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

Average Error: 0.4 → 0.5

Time: 23.1s

Precision: 64

Math

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

↓

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

C

double f(double x, double y) {
        double r17051309 = 3.0;
        double r17051310 = x;
        double r17051311 = sqrt(r17051310);
        double r17051312 = r17051309 * r17051311;
        double r17051313 = y;
        double r17051314 = 1.0;
        double r17051315 = 9.0;
        double r17051316 = r17051310 * r17051315;
        double r17051317 = r17051314 / r17051316;
        double r17051318 = r17051313 + r17051317;
        double r17051319 = r17051318 - r17051314;
        double r17051320 = r17051312 * r17051319;
        return r17051320;
}

↓

double f(double x, double y) {
        double r17051321 = x;
        double r17051322 = sqrt(r17051321);
        double r17051323 = 1.0;
        double r17051324 = -1.0;
        double r17051325 = fma(r17051323, r17051324, r17051323);
        double r17051326 = 3.0;
        double r17051327 = r17051325 * r17051326;
        double r17051328 = r17051322 * r17051327;
        double r17051329 = sqrt(r17051323);
        double r17051330 = r17051329 / r17051322;
        double r17051331 = 9.0;
        double r17051332 = r17051330 / r17051331;
        double r17051333 = r17051330 * r17051332;
        double r17051334 = r17051333 - r17051323;
        double r17051335 = y;
        double r17051336 = r17051334 + r17051335;
        double r17051337 = r17051336 * r17051322;
        double r17051338 = r17051337 * r17051326;
        double r17051339 = r17051328 + r17051338;
        return r17051339;
}

Error

Bits error versus x

Bits error versus y

Target

Original	0.4
Target	0.4
Herbie	0.5

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

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. Using strategy rm

3. Applied add-cube-cbrt 0.4

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

4. Applied add-sqr-sqrt 15.4

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

5. Applied prod-diff 15.4

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

6. Applied distribute-lft-in 15.4

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

7. Simplified 0.4

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

8. Simplified 0.4

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

10. Applied *-un-lft-identity 0.4

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

11. Applied add-sqr-sqrt 0.4

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

12. Applied add-sqr-sqrt 0.4

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

13. Applied times-frac 0.5

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

14. Applied times-frac 0.5

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

15. Simplified 0.5

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

16. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(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)))