Average Error: 0.4 → 0.4
Time: 23.7s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{\frac{1}{x}}{9} - 1\right)\right)\right) + \sqrt{x} \cdot \left(\mathsf{fma}\left(1, -1, 1\right) \cdot 3\right)\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{\frac{1}{x}}{9} - 1\right)\right)\right) + \sqrt{x} \cdot \left(\mathsf{fma}\left(1, -1, 1\right) \cdot 3\right)
double f(double x, double y) {
        double r18169275 = 3.0;
        double r18169276 = x;
        double r18169277 = sqrt(r18169276);
        double r18169278 = r18169275 * r18169277;
        double r18169279 = y;
        double r18169280 = 1.0;
        double r18169281 = 9.0;
        double r18169282 = r18169276 * r18169281;
        double r18169283 = r18169280 / r18169282;
        double r18169284 = r18169279 + r18169283;
        double r18169285 = r18169284 - r18169280;
        double r18169286 = r18169278 * r18169285;
        return r18169286;
}
↓
double f(double x, double y) {
        double r18169287 = 3.0;
        double r18169288 = x;
        double r18169289 = sqrt(r18169288);
        double r18169290 = y;
        double r18169291 = 1.0;
        double r18169292 = r18169291 / r18169288;
        double r18169293 = 9.0;
        double r18169294 = r18169292 / r18169293;
        double r18169295 = r18169294 - r18169291;
        double r18169296 = r18169290 + r18169295;
        double r18169297 = r18169289 * r18169296;
        double r18169298 = r18169287 * r18169297;
        double r18169299 = -1.0;
        double r18169300 = fma(r18169291, r18169299, r18169291);
        double r18169301 = r18169300 * r18169287;
        double r18169302 = r18169289 * r18169301;
        double r18169303 = r18169298 + r18169302;
        return r18169303;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.4
Target0.4
Herbie0.4
\[3 \cdot \left(y \cdot \sqrt{x} + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \sqrt{x}\right)\]

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 add-cube-cbrt0.4

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

  4. Applied add-sqr-sqrt15.4

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

  5. Applied prod-diff15.4

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

  6. Applied distribute-lft-in15.4

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

  7. Simplified0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019192 +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)))