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

Average Error: 0.4 → 0.4

Time: 21.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double y) {
        double r17478075 = 3.0;
        double r17478076 = x;
        double r17478077 = sqrt(r17478076);
        double r17478078 = r17478075 * r17478077;
        double r17478079 = y;
        double r17478080 = 1.0;
        double r17478081 = 9.0;
        double r17478082 = r17478076 * r17478081;
        double r17478083 = r17478080 / r17478082;
        double r17478084 = r17478079 + r17478083;
        double r17478085 = r17478084 - r17478080;
        double r17478086 = r17478078 * r17478085;
        return r17478086;
}

↓

double f(double x, double y) {
        double r17478087 = 3.0;
        double r17478088 = 1.0;
        double r17478089 = 9.0;
        double r17478090 = r17478088 / r17478089;
        double r17478091 = x;
        double r17478092 = r17478090 / r17478091;
        double r17478093 = y;
        double r17478094 = r17478093 - r17478088;
        double r17478095 = r17478092 + r17478094;
        double r17478096 = r17478087 * r17478095;
        double r17478097 = sqrt(r17478091);
        double r17478098 = r17478096 * r17478097;
        double r17478099 = r17478087 * r17478097;
        double r17478100 = -1.0;
        double r17478101 = fma(r17478088, r17478100, r17478088);
        double r17478102 = r17478099 * r17478101;
        double r17478103 = r17478098 + r17478102;
        return r17478103;
}

Error

Bits error versus x

Bits error versus y

Target

Original	0.4
Target	0.4
Herbie	0.4

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

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 *-un-lft-identity 0.4

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

4. Applied times-frac 0.4

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

6. Applied add-cube-cbrt 0.4

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

7. Applied add-sqr-sqrt 15.9

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

8. Applied prod-diff 15.9

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

9. Applied distribute-lft-in 15.9

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

10. Simplified 0.4

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

11. Simplified 0.4

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

12. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019169 +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)))