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

Percentage Accurate: 72.3% → 98.1%

Time: 5.7s

Alternatives: 6

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: 72.3% 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: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+31}:\\ \;\;\;\;\left(1 - \mathsf{log1p}\left(-x\right)\right) - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+76}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9e+31)
   (- (- 1.0 (log1p (- x))) (log (/ -1.0 y)))
   (if (<= y 7.2e+76)
     (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
     (+ 1.0 (- (log y) (log (+ x -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9e+31) {
		tmp = (1.0 - log1p(-x)) - log((-1.0 / y));
	} else if (y <= 7.2e+76) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 + (log(y) - log((x + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9e+31) {
		tmp = (1.0 - Math.log1p(-x)) - Math.log((-1.0 / y));
	} else if (y <= 7.2e+76) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 + (Math.log(y) - Math.log((x + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9e+31:
		tmp = (1.0 - math.log1p(-x)) - math.log((-1.0 / y))
	elif y <= 7.2e+76:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = 1.0 + (math.log(y) - math.log((x + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9e+31)
		tmp = Float64(Float64(1.0 - log1p(Float64(-x))) - log(Float64(-1.0 / y)));
	elseif (y <= 7.2e+76)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 + Float64(log(y) - log(Float64(x + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9e+31], N[(N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.2e+76], N[(1.0 - N[Log[1 + N[(N[(y - x), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Log[y], $MachinePrecision] - N[Log[N[(x + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.9999999999999992e31

   1. Initial program 19.2%

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

      1. sub-neg 19.2%

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

      2. log1p-def 19.3%

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

      3. distribute-neg-frac 19.3%

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

      4. sub-neg 19.3%

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

      5. distribute-neg-in 19.3%

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

      6. remove-double-neg 19.3%

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

      7. +-commutative 19.3%

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

      8. sub-neg 19.3%

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

   3. Simplified 19.3%

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

   5. Taylor expanded in y around -inf 99.5%

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

      1. associate--r+ 99.5%

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

      2. sub-neg 99.5%

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

      3. metadata-eval 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. metadata-eval 99.5%

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

      6. +-commutative 99.5%

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

      7. log1p-def 99.5%

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

      8. mul-1-neg 99.5%

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

   7. Simplified 99.5%

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

   if -8.9999999999999992e31 < y < 7.2000000000000006e76

   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-def 100.0%

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

      3. distribute-neg-frac 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   if 7.2000000000000006e76 < y

   1. Initial program 33.3%

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

      1. sub-neg 33.3%

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

      2. log1p-def 33.3%

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

      3. distribute-neg-frac 33.3%

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

      4. sub-neg 33.3%

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

      5. distribute-neg-in 33.3%

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

      6. remove-double-neg 33.3%

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

      7. +-commutative 33.3%

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

      8. sub-neg 33.3%

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

   3. Simplified 33.3%

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

   5. Taylor expanded in y around inf 98.3%

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

      1. log-rec 98.3%

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

      2. unsub-neg 98.3%

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

      3. sub-neg 98.3%

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

      4. metadata-eval 98.3%

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

      5. +-commutative 98.3%

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

   7. Simplified 98.3%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+31}:\\ \;\;\;\;\left(1 - \mathsf{log1p}\left(-x\right)\right) - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+76}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{+76}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\left(y - x\right) \cdot \frac{1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.2e+76)
   (- 1.0 (log1p (* (- y x) (/ 1.0 (- 1.0 y)))))
   (+ 1.0 (- (log y) (log (+ x -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.2e+76) {
		tmp = 1.0 - log1p(((y - x) * (1.0 / (1.0 - y))));
	} else {
		tmp = 1.0 + (log(y) - log((x + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7.2e+76) {
		tmp = 1.0 - Math.log1p(((y - x) * (1.0 / (1.0 - y))));
	} else {
		tmp = 1.0 + (Math.log(y) - Math.log((x + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7.2e+76:
		tmp = 1.0 - math.log1p(((y - x) * (1.0 / (1.0 - y))))
	else:
		tmp = 1.0 + (math.log(y) - math.log((x + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.2e+76)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) * Float64(1.0 / Float64(1.0 - y)))));
	else
		tmp = Float64(1.0 + Float64(log(y) - log(Float64(x + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.2e+76], N[(1.0 - N[Log[1 + N[(N[(y - x), $MachinePrecision] * N[(1.0 / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Log[y], $MachinePrecision] - N[Log[N[(x + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.2000000000000006e76

   1. Initial program 79.3%

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

      1. sub-neg 79.3%

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

      2. log1p-def 79.3%

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

      3. distribute-neg-frac 79.3%

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

      4. sub-neg 79.3%

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

      5. distribute-neg-in 79.3%

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

      6. remove-double-neg 79.3%

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

      7. +-commutative 79.3%

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

      8. sub-neg 79.3%

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

   3. Simplified 79.3%

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

      1. clear-num 79.3%

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

      2. associate-/r/ 79.9%

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

   6. Applied egg-rr 79.9%

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

   if 7.2000000000000006e76 < y

   1. Initial program 33.3%

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

      1. sub-neg 33.3%

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

      2. log1p-def 33.3%

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

      3. distribute-neg-frac 33.3%

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

      4. sub-neg 33.3%

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

      5. distribute-neg-in 33.3%

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

      6. remove-double-neg 33.3%

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

      7. +-commutative 33.3%

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

      8. sub-neg 33.3%

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

   3. Simplified 33.3%

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

   5. Taylor expanded in y around inf 98.3%

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

      1. log-rec 98.3%

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

      2. unsub-neg 98.3%

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

      3. sub-neg 98.3%

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

      4. metadata-eval 98.3%

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

      5. +-commutative 98.3%

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

   7. Simplified 98.3%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{+76}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\left(y - x\right) \cdot \frac{1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 1

   1. Initial program 75.3%

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

      1. sub-neg 75.3%

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

      2. log1p-def 75.3%

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

      3. distribute-neg-frac 75.3%

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

      4. sub-neg 75.3%

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

      5. distribute-neg-in 75.3%

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

      6. remove-double-neg 75.3%

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

      7. +-commutative 75.3%

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

      8. sub-neg 75.3%

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

   3. Simplified 75.3%

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

   if 1 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 75.3%

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

      1. sub-neg 75.3%

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

      2. log1p-def 75.3%

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

      3. distribute-neg-frac 75.3%

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

      4. sub-neg 75.3%

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

      5. distribute-neg-in 75.3%

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

      6. remove-double-neg 75.3%

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

      7. +-commutative 75.3%

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

      8. sub-neg 75.3%

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

   3. Simplified 75.3%

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

   5. Taylor expanded in x around inf 75.3%

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

      1. neg-mul-1 75.3%

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

      2. distribute-neg-frac 75.3%

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

   7. Simplified 75.3%

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

4. Final simplification 75.3%

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

Alternative 4: 73.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

   1. sub-neg 75.3%

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

   2. log1p-def 75.3%

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

   3. distribute-neg-frac 75.3%

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

   4. sub-neg 75.3%

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

   5. distribute-neg-in 75.3%

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

   6. remove-double-neg 75.3%

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

   7. +-commutative 75.3%

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

   8. sub-neg 75.3%

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

3. Simplified 75.3%

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

5. Taylor expanded in x around inf 75.3%

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

   1. neg-mul-1 75.3%

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

   2. distribute-neg-frac 75.3%

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

7. Simplified 75.3%

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

8. Final simplification 75.3%

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

Alternative 5: 62.6% 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 75.3%

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

   1. sub-neg 75.3%

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

   2. log1p-def 75.3%

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

   3. distribute-neg-frac 75.3%

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

   4. sub-neg 75.3%

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

   5. distribute-neg-in 75.3%

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

   6. remove-double-neg 75.3%

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

   7. +-commutative 75.3%

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

   8. sub-neg 75.3%

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

3. Simplified 75.3%

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

5. Taylor expanded in y around 0 65.3%

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

   1. log1p-def 65.3%

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

   2. mul-1-neg 65.3%

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

7. Simplified 65.3%

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

8. Final simplification 65.3%

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

Alternative 6: 2.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

   1. sub-neg 75.3%

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

   2. log1p-def 75.3%

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

   3. distribute-neg-frac 75.3%

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

   4. sub-neg 75.3%

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

   5. distribute-neg-in 75.3%

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

   6. remove-double-neg 75.3%

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

   7. +-commutative 75.3%

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

   8. sub-neg 75.3%

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

3. Simplified 75.3%

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

5. Taylor expanded in y around inf 2.3%

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

6. Final simplification 2.3%

   \[\leadsto 1 - \mathsf{log1p}\left(-1\right) \]
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 2024036 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (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))))))