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

Average Error: 18.5 → 0.2

Time: 13.6s

Precision: binary64

Cost: 7492

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.99:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.99)
   (- 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 (((x - y) / (1.0 - y)) <= 0.99) {
		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 (((x - y) / (1.0 - y)) <= 0.99) {
		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 ((x - y) / (1.0 - y)) <= 0.99:
		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 (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.99)
		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[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.99], 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	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 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 0.98999999999999999

   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
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 y x) (-.f64 1 y)))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite=> div-sub_binary64 (-.f64 (/.f64 y (-.f64 1 y)) (/.f64 x (-.f64 1 y)))))): 0 points increase in error, 1 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 y (-.f64 1 y)) (neg.f64 (/.f64 x (-.f64 1 y))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 (/.f64 x (-.f64 1 y))) (/.f64 y (-.f64 1 y)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (+.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 (/.f64 x (-.f64 1 y)))) (/.f64 y (-.f64 1 y))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 (/.f64 x (-.f64 1 y)) (/.f64 y (-.f64 1 y))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (-.f64 0 (Rewrite<= div-sub_binary64 (/.f64 (-.f64 x y) (-.f64 1 y)))))): 1 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (/.f64 (-.f64 x y) (-.f64 1 y)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 (neg.f64 (/.f64 (-.f64 x y) (-.f64 1 y))))))): 3 points increase in error, 0 points decrease in error
      (-.f64 1 (log.f64 (Rewrite<= sub-neg_binary64 (-.f64 1 (/.f64 (-.f64 x y) (-.f64 1 y)))))): 0 points increase in error, 0 points decrease in error

   if 0.98999999999999999 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 61.8

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

   2. Simplified 61.8

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)} \]
      Proof
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 y x) (-.f64 1 y)))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite=> div-sub_binary64 (-.f64 (/.f64 y (-.f64 1 y)) (/.f64 x (-.f64 1 y)))))): 0 points increase in error, 1 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 y (-.f64 1 y)) (neg.f64 (/.f64 x (-.f64 1 y))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 (/.f64 x (-.f64 1 y))) (/.f64 y (-.f64 1 y)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (+.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 (/.f64 x (-.f64 1 y)))) (/.f64 y (-.f64 1 y))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 (/.f64 x (-.f64 1 y)) (/.f64 y (-.f64 1 y))))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (-.f64 0 (Rewrite<= div-sub_binary64 (/.f64 (-.f64 x y) (-.f64 1 y)))))): 1 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (/.f64 (-.f64 x y) (-.f64 1 y)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 (neg.f64 (/.f64 (-.f64 x y) (-.f64 1 y))))))): 3 points increase in error, 0 points decrease in error
      (-.f64 1 (log.f64 (Rewrite<= sub-neg_binary64 (-.f64 1 (/.f64 (-.f64 x y) (-.f64 1 y)))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in y around inf 53.6

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

   4. Simplified 53.6

      \[\leadsto \color{blue}{\left(1 + \log y\right) - \log \left(-1 + x\right)} \]
      Proof
      (-.f64 (+.f64 1 (log.f64 y)) (log.f64 (+.f64 -1 x))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (Rewrite<= metadata-eval (-.f64 1 0)) (log.f64 y)) (log.f64 (+.f64 -1 x))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate--r-_binary64 (-.f64 1 (-.f64 0 (log.f64 y)))) (log.f64 (+.f64 -1 x))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 1 (-.f64 (Rewrite<= metadata-eval (log.f64 1)) (log.f64 y))) (log.f64 (+.f64 -1 x))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 1 (Rewrite<= log-div_binary64 (log.f64 (/.f64 1 y)))) (log.f64 (+.f64 -1 x))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 1 (log.f64 (/.f64 1 y))) (log.f64 (Rewrite<= +-commutative_binary64 (+.f64 x -1)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 1 (log.f64 (/.f64 1 y))) (log.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 1 (log.f64 (/.f64 1 y))) (log.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--r+_binary64 (-.f64 1 (+.f64 (log.f64 (/.f64 1 y)) (log.f64 (-.f64 x 1))))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in y around 0 53.6

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

   6. Simplified 0.5

      \[\leadsto \color{blue}{1 - \log \left(\frac{x + -1}{y}\right)} \]
      Proof
      (-.f64 1 (log.f64 (/.f64 (+.f64 x -1) y))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 -1 x)) y))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (Rewrite=> log-div_binary64 (-.f64 (log.f64 (+.f64 -1 x)) (log.f64 y)))): 158 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 1 (log.f64 (+.f64 -1 x))) (log.f64 y))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= unsub-neg_binary64 (+.f64 1 (neg.f64 (log.f64 (+.f64 -1 x))))) (log.f64 y)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-+l+_binary64 (+.f64 1 (+.f64 (neg.f64 (log.f64 (+.f64 -1 x))) (log.f64 y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (Rewrite<= +-commutative_binary64 (+.f64 (log.f64 y) (neg.f64 (log.f64 (+.f64 -1 x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (Rewrite=> unsub-neg_binary64 (-.f64 (log.f64 y) (log.f64 (+.f64 -1 x))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 1 (log.f64 y)) (log.f64 (+.f64 -1 x)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 1 (log.f64 y)) (log.f64 (Rewrite=> +-commutative_binary64 (+.f64 x -1)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 1 (log.f64 y)) (log.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 1 (log.f64 y)) (log.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)))): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.99:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	7.1
Cost	7244

\[\begin{array}{l} t_0 := 1 - \log \left(\frac{x}{y + -1}\right)\\ \mathbf{if}\;y \leq -9.5 \cdot 10^{+26}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.011:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	7.0
Cost	7180

\[\begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+28}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq -2.6:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 3
Error	0.8
Cost	7112

\[\begin{array}{l} t_0 := 1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{if}\;y \leq -1.8:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	0.9
Cost	7112

\[\begin{array}{l} t_0 := 1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{if}\;y \leq -2100000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 18000000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	7.3
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+16}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 6
Error	13.3
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+16}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]

Alternative 7
Error	23.6
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq -0.9:\\ \;\;\;\;1 - \log \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(-1 + \left(1 + \frac{x}{y + -1}\right)\right)\\ \end{array} \]

Alternative 8
Error	23.8
Cost	6656

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

Alternative 9
Error	35.1
Cost	704

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

Alternative 10
Error	35.1
Cost	448

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

Alternative 11
Error	36.2
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022325 
(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))))))