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

Percentage Accurate: 71.8% → 99.4%

Time: 11.0s

Alternatives: 11

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: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.0005:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1 - \left(\frac{\left(1 - x\right) + \frac{1 - x}{y}}{y} - x\right)}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.0005)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (- 1.0 (log (/ (- -1.0 (- (/ (+ (- 1.0 x) (/ (- 1.0 x) y)) y) x)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.0005) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - log(((-1.0 - ((((1.0 - x) + ((1.0 - x) / y)) / y) - x)) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.0005) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - Math.log(((-1.0 - ((((1.0 - x) + ((1.0 - x) / y)) / y) - x)) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.0005:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 - math.log(((-1.0 - ((((1.0 - x) + ((1.0 - x) / y)) / y) - x)) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.0005)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(-1.0 - Float64(Float64(Float64(Float64(1.0 - x) + Float64(Float64(1.0 - x) / y)) / y) - x)) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.0005], 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[(-1.0 - N[(N[(N[(N[(1.0 - x), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] - x), $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)) < 5.0000000000000001e-4

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 5.0000000000000001e-4 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 8.3%

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

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.0005:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1 - \left(\frac{\left(1 - x\right) + \frac{1 - x}{y}}{y} - x\right)}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.0005:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\left(x + -1\right) + \frac{x + -1}{y}}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.0005)
   (- 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)) <= 0.0005) {
		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)) <= 0.0005) {
		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)) <= 0.0005:
		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)) <= 0.0005)
		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], 0.0005], 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)) < 5.0000000000000001e-4

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 5.0000000000000001e-4 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 8.3%

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

   3. Taylor expanded in y around -inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-neg-frac2 99.8%

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

      3. associate--l+ 99.8%

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

      4. sub-neg 99.8%

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

      5. mul-1-neg 99.8%

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

      6. +-commutative 99.8%

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

      7. distribute-neg-frac2 99.8%

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

      8. distribute-neg-frac 99.8%

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

      9. mul-1-neg 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.0005:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\left(x + -1\right) + \frac{x + -1}{y}}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+40}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq -0.031 \lor \neg \left(y \leq 0.16\right):\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.7e+40)
   (- 1.0 (log (/ -1.0 y)))
   (if (or (<= y -0.031) (not (<= y 0.16)))
     (- 1.0 (log (/ x (+ y -1.0))))
     (- (- 1.0 y) (log1p (- x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.7e+40) {
		tmp = 1.0 - log((-1.0 / y));
	} else if ((y <= -0.031) || !(y <= 0.16)) {
		tmp = 1.0 - log((x / (y + -1.0)));
	} else {
		tmp = (1.0 - y) - log1p(-x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.7e+40) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if ((y <= -0.031) || !(y <= 0.16)) {
		tmp = 1.0 - Math.log((x / (y + -1.0)));
	} else {
		tmp = (1.0 - y) - Math.log1p(-x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.7e+40:
		tmp = 1.0 - math.log((-1.0 / y))
	elif (y <= -0.031) or not (y <= 0.16):
		tmp = 1.0 - math.log((x / (y + -1.0)))
	else:
		tmp = (1.0 - y) - math.log1p(-x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.7e+40)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif ((y <= -0.031) || !(y <= 0.16))
		tmp = Float64(1.0 - log(Float64(x / Float64(y + -1.0))));
	else
		tmp = Float64(Float64(1.0 - y) - log1p(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.7e+40], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -0.031], N[Not[LessEqual[y, 0.16]], $MachinePrecision]], N[(1.0 - N[Log[N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - y), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.70000000000000009e40

   1. Initial program 14.7%

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

   3. Taylor expanded in x around 0 2.9%

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

      1. log1p-define 2.9%

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

   5. Simplified 2.9%

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

   6. Taylor expanded in y around -inf 71.6%

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

   if -2.70000000000000009e40 < y < -0.031 or 0.160000000000000003 < y

   1. Initial program 64.3%

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

   3. Taylor expanded in x around inf 87.9%

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

      1. mul-1-neg 87.9%

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

      2. distribute-frac-neg2 87.9%

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

      3. sub-neg 87.9%

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

      4. distribute-neg-in 87.9%

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

      5. metadata-eval 87.9%

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

      6. remove-double-neg 87.9%

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

      7. +-commutative 87.9%

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

   5. Simplified 87.9%

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

   if -0.031 < y < 0.160000000000000003

   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 99.3%

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

   5. Simplified 99.3%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+40}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq -0.031 \lor \neg \left(y \leq 0.16\right):\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.0005:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.0005)
   (- 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)) <= 0.0005) {
		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)) <= 0.0005) {
		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)) <= 0.0005:
		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)) <= 0.0005)
		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], 0.0005], 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)) < 5.0000000000000001e-4

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 5.0000000000000001e-4 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 8.3%

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

   3. Taylor expanded in y around inf 98.9%

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

      1. associate-*r/ 98.9%

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

      2. distribute-lft-in 98.9%

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

      3. metadata-eval 98.9%

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

      4. neg-mul-1 98.9%

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

      5. mul-1-neg 98.9%

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

      6. remove-double-neg 98.9%

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

   5. Simplified 98.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.0005:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{elif}\;y \leq 0.44:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.6)
   (- 1.0 (log (/ (+ x -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 <= -1.6) {
		tmp = 1.0 - log(((x + -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 <= -1.6) {
		tmp = 1.0 - Math.log(((x + -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 <= -1.6:
		tmp = 1.0 - math.log(((x + -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 <= -1.6)
		tmp = Float64(1.0 - log(Float64(Float64(x + -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, -1.6], N[(1.0 - N[Log[N[(N[(x + -1.0), $MachinePrecision] / 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 < -1.6000000000000001

   1. Initial program 26.5%

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

   3. Taylor expanded in y around inf 96.0%

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

      1. associate-*r/ 96.0%

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

      2. distribute-lft-in 96.0%

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

      3. metadata-eval 96.0%

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

      4. neg-mul-1 96.0%

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

      5. mul-1-neg 96.0%

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

      6. remove-double-neg 96.0%

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

   5. Simplified 96.0%

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

   if -1.6000000000000001 < y < 0.440000000000000002

   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 99.3%

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

   5. Simplified 99.3%

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

   if 0.440000000000000002 < y

   1. Initial program 55.3%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-frac-neg2 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. metadata-eval 100.0%

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

      6. remove-double-neg 100.0%

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

      7. +-commutative 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{elif}\;y \leq 0.44:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -56:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -56.0) (- 1.0 (log (/ -1.0 y))) (- (- 1.0 y) (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -56.0) {
		tmp = 1.0 - log((-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 <= -56.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = (1.0 - y) - Math.log1p(-x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -56.0:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = (1.0 - y) - math.log1p(-x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -56.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = Float64(Float64(1.0 - y) - log1p(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -56.0], N[(1.0 - N[Log[N[(-1.0 / 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 < -56

   1. Initial program 24.6%

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

   3. Taylor expanded in x around 0 5.6%

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

      1. log1p-define 5.6%

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

   5. Simplified 5.6%

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

   6. Taylor expanded in y around -inf 66.0%

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

   if -56 < y

   1. Initial program 93.5%

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

   3. Taylor expanded in y around 0 84.1%

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

   5. Simplified 84.1%

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

Alternative 7: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -600000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -600000.0) (- 1.0 (log (/ -1.0 y))) (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -600000.0) {
		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 <= -600000.0) {
		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 <= -600000.0:
		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 <= -600000.0)
		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, -600000.0], 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 < -6e5

   1. Initial program 22.6%

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

   3. Taylor expanded in x around 0 5.7%

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

      1. log1p-define 5.7%

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

   5. Simplified 5.7%

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

   6. Taylor expanded in y around -inf 67.7%

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

   if -6e5 < y

   1. Initial program 93.5%

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

   3. Taylor expanded in y around 0 82.6%

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

      1. sub-neg 82.6%

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

      2. mul-1-neg 82.6%

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

      3. log1p-define 82.6%

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

      4. mul-1-neg 82.6%

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

   5. Simplified 82.6%

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

Alternative 8: 62.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7:\\ \;\;\;\;1 - \log \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + x \cdot \left(0.3333333333333333 - x \cdot -0.25\right)\right) - -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.7)
   (- 1.0 (log (- x)))
   (+
    1.0
    (* x (- (* x (+ 0.5 (* x (- 0.3333333333333333 (* x -0.25))))) -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.7) {
		tmp = 1.0 - log(-x);
	} else {
		tmp = 1.0 + (x * ((x * (0.5 + (x * (0.3333333333333333 - (x * -0.25))))) - -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.7d0)) then
        tmp = 1.0d0 - log(-x)
    else
        tmp = 1.0d0 + (x * ((x * (0.5d0 + (x * (0.3333333333333333d0 - (x * (-0.25d0)))))) - (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.7) {
		tmp = 1.0 - Math.log(-x);
	} else {
		tmp = 1.0 + (x * ((x * (0.5 + (x * (0.3333333333333333 - (x * -0.25))))) - -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.7:
		tmp = 1.0 - math.log(-x)
	else:
		tmp = 1.0 + (x * ((x * (0.5 + (x * (0.3333333333333333 - (x * -0.25))))) - -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.7)
		tmp = Float64(1.0 - log(Float64(-x)));
	else
		tmp = Float64(1.0 + Float64(x * Float64(Float64(x * Float64(0.5 + Float64(x * Float64(0.3333333333333333 - Float64(x * -0.25))))) - -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.7)
		tmp = 1.0 - log(-x);
	else
		tmp = 1.0 + (x * ((x * (0.5 + (x * (0.3333333333333333 - (x * -0.25))))) - -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.7], N[(1.0 - N[Log[(-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(N[(x * N[(0.5 + N[(x * N[(0.3333333333333333 - N[(x * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7000000000000002

   1. Initial program 85.1%

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

   3. Taylor expanded in x around inf 96.6%

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

      1. mul-1-neg 96.6%

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

      2. distribute-frac-neg2 96.6%

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

      3. sub-neg 96.6%

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

      4. distribute-neg-in 96.6%

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

      5. metadata-eval 96.6%

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

      6. remove-double-neg 96.6%

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

      7. +-commutative 96.6%

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

   5. Simplified 96.6%

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

   6. Taylor expanded in y around 0 65.8%

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

      1. neg-mul-1 65.8%

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

   8. Simplified 65.8%

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

   if -2.7000000000000002 < x

   1. Initial program 66.5%

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

   3. Taylor expanded in y around 0 58.4%

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

   4. Taylor expanded in x around 0 58.5%

      \[\leadsto 1 - \color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(-0.25 \cdot x - 0.3333333333333333\right) - 0.5\right) - 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7:\\ \;\;\;\;1 - \log \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 + x \cdot \left(0.3333333333333333 - x \cdot -0.25\right)\right) - -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.4% 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 72.7%

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

3. Taylor expanded in y around 0 62.0%

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

   1. sub-neg 62.0%

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

   2. mul-1-neg 62.0%

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

   3. log1p-define 62.0%

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

   4. mul-1-neg 62.0%

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

5. Simplified 62.0%

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

Alternative 10: 43.0% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

3. Taylor expanded in y around 0 62.0%

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

4. Taylor expanded in x around 0 40.2%

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

   1. neg-mul-1 40.2%

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

6. Simplified 40.2%

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

7. Final simplification 40.2%

   \[\leadsto x + 1 \]
8. Add Preprocessing

Alternative 11: 42.8% 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 72.7%

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

3. Taylor expanded in x around 0 38.1%

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

   1. log1p-define 38.1%

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

5. Simplified 38.1%

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

6. Taylor expanded in y around 0 39.8%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Developer target: 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 2024097 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (if (< y -81284752.61947241) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y))))) (if (< y 3.0094271212461764e+25) (log (/ (exp 1.0) (- 1.0 (/ (- x y) (- 1.0 y))))) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y)))))))

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