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

Average Error: 0.2 → 0.3

Time: 20.7s

Precision: 64

Math

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

↓

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

C

double f(double x, double y) {
        double r309674 = 1.0;
        double r309675 = x;
        double r309676 = 9.0;
        double r309677 = r309675 * r309676;
        double r309678 = r309674 / r309677;
        double r309679 = r309674 - r309678;
        double r309680 = y;
        double r309681 = 3.0;
        double r309682 = sqrt(r309675);
        double r309683 = r309681 * r309682;
        double r309684 = r309680 / r309683;
        double r309685 = r309679 - r309684;
        return r309685;
}

↓

double f(double x, double y) {
        double r309686 = 1.0;
        double r309687 = cbrt(r309686);
        double r309688 = r309687 * r309687;
        double r309689 = x;
        double r309690 = 9.0;
        double r309691 = sqrt(r309690);
        double r309692 = r309689 * r309691;
        double r309693 = r309686 / r309692;
        double r309694 = -r309693;
        double r309695 = r309694 / r309691;
        double r309696 = fma(r309688, r309687, r309695);
        double r309697 = 1.0;
        double r309698 = r309697 / r309691;
        double r309699 = r309686 / r309689;
        double r309700 = r309699 / r309691;
        double r309701 = -r309700;
        double r309702 = r309701 + r309700;
        double r309703 = r309698 * r309702;
        double r309704 = r309696 + r309703;
        double r309705 = y;
        double r309706 = 3.0;
        double r309707 = sqrt(r309689);
        double r309708 = r309706 * r309707;
        double r309709 = r309705 / r309708;
        double r309710 = r309704 - r309709;
        return r309710;
}

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. Simplified 0.2

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

4. Applied add-sqr-sqrt 0.2

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

5. Applied *-un-lft-identity 0.2

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

6. Applied *-un-lft-identity 0.2

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

7. Applied times-frac 0.2

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

8. Applied times-frac 0.3

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

9. Applied add-cube-cbrt 0.3

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

10. Applied prod-diff 0.3

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

11. Simplified 0.3

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

12. Simplified 0.3

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

14. Applied div-inv 0.3

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

15. Applied associate-/l* 0.3

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

16. Simplified 0.3

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

17. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(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)))))