Average Error: 0.4 → 0.5
Time: 5.6s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\mathsf{fma}\left(y, 3, 0.3333333333333333148296162562473909929395 \cdot \frac{1}{x} - 3\right) \cdot \sqrt{x} + \left(\sqrt{x} \cdot \mathsf{fma}\left(-1, 1, 1\right)\right) \cdot 3\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\mathsf{fma}\left(y, 3, 0.3333333333333333148296162562473909929395 \cdot \frac{1}{x} - 3\right) \cdot \sqrt{x} + \left(\sqrt{x} \cdot \mathsf{fma}\left(-1, 1, 1\right)\right) \cdot 3
double f(double x, double y) {
        double r362239 = 3.0;
        double r362240 = x;
        double r362241 = sqrt(r362240);
        double r362242 = r362239 * r362241;
        double r362243 = y;
        double r362244 = 1.0;
        double r362245 = 9.0;
        double r362246 = r362240 * r362245;
        double r362247 = r362244 / r362246;
        double r362248 = r362243 + r362247;
        double r362249 = r362248 - r362244;
        double r362250 = r362242 * r362249;
        return r362250;
}
↓
double f(double x, double y) {
        double r362251 = y;
        double r362252 = 3.0;
        double r362253 = 0.3333333333333333;
        double r362254 = 1.0;
        double r362255 = x;
        double r362256 = r362254 / r362255;
        double r362257 = r362253 * r362256;
        double r362258 = r362257 - r362252;
        double r362259 = fma(r362251, r362252, r362258);
        double r362260 = sqrt(r362255);
        double r362261 = r362259 * r362260;
        double r362262 = 1.0;
        double r362263 = -r362262;
        double r362264 = fma(r362263, r362254, r362262);
        double r362265 = r362260 * r362264;
        double r362266 = r362265 * r362252;
        double r362267 = r362261 + r362266;
        return r362267;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.4
Target0.4
Herbie0.5
\[3 \cdot \left(y \cdot \sqrt{x} + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \sqrt{x}\right)\]

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 associate-*l*0.4

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

  5. Applied add-cube-cbrt0.4

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

  6. Applied add-sqr-sqrt15.8

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

  7. Applied prod-diff15.8

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

  8. Applied distribute-lft-in15.8

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

  9. Applied distribute-lft-in15.8

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

  10. Simplified0.4

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

  11. Simplified0.4

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

  13. Applied associate-*r*0.5

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

  14. Taylor expanded around 0 0.5

    \[\leadsto \color{blue}{\left(\left(3 \cdot y + 0.3333333333333333148296162562473909929395 \cdot \frac{1}{x}\right) - 3\right)} \cdot \sqrt{x} + \left(\sqrt{x} \cdot \mathsf{fma}\left(-1, 1, 1\right)\right) \cdot 3\]

  15. Simplified0.5

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3, 0.3333333333333333148296162562473909929395 \cdot \frac{1}{x} - 3\right)} \cdot \sqrt{x} + \left(\sqrt{x} \cdot \mathsf{fma}\left(-1, 1, 1\right)\right) \cdot 3\]

  16. Final simplification0.5

    \[\leadsto \mathsf{fma}\left(y, 3, 0.3333333333333333148296162562473909929395 \cdot \frac{1}{x} - 3\right) \cdot \sqrt{x} + \left(\sqrt{x} \cdot \mathsf{fma}\left(-1, 1, 1\right)\right) \cdot 3\]

Reproduce

herbie shell --seed 2019356 +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)))