Average Error: 0.2 → 0.2
Time: 14.5s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\mathsf{fma}\left(1, 1 - \frac{1}{x \cdot 9}, \frac{\frac{-y}{3}}{\sqrt{x}}\right)\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\mathsf{fma}\left(1, 1 - \frac{1}{x \cdot 9}, \frac{\frac{-y}{3}}{\sqrt{x}}\right)
double f(double x, double y) {
        double r224511 = 1.0;
        double r224512 = x;
        double r224513 = 9.0;
        double r224514 = r224512 * r224513;
        double r224515 = r224511 / r224514;
        double r224516 = r224511 - r224515;
        double r224517 = y;
        double r224518 = 3.0;
        double r224519 = sqrt(r224512);
        double r224520 = r224518 * r224519;
        double r224521 = r224517 / r224520;
        double r224522 = r224516 - r224521;
        return r224522;
}
↓
double f(double x, double y) {
        double r224523 = 1.0;
        double r224524 = 1.0;
        double r224525 = x;
        double r224526 = 9.0;
        double r224527 = r224525 * r224526;
        double r224528 = r224524 / r224527;
        double r224529 = r224524 - r224528;
        double r224530 = y;
        double r224531 = -r224530;
        double r224532 = 3.0;
        double r224533 = r224531 / r224532;
        double r224534 = sqrt(r224525);
        double r224535 = r224533 / r224534;
        double r224536 = fma(r224523, r224529, r224535);
        return r224536;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.2
Target0.2
Herbie0.2
\[\left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]

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 *-un-lft-identity0.2

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

  4. Applied fma-neg0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019347 +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)))))