Average Error: 0.2 → 0.2
Time: 1.0m
Precision: 64
\[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}}\]
↓
\[\mathsf{fma}\left(\frac{\frac{y}{3}}{\sqrt{x}}, -1, \frac{\frac{y}{3}}{\sqrt{x}}\right) + \left(1 - \left(\frac{0.1111111111111111049432054187491303309798}{x} + \frac{\frac{y}{3}}{\sqrt{x}}\right)\right)\]
\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(1 - \left(\frac{0.1111111111111111049432054187491303309798}{x} + \frac{\frac{y}{3}}{\sqrt{x}}\right)\right)
double f(double x, double y) {
        double r14496531 = 1.0;
        double r14496532 = x;
        double r14496533 = 9.0;
        double r14496534 = r14496532 * r14496533;
        double r14496535 = r14496531 / r14496534;
        double r14496536 = r14496531 - r14496535;
        double r14496537 = y;
        double r14496538 = 3.0;
        double r14496539 = sqrt(r14496532);
        double r14496540 = r14496538 * r14496539;
        double r14496541 = r14496537 / r14496540;
        double r14496542 = r14496536 - r14496541;
        return r14496542;
}
↓
double f(double x, double y) {
        double r14496543 = y;
        double r14496544 = 3.0;
        double r14496545 = r14496543 / r14496544;
        double r14496546 = x;
        double r14496547 = sqrt(r14496546);
        double r14496548 = r14496545 / r14496547;
        double r14496549 = -1.0;
        double r14496550 = fma(r14496548, r14496549, r14496548);
        double r14496551 = 1.0;
        double r14496552 = 0.1111111111111111;
        double r14496553 = r14496552 / r14496546;
        double r14496554 = r14496553 + r14496548;
        double r14496555 = r14496551 - r14496554;
        double r14496556 = r14496550 + r14496555;
        return r14496556;
}

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-sqrt29.9

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

  5. Applied prod-diff29.9

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

  6. Simplified0.2

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

  7. Simplified0.2

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

  8. Taylor expanded around 0 0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019168 +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)))))