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

Average Error: 18.2 → 11.1

Time: 9.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -6.305417564235616 \cdot 10^{-122} \lor \neg \left(x \leq 0.006781316404929303\right):\\ \;\;\;\;1 - \log \left(\left(1 + \frac{y}{1 - y}\right) - \frac{x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{\sqrt{1 - y}}}}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y \cdot y}} \cdot \frac{\sqrt[3]{1 + y}}{\sqrt[3]{\sqrt{1 - y}}}\right)\\ \end{array}\]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (or (<= x -6.305417564235616e-122) (not (<= x 0.006781316404929303)))
   (- 1.0 (log (- (+ 1.0 (/ y (- 1.0 y))) (/ x (- 1.0 y)))))
   (-
    1.0
    (log
     (-
      1.0
      (*
       (/
        (/ (- x y) (cbrt (sqrt (- 1.0 y))))
        (* (cbrt (- 1.0 y)) (cbrt (- 1.0 (* y y)))))
       (/ (cbrt (+ 1.0 y)) (cbrt (sqrt (- 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 <= -6.305417564235616e-122) || !(x <= 0.006781316404929303)) {
		tmp = 1.0 - log((1.0 + (y / (1.0 - y))) - (x / (1.0 - y)));
	} else {
		tmp = 1.0 - log(1.0 - ((((x - y) / cbrt(sqrt(1.0 - y))) / (cbrt(1.0 - y) * cbrt(1.0 - (y * y)))) * (cbrt(1.0 + y) / cbrt(sqrt(1.0 - y)))));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original	18.2
Target	0.1
Herbie	11.1

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

Derivation

1. Split input into 2 regimes

2. if x < -6.3054175642356161e-122 or 0.0067813164049293027 < x

   1. Initial program 15.9

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
   2. Using strategy rm

   3. Applied add-cube-cbrt_binary64_8297 14.7

      \[\leadsto 1 - \log \left(1 - \frac{x - y}{\color{blue}{\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}\right) \cdot \sqrt[3]{1 - y}}}\right)\]

   4. Applied associate-/r*_binary64_8206 14.7

      \[\leadsto 1 - \log \left(1 - \color{blue}{\frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}}\right)\]
   5. Using strategy rm

   6. Applied add-cube-cbrt_binary64_8297 14.7

      \[\leadsto 1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{\color{blue}{\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}\right) \cdot \sqrt[3]{1 - y}}}}\right)\]

   7. Applied cbrt-prod_binary64_8293 14.6

      \[\leadsto 1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\color{blue}{\sqrt[3]{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \sqrt[3]{\sqrt[3]{1 - y}}}}\right)\]

   8. Applied add-cube-cbrt_binary64_8297 14.8

      \[\leadsto 1 - \log \left(1 - \frac{\frac{\color{blue}{\left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right) \cdot \sqrt[3]{x - y}}}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \sqrt[3]{\sqrt[3]{1 - y}}}\right)\]

   9. Applied times-frac_binary64_8268 14.8

      \[\leadsto 1 - \log \left(1 - \frac{\color{blue}{\frac{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}{\sqrt[3]{1 - y}} \cdot \frac{\sqrt[3]{x - y}}{\sqrt[3]{1 - y}}}}{\sqrt[3]{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \sqrt[3]{\sqrt[3]{1 - y}}}\right)\]

   10. Applied times-frac_binary64_8268 14.8

       \[\leadsto 1 - \log \left(1 - \color{blue}{\frac{\frac{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}{\sqrt[3]{1 - y}}}{\sqrt[3]{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}} \cdot \frac{\frac{\sqrt[3]{x - y}}{\sqrt[3]{1 - y}}}{\sqrt[3]{\sqrt[3]{1 - y}}}}\right)\]

   11. Applied cancel-sign-sub-inv_binary64_8228 14.8

       \[\leadsto 1 - \log \color{blue}{\left(1 + \left(-\frac{\frac{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}{\sqrt[3]{1 - y}}}{\sqrt[3]{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}\right) \cdot \frac{\frac{\sqrt[3]{x - y}}{\sqrt[3]{1 - y}}}{\sqrt[3]{\sqrt[3]{1 - y}}}\right)}\]

   12. Simplified 14.8

       \[\leadsto 1 - \log \left(1 + \color{blue}{\frac{\frac{\sqrt[3]{x - y}}{\sqrt[3]{1 - y}}}{\sqrt[3]{\sqrt[3]{1 - y}}} \cdot \left(-\frac{\frac{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}{\sqrt[3]{1 - y}}}{\sqrt[3]{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}\right)}\right)\]

   13. Taylor expanded around 0 4.4

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

   14. Simplified 4.4

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

   if -6.3054175642356161e-122 < x < 0.0067813164049293027

   1. Initial program 20.8

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
   2. Using strategy rm

   3. Applied add-cube-cbrt_binary64_8297 19.0

      \[\leadsto 1 - \log \left(1 - \frac{x - y}{\color{blue}{\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}\right) \cdot \sqrt[3]{1 - y}}}\right)\]

   4. Applied associate-/r*_binary64_8206 19.0

      \[\leadsto 1 - \log \left(1 - \color{blue}{\frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}}\right)\]
   5. Using strategy rm

   6. Applied add-sqr-sqrt_binary64_8284 19.1

      \[\leadsto 1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{\color{blue}{\sqrt{1 - y} \cdot \sqrt{1 - y}}}}\right)\]

   7. Applied cbrt-prod_binary64_8293 18.9

      \[\leadsto 1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\color{blue}{\sqrt[3]{\sqrt{1 - y}} \cdot \sqrt[3]{\sqrt{1 - y}}}}\right)\]

   8. Applied flip--_binary64_8237 18.4

      \[\leadsto 1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{\color{blue}{\frac{1 \cdot 1 - y \cdot y}{1 + y}}} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{\sqrt{1 - y}} \cdot \sqrt[3]{\sqrt{1 - y}}}\right)\]

   9. Applied cbrt-div_binary64_8294 18.4

      \[\leadsto 1 - \log \left(1 - \frac{\frac{x - y}{\color{blue}{\frac{\sqrt[3]{1 \cdot 1 - y \cdot y}}{\sqrt[3]{1 + y}}} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{\sqrt{1 - y}} \cdot \sqrt[3]{\sqrt{1 - y}}}\right)\]

   10. Applied associate-*l/_binary64_8205 18.4

       \[\leadsto 1 - \log \left(1 - \frac{\frac{x - y}{\color{blue}{\frac{\sqrt[3]{1 \cdot 1 - y \cdot y} \cdot \sqrt[3]{1 - y}}{\sqrt[3]{1 + y}}}}}{\sqrt[3]{\sqrt{1 - y}} \cdot \sqrt[3]{\sqrt{1 - y}}}\right)\]

   11. Applied associate-/r/_binary64_8208 18.5

       \[\leadsto 1 - \log \left(1 - \frac{\color{blue}{\frac{x - y}{\sqrt[3]{1 \cdot 1 - y \cdot y} \cdot \sqrt[3]{1 - y}} \cdot \sqrt[3]{1 + y}}}{\sqrt[3]{\sqrt{1 - y}} \cdot \sqrt[3]{\sqrt{1 - y}}}\right)\]

   12. Applied times-frac_binary64_8268 18.4

       \[\leadsto 1 - \log \left(1 - \color{blue}{\frac{\frac{x - y}{\sqrt[3]{1 \cdot 1 - y \cdot y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{\sqrt{1 - y}}} \cdot \frac{\sqrt[3]{1 + y}}{\sqrt[3]{\sqrt{1 - y}}}}\right)\]

   13. Applied cancel-sign-sub-inv_binary64_8228 18.4

       \[\leadsto 1 - \log \color{blue}{\left(1 + \left(-\frac{\frac{x - y}{\sqrt[3]{1 \cdot 1 - y \cdot y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{\sqrt{1 - y}}}\right) \cdot \frac{\sqrt[3]{1 + y}}{\sqrt[3]{\sqrt{1 - y}}}\right)}\]

   14. Simplified 18.3

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

4. Final simplification 11.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.305417564235616 \cdot 10^{-122} \lor \neg \left(x \leq 0.006781316404929303\right):\\ \;\;\;\;1 - \log \left(\left(1 + \frac{y}{1 - y}\right) - \frac{x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{\sqrt{1 - y}}}}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y \cdot y}} \cdot \frac{\sqrt[3]{1 + y}}{\sqrt[3]{\sqrt{1 - y}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020344 
(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))))))