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

Average Error: 28.1% → 0.09%

Time: 13.0s

Precision: binary64

Cost: 13828

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + -1}{y}\\ \mathbf{if}\;y \leq -750000:\\ \;\;\;\;\log \left(\frac{e}{\frac{t_0}{y} + t_0}\right)\\ \mathbf{elif}\;y \leq 54000000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y \cdot e}{x + -1}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x -1.0) y)))
   (if (<= y -750000.0)
     (log (/ E (+ (/ t_0 y) t_0)))
     (if (<= y 54000000000000.0)
       (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
       (log (/ (* y E) (+ x -1.0)))))))

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 t_0 = (x + -1.0) / y;
	double tmp;
	if (y <= -750000.0) {
		tmp = log((((double) M_E) / ((t_0 / y) + t_0)));
	} else if (y <= 54000000000000.0) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = log(((y * ((double) M_E)) / (x + -1.0)));
	}
	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 t_0 = (x + -1.0) / y;
	double tmp;
	if (y <= -750000.0) {
		tmp = Math.log((Math.E / ((t_0 / y) + t_0)));
	} else if (y <= 54000000000000.0) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = Math.log(((y * Math.E) / (x + -1.0)));
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	t_0 = (x + -1.0) / y
	tmp = 0
	if y <= -750000.0:
		tmp = math.log((math.e / ((t_0 / y) + t_0)))
	elif y <= 54000000000000.0:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = math.log(((y * math.e) / (x + -1.0)))
	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)
	t_0 = Float64(Float64(x + -1.0) / y)
	tmp = 0.0
	if (y <= -750000.0)
		tmp = log(Float64(exp(1) / Float64(Float64(t_0 / y) + t_0)));
	elseif (y <= 54000000000000.0)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = log(Float64(Float64(y * exp(1)) / Float64(x + -1.0)));
	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_] := Block[{t$95$0 = N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -750000.0], N[Log[N[(E / N[(N[(t$95$0 / y), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 54000000000000.0], N[(1.0 - N[Log[1 + N[(N[(y - x), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(y * E), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Target

Original	28.1%
Target	0.17%
Herbie	0.09%

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

Derivation

1. Split input into 3 regimes

2. if y < -7.5e5

   1. Initial program 81.45

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

   2. Simplified 81.45

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

      [Start] 81.45	\[ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
      sub-neg [=>] 81.45	\[ 1 - \log \color{blue}{\left(1 + \left(-\frac{x - y}{1 - y}\right)\right)} \]
      log1p-def [=>] 81.45	\[ 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]
      div-sub [=>] 81.42	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} - \frac{y}{1 - y}\right)}\right) \]
      sub-neg [=>] 81.42	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} + \left(-\frac{y}{1 - y}\right)\right)}\right) \]
      +-commutative [=>] 81.42	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\left(-\frac{y}{1 - y}\right) + \frac{x}{1 - y}\right)}\right) \]
      distribute-neg-in [=>] 81.42	\[ 1 - \mathsf{log1p}\left(\color{blue}{\left(-\left(-\frac{y}{1 - y}\right)\right) + \left(-\frac{x}{1 - y}\right)}\right) \]
      remove-double-neg [=>] 81.42	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y}} + \left(-\frac{x}{1 - y}\right)\right) \]
      sub-neg [<=] 81.42	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y} - \frac{x}{1 - y}}\right) \]
      div-sub [<=] 81.45	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y - x}{1 - y}}\right) \]

   3. Applied egg-rr 81.44

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

   4. Taylor expanded in y around -inf 0.13

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

   5. Simplified 0.13

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

      [Start] 0.13	\[ \log \left(\frac{e}{\left(\frac{x}{y} + -1 \cdot \frac{1 - x}{{y}^{2}}\right) - \frac{1}{y}}\right) \]
      +-commutative [=>] 0.13	\[ \log \left(\frac{e}{\color{blue}{\left(-1 \cdot \frac{1 - x}{{y}^{2}} + \frac{x}{y}\right)} - \frac{1}{y}}\right) \]
      associate--l+ [=>] 0.13	\[ \log \left(\frac{e}{\color{blue}{-1 \cdot \frac{1 - x}{{y}^{2}} + \left(\frac{x}{y} - \frac{1}{y}\right)}}\right) \]
      associate-*r/ [=>] 0.13	\[ \log \left(\frac{e}{\color{blue}{\frac{-1 \cdot \left(1 - x\right)}{{y}^{2}}} + \left(\frac{x}{y} - \frac{1}{y}\right)}\right) \]
      unpow2 [=>] 0.13	\[ \log \left(\frac{e}{\frac{-1 \cdot \left(1 - x\right)}{\color{blue}{y \cdot y}} + \left(\frac{x}{y} - \frac{1}{y}\right)}\right) \]
      associate-/r* [=>] 0.13	\[ \log \left(\frac{e}{\color{blue}{\frac{\frac{-1 \cdot \left(1 - x\right)}{y}}{y}} + \left(\frac{x}{y} - \frac{1}{y}\right)}\right) \]
      sub-neg [=>] 0.13	\[ \log \left(\frac{e}{\frac{\frac{-1 \cdot \color{blue}{\left(1 + \left(-x\right)\right)}}{y}}{y} + \left(\frac{x}{y} - \frac{1}{y}\right)}\right) \]
      mul-1-neg [<=] 0.13	\[ \log \left(\frac{e}{\frac{\frac{-1 \cdot \left(1 + \color{blue}{-1 \cdot x}\right)}{y}}{y} + \left(\frac{x}{y} - \frac{1}{y}\right)}\right) \]
      associate-*r/ [<=] 0.13	\[ \log \left(\frac{e}{\frac{\color{blue}{-1 \cdot \frac{1 + -1 \cdot x}{y}}}{y} + \left(\frac{x}{y} - \frac{1}{y}\right)}\right) \]
      mul-1-neg [=>] 0.13	\[ \log \left(\frac{e}{\frac{\color{blue}{-\frac{1 + -1 \cdot x}{y}}}{y} + \left(\frac{x}{y} - \frac{1}{y}\right)}\right) \]
      distribute-neg-frac [=>] 0.13	\[ \log \left(\frac{e}{\frac{\color{blue}{\frac{-\left(1 + -1 \cdot x\right)}{y}}}{y} + \left(\frac{x}{y} - \frac{1}{y}\right)}\right) \]
      mul-1-neg [=>] 0.13	\[ \log \left(\frac{e}{\frac{\frac{-\left(1 + \color{blue}{\left(-x\right)}\right)}{y}}{y} + \left(\frac{x}{y} - \frac{1}{y}\right)}\right) \]
      sub-neg [<=] 0.13	\[ \log \left(\frac{e}{\frac{\frac{-\color{blue}{\left(1 - x\right)}}{y}}{y} + \left(\frac{x}{y} - \frac{1}{y}\right)}\right) \]
      div-sub [<=] 0.13	\[ \log \left(\frac{e}{\frac{\frac{-\left(1 - x\right)}{y}}{y} + \color{blue}{\frac{x - 1}{y}}}\right) \]
      sub-neg [=>] 0.13	\[ \log \left(\frac{e}{\frac{\frac{-\left(1 - x\right)}{y}}{y} + \frac{\color{blue}{x + \left(-1\right)}}{y}}\right) \]
      metadata-eval [=>] 0.13	\[ \log \left(\frac{e}{\frac{\frac{-\left(1 - x\right)}{y}}{y} + \frac{x + \color{blue}{-1}}{y}}\right) \]

   if -7.5e5 < y < 5.4e13

   1. Initial program 0.08

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

   2. Simplified 0.04

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

      [Start] 0.08	\[ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
      sub-neg [=>] 0.08	\[ 1 - \log \color{blue}{\left(1 + \left(-\frac{x - y}{1 - y}\right)\right)} \]
      log1p-def [=>] 0.04	\[ 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]
      div-sub [=>] 0.04	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} - \frac{y}{1 - y}\right)}\right) \]
      sub-neg [=>] 0.04	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} + \left(-\frac{y}{1 - y}\right)\right)}\right) \]
      +-commutative [=>] 0.04	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\left(-\frac{y}{1 - y}\right) + \frac{x}{1 - y}\right)}\right) \]
      distribute-neg-in [=>] 0.04	\[ 1 - \mathsf{log1p}\left(\color{blue}{\left(-\left(-\frac{y}{1 - y}\right)\right) + \left(-\frac{x}{1 - y}\right)}\right) \]
      remove-double-neg [=>] 0.04	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y}} + \left(-\frac{x}{1 - y}\right)\right) \]
      sub-neg [<=] 0.04	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y} - \frac{x}{1 - y}}\right) \]
      div-sub [<=] 0.04	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y - x}{1 - y}}\right) \]

   if 5.4e13 < y

   1. Initial program 46.89

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

   2. Simplified 46.89

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

      [Start] 46.89	\[ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
      sub-neg [=>] 46.89	\[ 1 - \log \color{blue}{\left(1 + \left(-\frac{x - y}{1 - y}\right)\right)} \]
      log1p-def [=>] 46.89	\[ 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]
      div-sub [=>] 46.85	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} - \frac{y}{1 - y}\right)}\right) \]
      sub-neg [=>] 46.85	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} + \left(-\frac{y}{1 - y}\right)\right)}\right) \]
      +-commutative [=>] 46.85	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\left(-\frac{y}{1 - y}\right) + \frac{x}{1 - y}\right)}\right) \]
      distribute-neg-in [=>] 46.85	\[ 1 - \mathsf{log1p}\left(\color{blue}{\left(-\left(-\frac{y}{1 - y}\right)\right) + \left(-\frac{x}{1 - y}\right)}\right) \]
      remove-double-neg [=>] 46.85	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y}} + \left(-\frac{x}{1 - y}\right)\right) \]
      sub-neg [<=] 46.85	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y} - \frac{x}{1 - y}}\right) \]
      div-sub [<=] 46.89	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y - x}{1 - y}}\right) \]

   3. Applied egg-rr 46.89

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

   4. Taylor expanded in y around -inf 0.26

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

4. Final simplification 0.09

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -750000:\\ \;\;\;\;\log \left(\frac{e}{\frac{\frac{x + -1}{y}}{y} + \frac{x + -1}{y}}\right)\\ \mathbf{elif}\;y \leq 54000000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y \cdot e}{x + -1}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2%
Cost	13449

\[\begin{array}{l} \mathbf{if}\;y \leq -370000000 \lor \neg \left(y \leq 1.18 \cdot 10^{+14}\right):\\ \;\;\;\;\log \left(\frac{y \cdot e}{x + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \end{array} \]

Alternative 2
Error	9.15%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.9999:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{y} - \log \left(\frac{-1}{y}\right)\right)\\ \end{array} \]

Alternative 3
Error	9.11%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;y \leq -8000000000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 5000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \end{array} \]

Alternative 4
Error	10.22%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -10.8:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 0.13:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \end{array} \]

Alternative 5
Error	14.68%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;y \leq -41:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 6
Error	15.3%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -85:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 0.00036:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 7
Error	20.93%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -29.5:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]

Alternative 8
Error	37.01%
Cost	6656

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

Alternative 9
Error	54.81%
Cost	448

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

Alternative 10
Error	56.26%
Cost	192

\[1 + x \]

Alternative 11
Error	56.45%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023089 
(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))))))