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

Average Error: 18.5 → 0.2

Time: 9.4s

Precision: binary64

Cost: 14084

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -460000:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + \left(\log \left(\frac{-1}{y}\right) - \frac{1 - x}{y \cdot \left(x + -1\right)}\right)\right)\\ \mathbf{elif}\;y \leq 10^{+15}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= y -460000.0)
   (-
    1.0
    (+ (log1p (- x)) (- (log (/ -1.0 y)) (/ (- 1.0 x) (* y (+ x -1.0))))))
   (if (<= y 1e+15)
     (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
     (- 1.0 (log (/ x 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 <= -460000.0) {
		tmp = 1.0 - (log1p(-x) + (log((-1.0 / y)) - ((1.0 - x) / (y * (x + -1.0)))));
	} else if (y <= 1e+15) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - log((x / 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 <= -460000.0) {
		tmp = 1.0 - (Math.log1p(-x) + (Math.log((-1.0 / y)) - ((1.0 - x) / (y * (x + -1.0)))));
	} else if (y <= 1e+15) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - Math.log((x / 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 <= -460000.0:
		tmp = 1.0 - (math.log1p(-x) + (math.log((-1.0 / y)) - ((1.0 - x) / (y * (x + -1.0)))))
	elif y <= 1e+15:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = 1.0 - math.log((x / 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 <= -460000.0)
		tmp = Float64(1.0 - Float64(log1p(Float64(-x)) + Float64(log(Float64(-1.0 / y)) - Float64(Float64(1.0 - x) / Float64(y * Float64(x + -1.0))))));
	elseif (y <= 1e+15)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 - log(Float64(x / 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, -460000.0], N[(1.0 - N[(N[Log[1 + (-x)], $MachinePrecision] + N[(N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision] - N[(N[(1.0 - x), $MachinePrecision] / N[(y * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+15], 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[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Target

Original	18.5
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 < -4.6e5

   1. Initial program 52.4

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

   2. Simplified 52.4

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

      [Start] 52.4	\[ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
      sub-neg [=>] 52.4	\[ 1 - \log \color{blue}{\left(1 + \left(-\frac{x - y}{1 - y}\right)\right)} \]
      log1p-def [=>] 52.4	\[ 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]
      div-sub [=>] 52.4	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} - \frac{y}{1 - y}\right)}\right) \]
      sub-neg [=>] 52.4	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} + \left(-\frac{y}{1 - y}\right)\right)}\right) \]
      +-commutative [=>] 52.4	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\left(-\frac{y}{1 - y}\right) + \frac{x}{1 - y}\right)}\right) \]
      distribute-neg-in [=>] 52.4	\[ 1 - \mathsf{log1p}\left(\color{blue}{\left(-\left(-\frac{y}{1 - y}\right)\right) + \left(-\frac{x}{1 - y}\right)}\right) \]
      remove-double-neg [=>] 52.4	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y}} + \left(-\frac{x}{1 - y}\right)\right) \]
      sub-neg [<=] 52.4	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y} - \frac{x}{1 - y}}\right) \]
      div-sub [<=] 52.4	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y - x}{1 - y}}\right) \]

   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 -4.6e5 < y < 1e15

   1. Initial program 0.1

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

   2. Simplified 0.0

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)} \]
      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 [=>] 0.0	\[ 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]
      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 1e15 < y

   1. Initial program 30.6

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

   2. Simplified 30.6

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

      [Start] 30.6	\[ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
      sub-neg [=>] 30.6	\[ 1 - \log \color{blue}{\left(1 + \left(-\frac{x - y}{1 - y}\right)\right)} \]
      log1p-def [=>] 30.6	\[ 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]
      div-sub [=>] 30.6	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} - \frac{y}{1 - y}\right)}\right) \]
      sub-neg [=>] 30.6	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} + \left(-\frac{y}{1 - y}\right)\right)}\right) \]
      +-commutative [=>] 30.6	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\left(-\frac{y}{1 - y}\right) + \frac{x}{1 - y}\right)}\right) \]
      distribute-neg-in [=>] 30.6	\[ 1 - \mathsf{log1p}\left(\color{blue}{\left(-\left(-\frac{y}{1 - y}\right)\right) + \left(-\frac{x}{1 - y}\right)}\right) \]
      remove-double-neg [=>] 30.6	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y}} + \left(-\frac{x}{1 - y}\right)\right) \]
      sub-neg [<=] 30.6	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y} - \frac{x}{1 - y}}\right) \]
      div-sub [<=] 30.6	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y - x}{1 - y}}\right) \]

   3. Applied egg-rr 56.6

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

   4. Taylor expanded in y around inf 29.8

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

   5. Applied egg-rr 32.8

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

   6. Simplified 0.7

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

      [Start] 32.8	\[ 1 - \left(e^{\mathsf{log1p}\left(\mathsf{log1p}\left(\frac{y - x}{-y}\right)\right)} - 1\right) \]
      expm1-def [=>] 32.8	\[ 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\frac{y - x}{-y}\right)\right)\right)} \]
      expm1-log1p [=>] 30.6	\[ 1 - \color{blue}{\mathsf{log1p}\left(\frac{y - x}{-y}\right)} \]
      log1p-def [<=] 30.6	\[ 1 - \color{blue}{\log \left(1 + \frac{y - x}{-y}\right)} \]
      div-sub [=>] 30.6	\[ 1 - \log \left(1 + \color{blue}{\left(\frac{y}{-y} - \frac{x}{-y}\right)}\right) \]
      *-lft-identity [<=] 30.6	\[ 1 - \log \left(1 + \left(\color{blue}{1 \cdot \frac{y}{-y}} - \frac{x}{-y}\right)\right) \]
      associate-+r- [=>] 0.7	\[ 1 - \log \color{blue}{\left(\left(1 + 1 \cdot \frac{y}{-y}\right) - \frac{x}{-y}\right)} \]
      associate-*r/ [=>] 0.7	\[ 1 - \log \left(\left(1 + \color{blue}{\frac{1 \cdot y}{-y}}\right) - \frac{x}{-y}\right) \]
      neg-mul-1 [=>] 0.7	\[ 1 - \log \left(\left(1 + \frac{1 \cdot y}{\color{blue}{-1 \cdot y}}\right) - \frac{x}{-y}\right) \]
      times-frac [=>] 0.7	\[ 1 - \log \left(\left(1 + \color{blue}{\frac{1}{-1} \cdot \frac{y}{y}}\right) - \frac{x}{-y}\right) \]
      metadata-eval [=>] 0.7	\[ 1 - \log \left(\left(1 + \color{blue}{-1} \cdot \frac{y}{y}\right) - \frac{x}{-y}\right) \]
      *-inverses [=>] 0.7	\[ 1 - \log \left(\left(1 + -1 \cdot \color{blue}{1}\right) - \frac{x}{-y}\right) \]
      metadata-eval [=>] 0.7	\[ 1 - \log \left(\left(1 + \color{blue}{-1}\right) - \frac{x}{-y}\right) \]
      metadata-eval [=>] 0.7	\[ 1 - \log \left(\color{blue}{0} - \frac{x}{-y}\right) \]
      neg-sub0 [<=] 0.7	\[ 1 - \log \color{blue}{\left(-\frac{x}{-y}\right)} \]
      distribute-frac-neg [<=] 0.7	\[ 1 - \log \color{blue}{\left(\frac{-x}{-y}\right)} \]
      mul-1-neg [<=] 0.7	\[ 1 - \log \left(\frac{\color{blue}{-1 \cdot x}}{-y}\right) \]
      *-commutative [=>] 0.7	\[ 1 - \log \left(\frac{\color{blue}{x \cdot -1}}{-y}\right) \]
      associate-/l* [=>] 0.7	\[ 1 - \log \color{blue}{\left(\frac{x}{\frac{-y}{-1}}\right)} \]
      neg-mul-1 [=>] 0.7	\[ 1 - \log \left(\frac{x}{\frac{\color{blue}{-1 \cdot y}}{-1}}\right) \]
      associate-/l* [=>] 0.7	\[ 1 - \log \left(\frac{x}{\color{blue}{\frac{-1}{\frac{-1}{y}}}}\right) \]
      metadata-eval [<=] 0.7	\[ 1 - \log \left(\frac{x}{\frac{-1}{\frac{\color{blue}{\frac{1}{-1}}}{y}}}\right) \]
      associate-/r* [<=] 0.7	\[ 1 - \log \left(\frac{x}{\frac{-1}{\color{blue}{\frac{1}{-1 \cdot y}}}}\right) \]
      neg-mul-1 [<=] 0.7	\[ 1 - \log \left(\frac{x}{\frac{-1}{\frac{1}{\color{blue}{-y}}}}\right) \]
      associate-/l* [<=] 0.7	\[ 1 - \log \left(\frac{x}{\color{blue}{\frac{-1 \cdot \left(-y\right)}{1}}}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -460000:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + \left(\log \left(\frac{-1}{y}\right) - \frac{1 - x}{y \cdot \left(x + -1\right)}\right)\right)\\ \mathbf{elif}\;y \leq 10^{+15}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	13444

\[\begin{array}{l} \mathbf{if}\;y \leq -265000000000:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+15}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 2
Error	6.1
Cost	7240

\[\begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+22}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+15}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 3
Error	6.9
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+22}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 4000000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 4
Error	6.6
Cost	7048

\[\begin{array}{l} \mathbf{if}\;y \leq -26:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 5
Error	7.0
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -108:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 6
Error	13.6
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -26:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]

Alternative 7
Error	24.1
Cost	6656

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

Alternative 8
Error	36.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023016 
(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))))))