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

Average Error: 18.4 → 0.1

Time: 5.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -546459.0231533627:\\ \;\;\;\;1 - \left(\left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + x}{y \cdot \left(1 + x\right)}\right)\right)\right)\\ \mathbf{elif}\;y \leq 5977358701217.452:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= y -546459.0231533627)
   (-
    1.0
    (+
     (+ (log1p (- x)) (log (/ -1.0 y)))
     (expm1 (log1p (/ (+ 1.0 x) (* y (+ 1.0 x)))))))
   (if (<= y 5977358701217.452)
     (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
     (- 1.0 (log (/ (+ x -1.0) y))))))

C

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

↓

double code(double x, double y) {
	double tmp;
	if (y <= -546459.0231533627) {
		tmp = 1.0 - ((log1p(-x) + log((-1.0 / y))) + expm1(log1p(((1.0 + x) / (y * (1.0 + x))))));
	} else if (y <= 5977358701217.452) {
		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) {
	return 1.0 - Math.log((1.0 - ((x - y) / (1.0 - y))));
}

↓

public static double code(double x, double y) {
	double tmp;
	if (y <= -546459.0231533627) {
		tmp = 1.0 - ((Math.log1p(-x) + Math.log((-1.0 / y))) + Math.expm1(Math.log1p(((1.0 + x) / (y * (1.0 + x))))));
	} else if (y <= 5977358701217.452) {
		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):
	return 1.0 - math.log((1.0 - ((x - y) / (1.0 - y))))

↓

def code(x, y):
	tmp = 0
	if y <= -546459.0231533627:
		tmp = 1.0 - ((math.log1p(-x) + math.log((-1.0 / y))) + math.expm1(math.log1p(((1.0 + x) / (y * (1.0 + x))))))
	elif y <= 5977358701217.452:
		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)
	return Float64(1.0 - log(Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y)))))
end

↓

function code(x, y)
	tmp = 0.0
	if (y <= -546459.0231533627)
		tmp = Float64(1.0 - Float64(Float64(log1p(Float64(-x)) + log(Float64(-1.0 / y))) + expm1(log1p(Float64(Float64(1.0 + x) / Float64(y * Float64(1.0 + x)))))));
	elseif (y <= 5977358701217.452)
		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_] := N[(1.0 - N[Log[N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := If[LessEqual[y, -546459.0231533627], N[(1.0 - N[(N[(N[Log[1 + (-x)], $MachinePrecision] + N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(Exp[N[Log[1 + N[(N[(1.0 + x), $MachinePrecision] / N[(y * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5977358701217.452], 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]]]

Error

Bits error versus x

Bits error versus y

Target

Original	18.4
Target	0.1
Herbie	0.1

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

Derivation

1. Split input into 3 regimes

2. if y < -546459.023153362679

   1. Initial program 52.1

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

   2. Simplified 52.1

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

   3. Taylor expanded in y around -inf 0.3

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

   4. Simplified 0.3

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

   5. Applied egg-rr 0.3

      \[\leadsto 1 - \left(\left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + x}{y \cdot \left(1 + x\right)}\right)\right)}\right) \]

   if -546459.023153362679 < y < 5977358701217.45215

   1. Initial program 0.1

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

   2. Simplified 0.0

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

   if 5977358701217.45215 < y

   1. Initial program 31.0

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

   2. Simplified 31.0

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

   3. Taylor expanded in y around inf 0.9

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

   4. Simplified 0.9

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

   5. Applied egg-rr 0.0

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -546459.0231533627:\\ \;\;\;\;1 - \left(\left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right) + \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + x}{y \cdot \left(1 + x\right)}\right)\right)\right)\\ \mathbf{elif}\;y \leq 5977358701217.452:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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))))))