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

Average Error: 18.1 → 0.2

Time: 7.1s

Precision: binary64

Cost: 14084

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -100000:\\ \;\;\;\;1 + \left(\left(\frac{-0.5}{y \cdot y} + \frac{-1}{y}\right) - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{-1 + x}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= y -100000.0)
   (+
    1.0
    (- (+ (/ -0.5 (* y y)) (/ -1.0 y)) (+ (log1p (- x)) (log (/ -1.0 y)))))
   (if (<= y 1.0)
     (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
     (+ 1.0 (log (/ y (+ -1.0 x)))))))

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 <= -100000.0) {
		tmp = 1.0 + (((-0.5 / (y * y)) + (-1.0 / y)) - (log1p(-x) + log((-1.0 / y))));
	} else if (y <= 1.0) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 + log((y / (-1.0 + x)));
	}
	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 <= -100000.0) {
		tmp = 1.0 + (((-0.5 / (y * y)) + (-1.0 / y)) - (Math.log1p(-x) + Math.log((-1.0 / y))));
	} else if (y <= 1.0) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 + Math.log((y / (-1.0 + x)));
	}
	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 <= -100000.0:
		tmp = 1.0 + (((-0.5 / (y * y)) + (-1.0 / y)) - (math.log1p(-x) + math.log((-1.0 / y))))
	elif y <= 1.0:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 + math.log((y / (-1.0 + x)))
	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 <= -100000.0)
		tmp = Float64(1.0 + Float64(Float64(Float64(-0.5 / Float64(y * y)) + Float64(-1.0 / y)) - Float64(log1p(Float64(-x)) + log(Float64(-1.0 / y)))));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 + log(Float64(y / Float64(-1.0 + x))));
	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, -100000.0], N[(1.0 + N[(N[(N[(-0.5 / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision] - N[(N[Log[1 + (-x)], $MachinePrecision] + N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], 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[(y / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Target

Original	18.1
Target	0.1
Herbie	0.2

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

Derivation

1. Split input into 3 regimes

2. if y < -1e5

   1. Initial program 51.5

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

   2. Simplified 51.5

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

   3. Taylor expanded in y around -inf 11.0

      \[\leadsto \color{blue}{\left(1 + \left(-1 \cdot \frac{\frac{x}{x - 1} - \frac{1}{x - 1}}{y} + -0.5 \cdot \frac{2 + -1 \cdot \frac{{\left(1 - x\right)}^{2}}{{\left(x - 1\right)}^{2}}}{{y}^{2}}\right)\right) - \left(\log \left(-1 \cdot \left(x - 1\right)\right) + \log \left(\frac{-1}{y}\right)\right)} \]

   4. Simplified 0.3

      \[\leadsto \color{blue}{1 + \left(\left(\frac{-0.5}{y \cdot y} + \frac{-1}{y}\right) - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\right)} \]

   if -1e5 < y < 1

   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 1 < y

   1. Initial program 29.6

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

   2. Simplified 29.6

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

   3. Taylor expanded in y around inf 2.1

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

   4. Simplified 2.1

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

   5. Applied egg-rr 1.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -100000:\\ \;\;\;\;1 + \left(\left(\frac{-0.5}{y \cdot y} + \frac{-1}{y}\right) - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{-1 + x}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	7240

\[\begin{array}{l} t_0 := 1 + \log \left(\frac{y}{-1 + x}\right)\\ \mathbf{if}\;y \leq -100000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.0
Cost	7112

\[\begin{array}{l} t_0 := 1 + \log \left(\frac{y}{-1 + x}\right)\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+16}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	9.9
Cost	7048

\[\begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-5}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 0.049:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 4
Error	10.4
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-5}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+16}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 5
Error	13.7
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-5}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]

Alternative 6
Error	23.9
Cost	6656

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

Alternative 7
Error	35.4
Cost	576

\[1 + x \cdot \frac{-1}{y + -1} \]

Alternative 8
Error	36.4
Cost	192

\[1 + x \]

Alternative 9
Error	61.4
Cost	64

\[x \]

Alternative 10
Error	36.6
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022228 
(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))))))