = (1.0 - y) - log1p(-x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.75) || !(y <= 1.0)) {
		tmp = 1.0 - Math.log(((x + -1.0) / y));
	} else {
		tmp = (1.0 - y) - Math.log1p(-x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.75) or not (y <= 1.0):
		tmp = 1.0 - math.log(((x + -1.0) / y))
	else:
		tmp = (1.0 - y) - math.log1p(-x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.75) || !(y <= 1.0))
		tmp = Float64(1.0 - log(Float64(Float64(x + -1.0) / y)));
	else
		tmp = Float64(Float64(1.0 - y) - log1p(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.75], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(1.0 - N[Log[N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - y), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.75 or 1 < y

   1. Initial program 31.0%

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

   3. Taylor expanded in y around inf 98.1%

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

      1. associate-*r/ 98.1%

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

      2. neg-mul-1 98.1%

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

      3. +-commutative 98.1%

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

      4. distribute-neg-in 98.1%

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

      5. mul-1-neg 98.1%

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

      6. remove-double-neg 98.1%

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

      7. metadata-eval 98.1%

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

   5. Simplified 98.1%

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

   if -1.75 < 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 97.7%

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

   5. Simplified 97.7%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.75 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -20:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 0.44:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -20.0)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 0.44)
     (- (- 1.0 y) (log1p (- x)))
     (- 1.0 (log (/ x (+ y -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -20.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 0.44) {
		tmp = (1.0 - y) - log1p(-x);
	} else {
		tmp = 1.0 - log((x / (y + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -20.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 0.44) {
		tmp = (1.0 - y) - Math.log1p(-x);
	} else {
		tmp = 1.0 - Math.log((x / (y + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -20.0:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 0.44:
		tmp = (1.0 - y) - math.log1p(-x)
	else:
		tmp = 1.0 - math.log((x / (y + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -20.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 0.44)
		tmp = Float64(Float64(1.0 - y) - log1p(Float64(-x)));
	else
		tmp = Float64(1.0 - log(Float64(x / Float64(y + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -20.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.44], N[(N[(1.0 - y), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -20

   1. Initial program 18.0%

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

   3. Taylor expanded in y around inf 97.8%

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

      1. associate-*r/ 97.8%

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

      2. neg-mul-1 97.8%

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

      3. +-commutative 97.8%

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

      4. distribute-neg-in 97.8%

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

      5. mul-1-neg 97.8%

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

      6. remove-double-neg 97.8%

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

      7. metadata-eval 97.8%

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

   5. Simplified 97.8%

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

   6. Taylor expanded in x around 0 76.2%

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

   if -20 < y < 0.440000000000000002

   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 97.7%

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

   5. Simplified 97.7%

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

   if 0.440000000000000002 < y

   1. Initial program 70.2%

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

   3. Taylor expanded in x around inf 97.6%

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

      1. mul-1-neg 97.6%

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

      2. distribute-neg-frac2 97.6%

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

      3. sub-neg 97.6%

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

      4. distribute-neg-in 97.6%

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

      5. metadata-eval 97.6%

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

      6. remove-double-neg 97.6%

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

      7. +-commutative 97.6%

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

   5. Simplified 97.6%

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

Alternative 6: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.4)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 1.0) (- (- 1.0 y) (log1p (- x))) (- 1.0 (log (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.4) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = (1.0 - y) - log1p(-x);
	} else {
		tmp = 1.0 - log((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.4) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = (1.0 - y) - Math.log1p(-x);
	} else {
		tmp = 1.0 - Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.4:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 1.0:
		tmp = (1.0 - y) - math.log1p(-x)
	else:
		tmp = 1.0 - math.log((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.4)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 1.0)
		tmp = Float64(Float64(1.0 - y) - log1p(Float64(-x)));
	else
		tmp = Float64(1.0 - log(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.4], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(N[(1.0 - y), $MachinePrecision] - 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 < -3.39999999999999991

   1. Initial program 18.0%

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

   3. Taylor expanded in y around inf 97.8%

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

      1. associate-*r/ 97.8%

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

      2. neg-mul-1 97.8%

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

      3. +-commutative 97.8%

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

      4. distribute-neg-in 97.8%

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

      5. mul-1-neg 97.8%

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

      6. remove-double-neg 97.8%

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

      7. metadata-eval 97.8%

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

   5. Simplified 97.8%

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

   6. Taylor expanded in x around 0 76.2%

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

   if -3.39999999999999991 < 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 97.7%

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

   5. Simplified 97.7%

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

   if 1 < y

   1. Initial program 70.2%

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

   3. Taylor expanded in y around inf 98.8%

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

      1. associate-*r/ 98.8%

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

      2. neg-mul-1 98.8%

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

      3. +-commutative 98.8%

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

      4. distribute-neg-in 98.8%

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

      5. mul-1-neg 98.8%

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

      6. remove-double-neg 98.8%

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

      7. metadata-eval 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in x around inf 96.3%

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

Alternative 7: 89.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -255000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -255000.0)
   (- 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 <= -255000.0) {
		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 <= -255000.0) {
		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 <= -255000.0:
		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 <= -255000.0)
		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, -255000.0], 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 < -255000

   1. Initial program 15.9%

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

   3. Taylor expanded in y around inf 99.1%

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

      1. associate-*r/ 99.1%

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

      2. neg-mul-1 99.1%

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

      3. +-commutative 99.1%

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

      4. distribute-neg-in 99.1%

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

      5. mul-1-neg 99.1%

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

      6. remove-double-neg 99.1%

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

      7. metadata-eval 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in x around 0 77.7%

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

   if -255000 < 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 95.4%

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

      1. sub-neg 95.4%

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

      2. mul-1-neg 95.4%

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

      3. log1p-define 95.5%

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

      4. mul-1-neg 95.5%

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

   5. Simplified 95.5%

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

   if 1 < y

   1. Initial program 70.2%

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

   3. Taylor expanded in y around inf 98.8%

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

      1. associate-*r/ 98.8%

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

      2. neg-mul-1 98.8%

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

      3. +-commutative 98.8%

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

      4. distribute-neg-in 98.8%

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

      5. mul-1-neg 98.8%

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

      6. remove-double-neg 98.8%

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

      7. metadata-eval 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in x around inf 96.3%

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

Alternative 8: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -255000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -255000.0) (- 1.0 (log (/ -1.0 y))) (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -255000.0) {
		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 (y <= -255000.0) {
		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 y <= -255000.0:
		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 (y <= -255000.0)
		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[y, -255000.0], 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 y < -255000

   1. Initial program 15.9%

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

   3. Taylor expanded in y around inf 99.1%

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

      1. associate-*r/ 99.1%

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

      2. neg-mul-1 99.1%

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

      3. +-commutative 99.1%

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

      4. distribute-neg-in 99.1%

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

      5. mul-1-neg 99.1%

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

      6. remove-double-neg 99.1%

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

      7. metadata-eval 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in x around 0 77.7%

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

   if -255000 < y

   1. Initial program 95.6%

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

   3. Taylor expanded in y around 0 81.7%

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

      1. sub-neg 81.7%

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

      2. mul-1-neg 81.7%

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

      3. log1p-define 81.7%

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

      4. mul-1-neg 81.7%

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

   5. Simplified 81.7%

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

Alternative 9: 62.5% accurate, 1.1× 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 72.0%

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

3. Taylor expanded in y around 0 61.2%

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

   1. sub-neg 61.2%

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

   2. mul-1-neg 61.2%

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

   3. log1p-define 61.2%

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

   4. mul-1-neg 61.2%

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

5. Simplified 61.2%

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

Alternative 10: 43.2% accurate, 111.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 72.0%

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

   1. sub-neg 72.0%

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

   2. log1p-define 72.0%

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

   3. distribute-neg-frac2 72.0%

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

   4. neg-sub0 72.0%

      \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{0 - \left(1 - y\right)}}\right) \]

   5. associate--r- 72.0%

      \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{\left(0 - 1\right) + y}}\right) \]

   6. metadata-eval 72.0%

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

   7. +-commutative 72.0%

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

3. Simplified 72.0%

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

5. Taylor expanded in x around 0 44.8%

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

6. Taylor expanded in y around 0 43.3%

   \[\leadsto 1 - \color{blue}{y \cdot \left(1 + y \cdot \left(0.5 + 0.3333333333333333 \cdot y\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative 43.3%

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

8. Simplified 43.3%

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

9. Taylor expanded in y around 0 45.6%

   \[\leadsto \color{blue}{1} \]
10. 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 2024170 
(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))))))