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

Average Error: 19.4 → 0.1

Time: 12.2s

Precision: binary64

Cost: 14084

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -480000:\\ \;\;\;\;1 + \left(\left(\frac{1 - x}{y \cdot \left(x + -1\right)} - \log \left(\frac{-1}{y}\right)\right) - \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{elif}\;y \leq 14800000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= y -480000.0)
   (+
    1.0
    (- (- (/ (- 1.0 x) (* y (+ x -1.0))) (log (/ -1.0 y))) (log1p (- x))))
   (if (<= y 14800000000000.0)
     (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
     (- 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 <= -480000.0) {
		tmp = 1.0 + ((((1.0 - x) / (y * (x + -1.0))) - log((-1.0 / y))) - log1p(-x));
	} else if (y <= 14800000000000.0) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} 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 <= -480000.0) {
		tmp = 1.0 + ((((1.0 - x) / (y * (x + -1.0))) - Math.log((-1.0 / y))) - Math.log1p(-x));
	} else if (y <= 14800000000000.0) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} 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 <= -480000.0:
		tmp = 1.0 + ((((1.0 - x) / (y * (x + -1.0))) - math.log((-1.0 / y))) - math.log1p(-x))
	elif y <= 14800000000000.0:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	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 <= -480000.0)
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(1.0 - x) / Float64(y * Float64(x + -1.0))) - log(Float64(-1.0 / y))) - log1p(Float64(-x))));
	elseif (y <= 14800000000000.0)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	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, -480000.0], N[(1.0 + N[(N[(N[(N[(1.0 - x), $MachinePrecision] / N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 14800000000000.0], N[(1.0 - N[Log[1 + N[(N[(y - x), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Target

Original	19.4
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 < -4.8e5

   1. Initial program 53.3

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

   2. Simplified 53.3

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

      [Start] 53.3	\[ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
      sub-neg [=>] 53.3	\[ 1 - \log \color{blue}{\left(1 + \left(-\frac{x - y}{1 - y}\right)\right)} \]
      log1p-def [=>] 53.3	\[ 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]
      div-sub [=>] 53.3	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} - \frac{y}{1 - y}\right)}\right) \]
      sub-neg [=>] 53.3	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} + \left(-\frac{y}{1 - y}\right)\right)}\right) \]
      +-commutative [=>] 53.3	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\left(-\frac{y}{1 - y}\right) + \frac{x}{1 - y}\right)}\right) \]
      distribute-neg-in [=>] 53.3	\[ 1 - \mathsf{log1p}\left(\color{blue}{\left(-\left(-\frac{y}{1 - y}\right)\right) + \left(-\frac{x}{1 - y}\right)}\right) \]
      remove-double-neg [=>] 53.3	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y}} + \left(-\frac{x}{1 - y}\right)\right) \]
      sub-neg [<=] 53.3	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y} - \frac{x}{1 - y}}\right) \]
      div-sub [<=] 53.3	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y - x}{1 - y}}\right) \]

   3. Taylor expanded in y around -inf 0.3

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

   4. Simplified 0.3

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

      [Start] 0.3	\[ 1 - \left(\log \left(-1 \cdot \left(x - 1\right)\right) + \left(\log \left(\frac{-1}{y}\right) + -1 \cdot \frac{\frac{1}{x - 1} - \frac{x}{x - 1}}{y}\right)\right) \]
      sub-neg [=>] 0.3	\[ 1 - \left(\log \left(-1 \cdot \color{blue}{\left(x + \left(-1\right)\right)}\right) + \left(\log \left(\frac{-1}{y}\right) + -1 \cdot \frac{\frac{1}{x - 1} - \frac{x}{x - 1}}{y}\right)\right) \]
      metadata-eval [=>] 0.3	\[ 1 - \left(\log \left(-1 \cdot \left(x + \color{blue}{-1}\right)\right) + \left(\log \left(\frac{-1}{y}\right) + -1 \cdot \frac{\frac{1}{x - 1} - \frac{x}{x - 1}}{y}\right)\right) \]
      distribute-lft-in [=>] 0.3	\[ 1 - \left(\log \color{blue}{\left(-1 \cdot x + -1 \cdot -1\right)} + \left(\log \left(\frac{-1}{y}\right) + -1 \cdot \frac{\frac{1}{x - 1} - \frac{x}{x - 1}}{y}\right)\right) \]
      metadata-eval [=>] 0.3	\[ 1 - \left(\log \left(-1 \cdot x + \color{blue}{1}\right) + \left(\log \left(\frac{-1}{y}\right) + -1 \cdot \frac{\frac{1}{x - 1} - \frac{x}{x - 1}}{y}\right)\right) \]
      +-commutative [<=] 0.3	\[ 1 - \left(\log \color{blue}{\left(1 + -1 \cdot x\right)} + \left(\log \left(\frac{-1}{y}\right) + -1 \cdot \frac{\frac{1}{x - 1} - \frac{x}{x - 1}}{y}\right)\right) \]
      log1p-def [=>] 0.3	\[ 1 - \left(\color{blue}{\mathsf{log1p}\left(-1 \cdot x\right)} + \left(\log \left(\frac{-1}{y}\right) + -1 \cdot \frac{\frac{1}{x - 1} - \frac{x}{x - 1}}{y}\right)\right) \]
      mul-1-neg [=>] 0.3	\[ 1 - \left(\mathsf{log1p}\left(\color{blue}{-x}\right) + \left(\log \left(\frac{-1}{y}\right) + -1 \cdot \frac{\frac{1}{x - 1} - \frac{x}{x - 1}}{y}\right)\right) \]
      mul-1-neg [=>] 0.3	\[ 1 - \left(\mathsf{log1p}\left(-x\right) + \left(\log \left(\frac{-1}{y}\right) + \color{blue}{\left(-\frac{\frac{1}{x - 1} - \frac{x}{x - 1}}{y}\right)}\right)\right) \]
      unsub-neg [=>] 0.3	\[ 1 - \left(\mathsf{log1p}\left(-x\right) + \color{blue}{\left(\log \left(\frac{-1}{y}\right) - \frac{\frac{1}{x - 1} - \frac{x}{x - 1}}{y}\right)}\right) \]
      div-sub [<=] 0.3	\[ 1 - \left(\mathsf{log1p}\left(-x\right) + \left(\log \left(\frac{-1}{y}\right) - \frac{\color{blue}{\frac{1 - x}{x - 1}}}{y}\right)\right) \]
      associate-/l/ [=>] 0.3	\[ 1 - \left(\mathsf{log1p}\left(-x\right) + \left(\log \left(\frac{-1}{y}\right) - \color{blue}{\frac{1 - x}{y \cdot \left(x - 1\right)}}\right)\right) \]
      sub-neg [=>] 0.3	\[ 1 - \left(\mathsf{log1p}\left(-x\right) + \left(\log \left(\frac{-1}{y}\right) - \frac{1 - x}{y \cdot \color{blue}{\left(x + \left(-1\right)\right)}}\right)\right) \]
      metadata-eval [=>] 0.3	\[ 1 - \left(\mathsf{log1p}\left(-x\right) + \left(\log \left(\frac{-1}{y}\right) - \frac{1 - x}{y \cdot \left(x + \color{blue}{-1}\right)}\right)\right) \]
      +-commutative [=>] 0.3	\[ 1 - \left(\mathsf{log1p}\left(-x\right) + \left(\log \left(\frac{-1}{y}\right) - \frac{1 - x}{y \cdot \color{blue}{\left(-1 + x\right)}}\right)\right) \]

   if -4.8e5 < y < 1.48e13

   1. Initial program 0.0

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

   2. Simplified 0.0

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

      [Start] 0.0	\[ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
      sub-neg [=>] 0.0	\[ 1 - \log \color{blue}{\left(1 + \left(-\frac{x - y}{1 - y}\right)\right)} \]
      log1p-def [=>] 0.0	\[ 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]
      div-sub [=>] 0.0	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} - \frac{y}{1 - y}\right)}\right) \]
      sub-neg [=>] 0.0	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} + \left(-\frac{y}{1 - y}\right)\right)}\right) \]
      +-commutative [=>] 0.0	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\left(-\frac{y}{1 - y}\right) + \frac{x}{1 - y}\right)}\right) \]
      distribute-neg-in [=>] 0.0	\[ 1 - \mathsf{log1p}\left(\color{blue}{\left(-\left(-\frac{y}{1 - y}\right)\right) + \left(-\frac{x}{1 - y}\right)}\right) \]
      remove-double-neg [=>] 0.0	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y}} + \left(-\frac{x}{1 - y}\right)\right) \]
      sub-neg [<=] 0.0	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y} - \frac{x}{1 - y}}\right) \]
      div-sub [<=] 0.0	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y - x}{1 - y}}\right) \]

   if 1.48e13 < y

   1. Initial program 31.5

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

   2. Simplified 31.5

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

      [Start] 31.5	\[ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
      sub-neg [=>] 31.5	\[ 1 - \log \color{blue}{\left(1 + \left(-\frac{x - y}{1 - y}\right)\right)} \]
      log1p-def [=>] 31.5	\[ 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]
      div-sub [=>] 31.5	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} - \frac{y}{1 - y}\right)}\right) \]
      sub-neg [=>] 31.5	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} + \left(-\frac{y}{1 - y}\right)\right)}\right) \]
      +-commutative [=>] 31.5	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\left(-\frac{y}{1 - y}\right) + \frac{x}{1 - y}\right)}\right) \]
      distribute-neg-in [=>] 31.5	\[ 1 - \mathsf{log1p}\left(\color{blue}{\left(-\left(-\frac{y}{1 - y}\right)\right) + \left(-\frac{x}{1 - y}\right)}\right) \]
      remove-double-neg [=>] 31.5	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y}} + \left(-\frac{x}{1 - y}\right)\right) \]
      sub-neg [<=] 31.5	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y} - \frac{x}{1 - y}}\right) \]
      div-sub [<=] 31.5	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y - x}{1 - y}}\right) \]

   3. Taylor expanded in y around inf 1.0

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

   4. Simplified 1.0

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

      [Start] 1.0	\[ 1 - \left(\log \left(\frac{1}{y}\right) + \log \left(x - 1\right)\right) \]
      +-commutative [=>] 1.0	\[ 1 - \color{blue}{\left(\log \left(x - 1\right) + \log \left(\frac{1}{y}\right)\right)} \]
      log-rec [=>] 1.0	\[ 1 - \left(\log \left(x - 1\right) + \color{blue}{\left(-\log y\right)}\right) \]
      unsub-neg [=>] 1.0	\[ 1 - \color{blue}{\left(\log \left(x - 1\right) - \log y\right)} \]
      sub-neg [=>] 1.0	\[ 1 - \left(\log \color{blue}{\left(x + \left(-1\right)\right)} - \log y\right) \]
      metadata-eval [=>] 1.0	\[ 1 - \left(\log \left(x + \color{blue}{-1}\right) - \log y\right) \]
      +-commutative [=>] 1.0	\[ 1 - \left(\log \color{blue}{\left(-1 + x\right)} - \log y\right) \]

   5. Taylor expanded in y around 0 1.0

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

   6. Simplified 0.0

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

      [Start] 1.0	\[ 1 - \left(\log \left(x - 1\right) - \log y\right) \]
      sub-neg [=>] 1.0	\[ 1 - \left(\log \color{blue}{\left(x + \left(-1\right)\right)} - \log y\right) \]
      metadata-eval [=>] 1.0	\[ 1 - \left(\log \left(x + \color{blue}{-1}\right) - \log y\right) \]
      +-commutative [<=] 1.0	\[ 1 - \left(\log \color{blue}{\left(-1 + x\right)} - \log y\right) \]
      log-div [<=] 0.0	\[ 1 - \color{blue}{\log \left(\frac{-1 + x}{y}\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -480000:\\ \;\;\;\;1 + \left(\left(\frac{1 - x}{y \cdot \left(x + -1\right)} - \log \left(\frac{-1}{y}\right)\right) - \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{elif}\;y \leq 14800000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	7.9
Cost	7312

\[\begin{array}{l} t_0 := 1 - \log \left(\frac{-1}{y}\right)\\ t_1 := 1 - \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;y \leq -1.25 \cdot 10^{+234}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{+182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -20:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	8.3
Cost	7248

\[\begin{array}{l} t_0 := 1 - \log \left(\frac{-1}{y}\right)\\ t_1 := 1 - \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;y \leq -1.25 \cdot 10^{+234}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{+182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.8:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	0.1
Cost	7241

\[\begin{array}{l} \mathbf{if}\;y \leq -1320000000 \lor \neg \left(y \leq 11000000000000\right):\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \end{array} \]

Alternative 4
Error	0.7
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 5
Error	0.9
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -900 \lor \neg \left(y \leq 2400\right):\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \end{array} \]

Alternative 6
Error	13.6
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -4.8:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]

Alternative 7
Error	24.8
Cost	6656

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

Alternative 8
Error	37.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023018 
(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))))))