Average Error: 0.4 → 0.4
Time: 14.7s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\sqrt{x} \cdot \left(\left(\left(\frac{1}{x \cdot 9} + y\right) - 1\right) \cdot 3\right) + \left(\sqrt{x} \cdot \left(1 \cdot 0\right)\right) \cdot 3\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\sqrt{x} \cdot \left(\left(\left(\frac{1}{x \cdot 9} + y\right) - 1\right) \cdot 3\right) + \left(\sqrt{x} \cdot \left(1 \cdot 0\right)\right) \cdot 3
double f(double x, double y) {
        double r382264 = 3.0;
        double r382265 = x;
        double r382266 = sqrt(r382265);
        double r382267 = r382264 * r382266;
        double r382268 = y;
        double r382269 = 1.0;
        double r382270 = 9.0;
        double r382271 = r382265 * r382270;
        double r382272 = r382269 / r382271;
        double r382273 = r382268 + r382272;
        double r382274 = r382273 - r382269;
        double r382275 = r382267 * r382274;
        return r382275;
}
↓
double f(double x, double y) {
        double r382276 = x;
        double r382277 = sqrt(r382276);
        double r382278 = 1.0;
        double r382279 = 9.0;
        double r382280 = r382276 * r382279;
        double r382281 = r382278 / r382280;
        double r382282 = y;
        double r382283 = r382281 + r382282;
        double r382284 = r382283 - r382278;
        double r382285 = 3.0;
        double r382286 = r382284 * r382285;
        double r382287 = r382277 * r382286;
        double r382288 = 0.0;
        double r382289 = r382278 * r382288;
        double r382290 = r382277 * r382289;
        double r382291 = r382290 * r382285;
        double r382292 = r382287 + r382291;
        return r382292;
}

Error

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Results

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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-sqrt16.0

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

  5. Applied prod-diff16.0

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

  6. Applied distribute-lft-in16.0

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

  7. Simplified0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (* 3 (+ (* y (sqrt x)) (* (- (/ 1 (* x 9)) 1) (sqrt x))))

  (* (* 3 (sqrt x)) (- (+ y (/ 1 (* x 9))) 1)))