Average Error: 0.2 → 0.2
Time: 33.8s
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\mathsf{fma}\left(\frac{\frac{y}{3}}{\sqrt{x}}, -1, \frac{\frac{y}{3}}{\sqrt{x}}\right) + \left(\left(1 - \frac{0.1111111111111111049432054187491303309798}{x}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\right)\]
\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}
↓
\mathsf{fma}\left(\frac{\frac{y}{3}}{\sqrt{x}}, -1, \frac{\frac{y}{3}}{\sqrt{x}}\right) + \left(\left(1 - \frac{0.1111111111111111049432054187491303309798}{x}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\right)
double f(double x, double y) {
        double r13910612 = 1.0;
        double r13910613 = x;
        double r13910614 = 9.0;
        double r13910615 = r13910613 * r13910614;
        double r13910616 = r13910612 / r13910615;
        double r13910617 = r13910612 - r13910616;
        double r13910618 = y;
        double r13910619 = 3.0;
        double r13910620 = sqrt(r13910613);
        double r13910621 = r13910619 * r13910620;
        double r13910622 = r13910618 / r13910621;
        double r13910623 = r13910617 - r13910622;
        return r13910623;
}
↓
double f(double x, double y) {
        double r13910624 = y;
        double r13910625 = 3.0;
        double r13910626 = r13910624 / r13910625;
        double r13910627 = x;
        double r13910628 = sqrt(r13910627);
        double r13910629 = r13910626 / r13910628;
        double r13910630 = -1.0;
        double r13910631 = fma(r13910629, r13910630, r13910629);
        double r13910632 = 1.0;
        double r13910633 = 0.1111111111111111;
        double r13910634 = r13910633 / r13910627;
        double r13910635 = r13910632 - r13910634;
        double r13910636 = r13910635 - r13910629;
        double r13910637 = r13910631 + r13910636;
        return r13910637;
}

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 add-cube-cbrt0.5

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

  4. Applied add-sqr-sqrt30.3

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

  5. Applied prod-diff30.3

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

  6. Simplified0.2

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

  7. Simplified0.2

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

  8. Taylor expanded around 0 0.2

    \[\leadsto \left(\left(1 - \color{blue}{\frac{0.1111111111111111049432054187491303309798}{x}}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\right) + \mathsf{fma}\left(\frac{\frac{y}{3}}{\sqrt{x}}, -1, \frac{\frac{y}{3}}{\sqrt{x}}\right)\]

  9. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(\frac{\frac{y}{3}}{\sqrt{x}}, -1, \frac{\frac{y}{3}}{\sqrt{x}}\right) + \left(\left(1 - \frac{0.1111111111111111049432054187491303309798}{x}\right) - \frac{\frac{y}{3}}{\sqrt{x}}\right)\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"

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

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