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

Average Error: 0.4 → 0.3

Time: 17.0s

Precision: 64

Math

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

↓

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

C

double f(double x, double y) {
        double r306560 = 3.0;
        double r306561 = x;
        double r306562 = sqrt(r306561);
        double r306563 = r306560 * r306562;
        double r306564 = y;
        double r306565 = 1.0;
        double r306566 = 9.0;
        double r306567 = r306561 * r306566;
        double r306568 = r306565 / r306567;
        double r306569 = r306564 + r306568;
        double r306570 = r306569 - r306565;
        double r306571 = r306563 * r306570;
        return r306571;
}

↓

double f(double x, double y) {
        double r306572 = 1.0;
        double r306573 = x;
        double r306574 = sqrt(r306573);
        double r306575 = r306572 * r306574;
        double r306576 = 9.0;
        double r306577 = sqrt(r306576);
        double r306578 = r306575 / r306577;
        double r306579 = r306578 / r306573;
        double r306580 = 3.0;
        double r306581 = r306580 / r306577;
        double r306582 = r306579 * r306581;
        double r306583 = r306580 * r306574;
        double r306584 = y;
        double r306585 = r306583 * r306584;
        double r306586 = -r306583;
        double r306587 = r306572 * r306586;
        double r306588 = r306585 + r306587;
        double r306589 = r306582 + r306588;
        return r306589;
}

Error

Bits error versus x

Bits error versus y

Target

Original	0.4
Target	0.4
Herbie	0.3

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

Derivation

1. Initial program 0.4

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

2. Simplified 0.4

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

4. Applied distribute-lft-in 0.4

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

5. Simplified 0.4

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

6. Simplified 0.4

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

8. Applied associate-*r/ 0.4

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

9. Simplified 0.4

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

11. Applied add-sqr-sqrt 0.4

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

12. Applied associate-/r/ 0.4

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

13. Applied times-frac 0.3

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

14. Simplified 0.3

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

16. Applied sub-neg 0.3

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

17. Applied distribute-lft-in 0.3

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

18. Applied distribute-lft-in 0.3

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

19. Simplified 0.3

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

20. Simplified 0.3

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

21. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019195 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"

  :herbie-target
  (* 3.0 (+ (* y (sqrt x)) (* (- (/ 1.0 (* x 9.0)) 1.0) (sqrt x))))

  (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))