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

Average Error: 18.0 → 0.1

Time: 11.9s

Precision: binary64

Cost: 8393

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -2300000000 \lor \neg \left(y \leq 2.8 \cdot 10^{+14}\right):\\ \;\;\;\;1 - \log \left(\frac{-1 + x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y \cdot \left(1 - y\right) + x \cdot \left(y + -1\right)}{\left(y + -1\right) \cdot \left(-1 + y \cdot y\right)} \cdot \left(y + 1\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2300000000.0) (not (<= y 2.8e+14)))
   (- 1.0 (log (/ (+ -1.0 x) y)))
   (-
    1.0
    (log1p
     (*
      (/ (+ (* y (- 1.0 y)) (* x (+ y -1.0))) (* (+ y -1.0) (+ -1.0 (* y y))))
      (+ y 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 tmp;
	if ((y <= -2300000000.0) || !(y <= 2.8e+14)) {
		tmp = 1.0 - log(((-1.0 + x) / y));
	} else {
		tmp = 1.0 - log1p(((((y * (1.0 - y)) + (x * (y + -1.0))) / ((y + -1.0) * (-1.0 + (y * y)))) * (y + 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 tmp;
	if ((y <= -2300000000.0) || !(y <= 2.8e+14)) {
		tmp = 1.0 - Math.log(((-1.0 + x) / y));
	} else {
		tmp = 1.0 - Math.log1p(((((y * (1.0 - y)) + (x * (y + -1.0))) / ((y + -1.0) * (-1.0 + (y * y)))) * (y + 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):
	tmp = 0
	if (y <= -2300000000.0) or not (y <= 2.8e+14):
		tmp = 1.0 - math.log(((-1.0 + x) / y))
	else:
		tmp = 1.0 - math.log1p(((((y * (1.0 - y)) + (x * (y + -1.0))) / ((y + -1.0) * (-1.0 + (y * y)))) * (y + 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)
	tmp = 0.0
	if ((y <= -2300000000.0) || !(y <= 2.8e+14))
		tmp = Float64(1.0 - log(Float64(Float64(-1.0 + x) / y)));
	else
		tmp = Float64(1.0 - log1p(Float64(Float64(Float64(Float64(y * Float64(1.0 - y)) + Float64(x * Float64(y + -1.0))) / Float64(Float64(y + -1.0) * Float64(-1.0 + Float64(y * y)))) * Float64(y + 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_] := If[Or[LessEqual[y, -2300000000.0], N[Not[LessEqual[y, 2.8e+14]], $MachinePrecision]], N[(1.0 - N[Log[N[(N[(-1.0 + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(N[(N[(N[(y * N[(1.0 - y), $MachinePrecision]), $MachinePrecision] + N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y + -1.0), $MachinePrecision] * N[(-1.0 + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Target

Original	18.0
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 2 regimes

2. if y < -2.3e9 or 2.8e14 < y

   1. Initial program 47.3

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

   2. Simplified 47.3

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

      [Start] 47.3	\[ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
      sub-neg [=>] 47.3	\[ 1 - \log \color{blue}{\left(1 + \left(-\frac{x - y}{1 - y}\right)\right)} \]
      log1p-def [=>] 47.3	\[ 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]
      div-sub [=>] 47.3	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} - \frac{y}{1 - y}\right)}\right) \]
      sub-neg [=>] 47.3	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} + \left(-\frac{y}{1 - y}\right)\right)}\right) \]
      +-commutative [=>] 47.3	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\left(-\frac{y}{1 - y}\right) + \frac{x}{1 - y}\right)}\right) \]
      distribute-neg-in [=>] 47.3	\[ 1 - \mathsf{log1p}\left(\color{blue}{\left(-\left(-\frac{y}{1 - y}\right)\right) + \left(-\frac{x}{1 - y}\right)}\right) \]
      remove-double-neg [=>] 47.3	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y}} + \left(-\frac{x}{1 - y}\right)\right) \]
      sub-neg [<=] 47.3	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y} - \frac{x}{1 - y}}\right) \]
      div-sub [<=] 47.3	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y - x}{1 - y}}\right) \]

   3. Taylor expanded in y around inf 48.1

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

   4. Simplified 48.1

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

      [Start] 48.1	\[ 1 - \left(\log \left(\frac{1}{y}\right) + \log \left(x - 1\right)\right) \]
      +-commutative [=>] 48.1	\[ 1 - \color{blue}{\left(\log \left(x - 1\right) + \log \left(\frac{1}{y}\right)\right)} \]
      log-rec [=>] 48.1	\[ 1 - \left(\log \left(x - 1\right) + \color{blue}{\left(-\log y\right)}\right) \]
      unsub-neg [=>] 48.1	\[ 1 - \color{blue}{\left(\log \left(x - 1\right) - \log y\right)} \]
      sub-neg [=>] 48.1	\[ 1 - \left(\log \color{blue}{\left(x + \left(-1\right)\right)} - \log y\right) \]
      metadata-eval [=>] 48.1	\[ 1 - \left(\log \left(x + \color{blue}{-1}\right) - \log y\right) \]
      +-commutative [=>] 48.1	\[ 1 - \left(\log \color{blue}{\left(-1 + x\right)} - \log y\right) \]

   5. Taylor expanded in y around 0 48.1

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

   6. Simplified 0.1

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

      [Start] 48.1	\[ 1 - \left(\log \left(x - 1\right) - \log y\right) \]
      sub-neg [=>] 48.1	\[ 1 - \left(\log \color{blue}{\left(x + \left(-1\right)\right)} - \log y\right) \]
      metadata-eval [=>] 48.1	\[ 1 - \left(\log \left(x + \color{blue}{-1}\right) - \log y\right) \]
      +-commutative [<=] 48.1	\[ 1 - \left(\log \color{blue}{\left(-1 + x\right)} - \log y\right) \]
      log-div [<=] 0.1	\[ 1 - \color{blue}{\log \left(\frac{-1 + x}{y}\right)} \]

   if -2.3e9 < y < 2.8e14

   1. Initial program 0.1

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

   2. Simplified 0.1

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

      [Start] 0.1	\[ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
      sub-neg [=>] 0.1	\[ 1 - \log \color{blue}{\left(1 + \left(-\frac{x - y}{1 - y}\right)\right)} \]
      log1p-def [=>] 0.1	\[ 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]
      div-sub [=>] 0.1	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} - \frac{y}{1 - y}\right)}\right) \]
      sub-neg [=>] 0.1	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} + \left(-\frac{y}{1 - y}\right)\right)}\right) \]
      +-commutative [=>] 0.1	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\left(-\frac{y}{1 - y}\right) + \frac{x}{1 - y}\right)}\right) \]
      distribute-neg-in [=>] 0.1	\[ 1 - \mathsf{log1p}\left(\color{blue}{\left(-\left(-\frac{y}{1 - y}\right)\right) + \left(-\frac{x}{1 - y}\right)}\right) \]
      remove-double-neg [=>] 0.1	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y}} + \left(-\frac{x}{1 - y}\right)\right) \]
      sub-neg [<=] 0.1	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y} - \frac{x}{1 - y}}\right) \]
      div-sub [<=] 0.1	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y - x}{1 - y}}\right) \]

   3. Applied egg-rr 0.1

      \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\frac{y \cdot \left(1 - y\right) - \left(1 - y\right) \cdot x}{\left(1 - y \cdot y\right) \cdot \left(1 - y\right)} \cdot \left(y + 1\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2300000000 \lor \neg \left(y \leq 2.8 \cdot 10^{+14}\right):\\ \;\;\;\;1 - \log \left(\frac{-1 + x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y \cdot \left(1 - y\right) + x \cdot \left(y + -1\right)}{\left(y + -1\right) \cdot \left(-1 + y \cdot y\right)} \cdot \left(y + 1\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	7492

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

Alternative 2
Error	7.6
Cost	7248

\[\begin{array}{l} t_0 := 1 - \log \left(\frac{-1}{y}\right)\\ t_1 := 1 - \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;y \leq -2.9 \cdot 10^{+74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3.7 \cdot 10^{+33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1500000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	0.8
Cost	7177

\[\begin{array}{l} \mathbf{if}\;y \leq -1500000000 \lor \neg \left(y \leq 460000000000\right):\\ \;\;\;\;1 - \log \left(\frac{-1 + x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{-x}{1 - y}\right)\\ \end{array} \]

Alternative 4
Error	0.7
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -1.8 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \log \left(\frac{-1 + x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \]

Alternative 5
Error	13.1
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -1500000000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]

Alternative 6
Error	23.5
Cost	6656

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

Alternative 7
Error	35.1
Cost	448

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

Alternative 8
Error	36.2
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023039 
(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))))))