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

Average Error: 28.4% → 0.17%

Time: 14.1s

Precision: binary64

Cost: 13448

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + -1}{y}\\ \mathbf{if}\;y \leq -410000000:\\ \;\;\;\;1 - \log t_0\\ \mathbf{elif}\;y \leq 33000000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e}{t_0}\right)\\ \end{array} \]

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 -410000000.0)
     (- 1.0 (log t_0))
     (if (<= y 33000000000000.0)
       (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
       (log (/ E t_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 <= -410000000.0) {
		tmp = 1.0 - log(t_0);
	} else if (y <= 33000000000000.0) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = log((((double) M_E) / t_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 <= -410000000.0) {
		tmp = 1.0 - Math.log(t_0);
	} else if (y <= 33000000000000.0) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = Math.log((Math.E / t_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 <= -410000000.0:
		tmp = 1.0 - math.log(t_0)
	elif y <= 33000000000000.0:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = math.log((math.e / t_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 <= -410000000.0)
		tmp = Float64(1.0 - log(t_0));
	elseif (y <= 33000000000000.0)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = log(Float64(exp(1) / t_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, -410000000.0], N[(1.0 - N[Log[t$95$0], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 33000000000000.0], N[(1.0 - N[Log[1 + N[(N[(y - x), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[(E / t$95$0), $MachinePrecision]], $MachinePrecision]]]]

Target

Original	28.4%
Target	0.19%
Herbie	0.17%

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

Derivation

1. Split input into 3 regimes

2. if y < -4.1e8

   1. Initial program 82.2

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

   2. Simplified 82.2

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

      [Start] 82.2	\[ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
      sub-neg [=>] 82.2	\[ 1 - \log \color{blue}{\left(1 + \left(-\frac{x - y}{1 - y}\right)\right)} \]
      log1p-def [=>] 82.2	\[ 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]
      div-sub [=>] 82.18	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} - \frac{y}{1 - y}\right)}\right) \]
      sub-neg [=>] 82.18	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} + \left(-\frac{y}{1 - y}\right)\right)}\right) \]
      +-commutative [=>] 82.18	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\left(-\frac{y}{1 - y}\right) + \frac{x}{1 - y}\right)}\right) \]
      distribute-neg-in [=>] 82.18	\[ 1 - \mathsf{log1p}\left(\color{blue}{\left(-\left(-\frac{y}{1 - y}\right)\right) + \left(-\frac{x}{1 - y}\right)}\right) \]
      remove-double-neg [=>] 82.18	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y}} + \left(-\frac{x}{1 - y}\right)\right) \]
      sub-neg [<=] 82.18	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y} - \frac{x}{1 - y}}\right) \]
      div-sub [<=] 82.2	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y - x}{1 - y}}\right) \]

   3. Applied egg-rr 80.15

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

   4. Taylor expanded in y around inf 100

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

   5. Simplified 100

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

      [Start] 100	\[ 1 - \left(\log \left(\frac{1}{y}\right) + \log \left(x - 1\right)\right) \]
      associate--r+ [=>] 100	\[ \color{blue}{\left(1 - \log \left(\frac{1}{y}\right)\right) - \log \left(x - 1\right)} \]
      log-div [=>] 100	\[ \left(1 - \color{blue}{\left(\log 1 - \log y\right)}\right) - \log \left(x - 1\right) \]
      metadata-eval [=>] 100	\[ \left(1 - \left(\color{blue}{0} - \log y\right)\right) - \log \left(x - 1\right) \]
      associate--r- [=>] 100	\[ \color{blue}{\left(\left(1 - 0\right) + \log y\right)} - \log \left(x - 1\right) \]
      metadata-eval [=>] 100	\[ \left(\color{blue}{1} + \log y\right) - \log \left(x - 1\right) \]
      sub-neg [=>] 100	\[ \left(1 + \log y\right) - \log \color{blue}{\left(x + \left(-1\right)\right)} \]
      metadata-eval [=>] 100	\[ \left(1 + \log y\right) - \log \left(x + \color{blue}{-1}\right) \]
      +-commutative [=>] 100	\[ \left(1 + \log y\right) - \log \color{blue}{\left(-1 + x\right)} \]

   6. Taylor expanded in y around 0 100

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

   7. Simplified 0.31

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

      [Start] 100	\[ \left(1 + \log y\right) - \log \left(x - 1\right) \]
      sub-neg [=>] 100	\[ \left(1 + \log y\right) - \log \color{blue}{\left(x + \left(-1\right)\right)} \]
      metadata-eval [=>] 100	\[ \left(1 + \log y\right) - \log \left(x + \color{blue}{-1}\right) \]
      +-commutative [<=] 100	\[ \left(1 + \log y\right) - \log \color{blue}{\left(-1 + x\right)} \]
      +-commutative [=>] 100	\[ \color{blue}{\left(\log y + 1\right)} - \log \left(-1 + x\right) \]
      associate-+r- [<=] 100	\[ \color{blue}{\log y + \left(1 - \log \left(-1 + x\right)\right)} \]
      +-commutative [=>] 100	\[ \color{blue}{\left(1 - \log \left(-1 + x\right)\right) + \log y} \]
      associate-+l- [=>] 100	\[ \color{blue}{1 - \left(\log \left(-1 + x\right) - \log y\right)} \]
      log-div [<=] 0.31	\[ 1 - \color{blue}{\log \left(\frac{-1 + x}{y}\right)} \]
      +-commutative [=>] 0.31	\[ 1 - \log \left(\frac{\color{blue}{x + -1}}{y}\right) \]

   if -4.1e8 < y < 3.3e13

   1. Initial program 0.17

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

   2. Simplified 0.13

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

      [Start] 0.17	\[ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
      sub-neg [=>] 0.17	\[ 1 - \log \color{blue}{\left(1 + \left(-\frac{x - y}{1 - y}\right)\right)} \]
      log1p-def [=>] 0.13	\[ 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]
      div-sub [=>] 0.13	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} - \frac{y}{1 - y}\right)}\right) \]
      sub-neg [=>] 0.13	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} + \left(-\frac{y}{1 - y}\right)\right)}\right) \]
      +-commutative [=>] 0.13	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\left(-\frac{y}{1 - y}\right) + \frac{x}{1 - y}\right)}\right) \]
      distribute-neg-in [=>] 0.13	\[ 1 - \mathsf{log1p}\left(\color{blue}{\left(-\left(-\frac{y}{1 - y}\right)\right) + \left(-\frac{x}{1 - y}\right)}\right) \]
      remove-double-neg [=>] 0.13	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y}} + \left(-\frac{x}{1 - y}\right)\right) \]
      sub-neg [<=] 0.13	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y} - \frac{x}{1 - y}}\right) \]
      div-sub [<=] 0.13	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y - x}{1 - y}}\right) \]

   if 3.3e13 < y

   1. Initial program 47.13

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

   2. Simplified 47.13

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

      [Start] 47.13	\[ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
      sub-neg [=>] 47.13	\[ 1 - \log \color{blue}{\left(1 + \left(-\frac{x - y}{1 - y}\right)\right)} \]
      log1p-def [=>] 47.13	\[ 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]
      div-sub [=>] 47.1	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} - \frac{y}{1 - y}\right)}\right) \]
      sub-neg [=>] 47.1	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} + \left(-\frac{y}{1 - y}\right)\right)}\right) \]
      +-commutative [=>] 47.1	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\left(-\frac{y}{1 - y}\right) + \frac{x}{1 - y}\right)}\right) \]
      distribute-neg-in [=>] 47.1	\[ 1 - \mathsf{log1p}\left(\color{blue}{\left(-\left(-\frac{y}{1 - y}\right)\right) + \left(-\frac{x}{1 - y}\right)}\right) \]
      remove-double-neg [=>] 47.1	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y}} + \left(-\frac{x}{1 - y}\right)\right) \]
      sub-neg [<=] 47.1	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y} - \frac{x}{1 - y}}\right) \]
      div-sub [<=] 47.13	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y - x}{1 - y}}\right) \]

   3. Applied egg-rr 46.1

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

   4. Taylor expanded in y around inf 1.48

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

   5. Simplified 1.48

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

      [Start] 1.48	\[ 1 - \left(\log \left(\frac{1}{y}\right) + \log \left(x - 1\right)\right) \]
      associate--r+ [=>] 1.48	\[ \color{blue}{\left(1 - \log \left(\frac{1}{y}\right)\right) - \log \left(x - 1\right)} \]
      log-div [=>] 1.48	\[ \left(1 - \color{blue}{\left(\log 1 - \log y\right)}\right) - \log \left(x - 1\right) \]
      metadata-eval [=>] 1.48	\[ \left(1 - \left(\color{blue}{0} - \log y\right)\right) - \log \left(x - 1\right) \]
      associate--r- [=>] 1.48	\[ \color{blue}{\left(\left(1 - 0\right) + \log y\right)} - \log \left(x - 1\right) \]
      metadata-eval [=>] 1.48	\[ \left(\color{blue}{1} + \log y\right) - \log \left(x - 1\right) \]
      sub-neg [=>] 1.48	\[ \left(1 + \log y\right) - \log \color{blue}{\left(x + \left(-1\right)\right)} \]
      metadata-eval [=>] 1.48	\[ \left(1 + \log y\right) - \log \left(x + \color{blue}{-1}\right) \]
      +-commutative [=>] 1.48	\[ \left(1 + \log y\right) - \log \color{blue}{\left(-1 + x\right)} \]

   6. Applied egg-rr 0.02

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

4. Final simplification 0.17

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -410000000:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{elif}\;y \leq 33000000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e}{\frac{x + -1}{y}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.22%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.99996:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \]

Alternative 2
Error	1.25%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -1.7 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \]

Alternative 3
Error	1.53%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -13.5 \lor \neg \left(y \leq 900000000000\right):\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \end{array} \]

Alternative 4
Error	14.6%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;y \leq -11.6:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 5
Error	15.88%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -15.5:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-17}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 6
Error	20.36%
Cost	6920

\[\begin{array}{l} \mathbf{if}\;y \leq -15.5:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-17}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(x\right)\\ \end{array} \]

Alternative 7
Error	36.25%
Cost	6788

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

Alternative 8
Error	55.41%
Cost	448

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

Alternative 9
Error	56.86%
Cost	192

\[1 + x \]

Alternative 10
Error	57.09%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023090 
(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))))))