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

Average Error: 0.2 → 0.2

Time: 12.1s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\left(\left(1 - \frac{\frac{1}{9}}{x}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\right) + \frac{\frac{y}{3}}{\sqrt{x}} \cdot 0\]

C

double f(double x, double y) {
        double r417032 = 1.0;
        double r417033 = x;
        double r417034 = 9.0;
        double r417035 = r417033 * r417034;
        double r417036 = r417032 / r417035;
        double r417037 = r417032 - r417036;
        double r417038 = y;
        double r417039 = 3.0;
        double r417040 = sqrt(r417033);
        double r417041 = r417039 * r417040;
        double r417042 = r417038 / r417041;
        double r417043 = r417037 - r417042;
        return r417043;
}

↓

double f(double x, double y) {
        double r417044 = 1.0;
        double r417045 = 9.0;
        double r417046 = r417044 / r417045;
        double r417047 = x;
        double r417048 = r417046 / r417047;
        double r417049 = r417044 - r417048;
        double r417050 = y;
        double r417051 = 3.0;
        double r417052 = r417050 / r417051;
        double r417053 = sqrt(r417047);
        double r417054 = r417052 / r417053;
        double r417055 = r417049 - r417054;
        double r417056 = 0.0;
        double r417057 = r417054 * r417056;
        double r417058 = r417055 + r417057;
        return r417058;
}

Error

Bits error versus x

Bits error versus y

Target

Original	0.2
Target	0.3
Herbie	0.2

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

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 add-cube-cbrt 0.5

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

4. Applied add-sqr-sqrt 30.1

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

5. Applied prod-diff 30.1

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

6. Simplified 0.2

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

7. Simplified 0.2

   \[\leadsto \left(\left(1 - \frac{\frac{1}{9}}{x}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\right) + \color{blue}{\frac{\frac{y}{3}}{\sqrt{x}} \cdot 0}\]

8. Final simplification 0.2

   \[\leadsto \left(\left(1 - \frac{\frac{1}{9}}{x}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\right) + \frac{\frac{y}{3}}{\sqrt{x}} \cdot 0\]

Reproduce

herbie shell --seed 2020047 +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)))))