Average Error: 0.4 → 0.4
Time: 18.7s
Precision: 64
\[\left(3.0 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1.0}{x \cdot 9.0}\right) - 1.0\right)\]
↓
\[\sqrt{x} \cdot \left(3.0 \cdot \left(\left(y - 1.0\right) + \frac{0.1111111111111111}{x}\right)\right) + \left(3.0 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(1.0, -1, 1.0\right)\]
\left(3.0 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1.0}{x \cdot 9.0}\right) - 1.0\right)
↓
\sqrt{x} \cdot \left(3.0 \cdot \left(\left(y - 1.0\right) + \frac{0.1111111111111111}{x}\right)\right) + \left(3.0 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(1.0, -1, 1.0\right)
double f(double x, double y) {
        double r16950312 = 3.0;
        double r16950313 = x;
        double r16950314 = sqrt(r16950313);
        double r16950315 = r16950312 * r16950314;
        double r16950316 = y;
        double r16950317 = 1.0;
        double r16950318 = 9.0;
        double r16950319 = r16950313 * r16950318;
        double r16950320 = r16950317 / r16950319;
        double r16950321 = r16950316 + r16950320;
        double r16950322 = r16950321 - r16950317;
        double r16950323 = r16950315 * r16950322;
        return r16950323;
}
↓
double f(double x, double y) {
        double r16950324 = x;
        double r16950325 = sqrt(r16950324);
        double r16950326 = 3.0;
        double r16950327 = y;
        double r16950328 = 1.0;
        double r16950329 = r16950327 - r16950328;
        double r16950330 = 0.1111111111111111;
        double r16950331 = r16950330 / r16950324;
        double r16950332 = r16950329 + r16950331;
        double r16950333 = r16950326 * r16950332;
        double r16950334 = r16950325 * r16950333;
        double r16950335 = r16950326 * r16950325;
        double r16950336 = -1.0;
        double r16950337 = fma(r16950328, r16950336, r16950328);
        double r16950338 = r16950335 * r16950337;
        double r16950339 = r16950334 + r16950338;
        return r16950339;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.4
Target0.4
Herbie0.4
\[3.0 \cdot \left(y \cdot \sqrt{x} + \left(\frac{1.0}{x \cdot 9.0} - 1.0\right) \cdot \sqrt{x}\right)\]

Derivation

  1. Initial program 0.4

    \[\left(3.0 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1.0}{x \cdot 9.0}\right) - 1.0\right)\]

  2. Taylor expanded around 0 0.4

    \[\leadsto \left(3.0 \cdot \sqrt{x}\right) \cdot \left(\left(y + \color{blue}{\frac{0.1111111111111111}{x}}\right) - 1.0\right)\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.4

    \[\leadsto \left(3.0 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{0.1111111111111111}{x}\right) - \color{blue}{\left(\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}\right) \cdot \sqrt[3]{1.0}}\right)\]

  5. Applied add-sqr-sqrt15.5

    \[\leadsto \left(3.0 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\sqrt{y + \frac{0.1111111111111111}{x}} \cdot \sqrt{y + \frac{0.1111111111111111}{x}}} - \left(\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}\right) \cdot \sqrt[3]{1.0}\right)\]

  6. Applied prod-diff15.5

    \[\leadsto \left(3.0 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{y + \frac{0.1111111111111111}{x}}, \sqrt{y + \frac{0.1111111111111111}{x}}, -\sqrt[3]{1.0} \cdot \left(\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{1.0}, \sqrt[3]{1.0} \cdot \sqrt[3]{1.0}, \sqrt[3]{1.0} \cdot \left(\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}\right)\right)\right)}\]

  7. Applied distribute-lft-in15.5

    \[\leadsto \color{blue}{\left(3.0 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\sqrt{y + \frac{0.1111111111111111}{x}}, \sqrt{y + \frac{0.1111111111111111}{x}}, -\sqrt[3]{1.0} \cdot \left(\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}\right)\right) + \left(3.0 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(-\sqrt[3]{1.0}, \sqrt[3]{1.0} \cdot \sqrt[3]{1.0}, \sqrt[3]{1.0} \cdot \left(\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}\right)\right)}\]

  8. Simplified0.4

    \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3.0 \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1.0\right)\right)\right)} + \left(3.0 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(-\sqrt[3]{1.0}, \sqrt[3]{1.0} \cdot \sqrt[3]{1.0}, \sqrt[3]{1.0} \cdot \left(\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}\right)\right)\]

  9. Simplified0.4

    \[\leadsto \sqrt{x} \cdot \left(3.0 \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1.0\right)\right)\right) + \color{blue}{\mathsf{fma}\left(1.0, -1, 1.0\right) \cdot \left(3.0 \cdot \sqrt{x}\right)}\]

  10. Final simplification0.4

    \[\leadsto \sqrt{x} \cdot \left(3.0 \cdot \left(\left(y - 1.0\right) + \frac{0.1111111111111111}{x}\right)\right) + \left(3.0 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(1.0, -1, 1.0\right)\]

Reproduce

herbie shell --seed 2019158 +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)))