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

Average Error: 0.2 → 0.3

Time: 5.1s

Precision: 64

Math

\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

↓

\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\frac{y}{\sqrt[3]{3}}}{\sqrt{x}}\]

C

double code(double x, double y) {
	return ((1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * sqrt(x))));
}

↓

double code(double x, double y) {
	return ((1.0 - (1.0 / (x * 9.0))) - ((1.0 / (cbrt(3.0) * cbrt(3.0))) * ((y / cbrt(3.0)) / sqrt(x))));
}

Error

Bits error versus x

Bits error versus y

Target

Original	0.2
Target	0.2
Herbie	0.3

\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

Derivation

1. Initial program 0.2

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
2. Using strategy rm

3. Applied associate-/r* 0.2

   \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\frac{\frac{y}{3}}{\sqrt{x}}}\]
4. Using strategy rm

5. Applied *-un-lft-identity 0.2

   \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{\frac{y}{3}}{\sqrt{\color{blue}{1 \cdot x}}}\]

6. Applied sqrt-prod 0.2

   \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{\frac{y}{3}}{\color{blue}{\sqrt{1} \cdot \sqrt{x}}}\]

7. Applied add-cube-cbrt 0.2

   \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{\frac{y}{\color{blue}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}}}{\sqrt{1} \cdot \sqrt{x}}\]

8. Applied *-un-lft-identity 0.2

   \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{\frac{\color{blue}{1 \cdot y}}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}}{\sqrt{1} \cdot \sqrt{x}}\]

9. Applied times-frac 0.2

   \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{\color{blue}{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{y}{\sqrt[3]{3}}}}{\sqrt{1} \cdot \sqrt{x}}\]

10. Applied times-frac 0.3

    \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\frac{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}}{\sqrt{1}} \cdot \frac{\frac{y}{\sqrt[3]{3}}}{\sqrt{x}}}\]

11. Simplified 0.3

    \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}} \cdot \frac{\frac{y}{\sqrt[3]{3}}}{\sqrt{x}}\]

12. Final simplification 0.3

    \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\frac{y}{\sqrt[3]{3}}}{\sqrt{x}}\]

Reproduce

herbie shell --seed 2020091 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (- (- 1 (/ (/ 1 x) 9)) (/ y (* 3 (sqrt x))))

  (- (- 1 (/ 1 (* x 9))) (/ y (* 3 (sqrt x)))))