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

Average Error: 18.5 → 0.9

Time: 8.7s

Precision: binary64

Cost: 13896

Math

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

↓

\[\begin{array}{l} t_0 := 1 - \log \left(\frac{-1 + x}{y}\right)\\ \mathbf{if}\;y \leq -1 \cdot 10^{+30}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{\mathsf{fma}\left(y, y, -1\right)} \cdot \left(y + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (/ (+ -1.0 x) y)))))
   (if (<= y -1e+30)
     t_0
     (if (<= y 1.0)
       (- 1.0 (log1p (* (/ (- x y) (fma y y -1.0)) (+ y 1.0))))
       t_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 t_0 = 1.0 - log(((-1.0 + x) / y));
	double tmp;
	if (y <= -1e+30) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = 1.0 - log1p((((x - y) / fma(y, y, -1.0)) * (y + 1.0)));
	} else {
		tmp = t_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)
	t_0 = Float64(1.0 - log(Float64(Float64(-1.0 + x) / y)))
	tmp = 0.0
	if (y <= -1e+30)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(1.0 - log1p(Float64(Float64(Float64(x - y) / fma(y, y, -1.0)) * Float64(y + 1.0))));
	else
		tmp = t_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_] := Block[{t$95$0 = N[(1.0 - N[Log[N[(N[(-1.0 + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1e+30], t$95$0, If[LessEqual[y, 1.0], N[(1.0 - N[Log[1 + N[(N[(N[(x - y), $MachinePrecision] / N[(y * y + -1.0), $MachinePrecision]), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	18.5
Target	0.1
Herbie	0.9

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

Derivation

1. Split input into 2 regimes

2. if y < -1e30 or 1 < y

   1. Initial program 47.2

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

   2. Simplified 47.2

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
      Proof
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (+.f64 y -1)))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (Rewrite=> +-commutative_binary64 (+.f64 -1 y))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (+.f64 (Rewrite<= metadata-eval (-.f64 0 1)) y)))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 1 y)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (Rewrite=> sub0-neg_binary64 (neg.f64 (-.f64 1 y)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 1 y)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 1 y) -1))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 (-.f64 x y) (-.f64 1 y)) -1)))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (-.f64 x y) (-.f64 1 y)))) -1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite=> associate-/l*_binary64 (/.f64 1 (/.f64 -1 (/.f64 (-.f64 x y) (-.f64 1 y))))))): 2 points increase in error, 1 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 1 -1) (/.f64 (-.f64 x y) (-.f64 1 y)))))): 1 points increase in error, 2 points decrease in error
      (-.f64 1 (log1p.f64 (*.f64 (Rewrite=> metadata-eval -1) (/.f64 (-.f64 x y) (-.f64 1 y))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite<= neg-mul-1_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))))))): 5 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 46.8

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

   4. Simplified 46.8

      \[\leadsto 1 - \color{blue}{\left(\log \left(-1 + x\right) - \log y\right)} \]
      Proof
      (-.f64 (log.f64 (+.f64 -1 x)) (log.f64 y)): 0 points increase in error, 0 points decrease in error
      (-.f64 (log.f64 (Rewrite<= +-commutative_binary64 (+.f64 x -1))) (log.f64 y)): 0 points increase in error, 0 points decrease in error
      (-.f64 (log.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1)))) (log.f64 y)): 0 points increase in error, 0 points decrease in error
      (-.f64 (log.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1))) (log.f64 y)): 0 points increase in error, 0 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<= log-rec_binary64 (log.f64 (/.f64 1 y)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (log.f64 (/.f64 1 y)) (log.f64 (-.f64 x 1)))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 0.3

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

   if -1e30 < y < 1

   1. Initial program 1.4

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

   2. Simplified 1.4

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
      Proof
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (+.f64 y -1)))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (Rewrite=> +-commutative_binary64 (+.f64 -1 y))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (+.f64 (Rewrite<= metadata-eval (-.f64 0 1)) y)))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 1 y)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (Rewrite=> sub0-neg_binary64 (neg.f64 (-.f64 1 y)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 1 y)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (-.f64 x y) (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 1 y) -1))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 (-.f64 x y) (-.f64 1 y)) -1)))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (-.f64 x y) (-.f64 1 y)))) -1))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite=> associate-/l*_binary64 (/.f64 1 (/.f64 -1 (/.f64 (-.f64 x y) (-.f64 1 y))))))): 2 points increase in error, 1 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 1 -1) (/.f64 (-.f64 x y) (-.f64 1 y)))))): 1 points increase in error, 2 points decrease in error
      (-.f64 1 (log1p.f64 (*.f64 (Rewrite=> metadata-eval -1) (/.f64 (-.f64 x y) (-.f64 1 y))))): 0 points increase in error, 0 points decrease in error
      (-.f64 1 (log1p.f64 (Rewrite<= neg-mul-1_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))))))): 5 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. Applied egg-rr 1.3

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+30}:\\ \;\;\;\;1 - \log \left(\frac{-1 + x}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{\mathsf{fma}\left(y, y, -1\right)} \cdot \left(y + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1 + x}{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	7492

\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.95:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1 + x}{y}\right)\\ \end{array} \]

Alternative 2
Error	1.7
Cost	7112

\[\begin{array}{l} t_0 := 1 - \log \left(\frac{-1 + x}{y}\right)\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{+48}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	9.7
Cost	7048

\[\begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-7}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 0.22:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 4
Error	10.8
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+27}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-10}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 5
Error	14.2
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+27}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]

Alternative 6
Error	24.0
Cost	6656

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

Alternative 7
Error	35.6
Cost	448

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

Alternative 8
Error	36.5
Cost	192

\[1 + x \]

Alternative 9
Error	36.6
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022317 
(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))))))