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

Average Error: 17.8 → 0.3

Time: 7.6s

Precision: binary64

Cost: 7492

Math

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

↓

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

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)) 0.5)
   (- 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)) <= 0.5) {
		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)) <= 0.5) {
		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)) <= 0.5:
		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)) <= 0.5)
		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], 0.5], 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	17.8
Target	0.1
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 0.5

   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))))))): 2 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 0.5 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 60.6

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

   2. Simplified 60.6

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)} \]
      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))))))): 2 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 61.1

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

   4. Taylor expanded in y around 0 53.0

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

   5. Simplified 0.8

      \[\leadsto 1 - \color{blue}{\log \left(\frac{-1 + x}{y}\right)} \]
      Proof
      (log.f64 (/.f64 (+.f64 -1 x) y)): 0 points increase in error, 0 points decrease in error
      (log.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 x -1)) y)): 0 points increase in error, 0 points decrease in error
      (log.f64 (/.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))) y)): 0 points increase in error, 0 points decrease in error
      (log.f64 (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)) y)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> log-div_binary64 (-.f64 (log.f64 (-.f64 x 1)) (log.f64 y))): 127 points increase in error, 14 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (log.f64 (-.f64 x 1)) (neg.f64 (log.f64 y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (log.f64 (-.f64 x 1)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (log.f64 y)))): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

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

Alternatives

Alternative 1
Error	1.1
Cost	7112

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

Alternative 2
Error	0.8
Cost	7112

\[\begin{array}{l} t_0 := 1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{if}\;y \leq -200000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 420000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	10.0
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -22.5:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-5}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 4
Error	25.1
Cost	6788

\[\begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-5}:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{1 - y}\\ \end{array} \]

Alternative 5
Error	13.3
Cost	6788

\[\begin{array}{l} \mathbf{if}\;y \leq -50:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]

Alternative 6
Error	35.0
Cost	448

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

Alternative 7
Error	36.0
Cost	192

\[x + 1 \]

Alternative 8
Error	36.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022330 
(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))))))