Average Error: 0.4 → 0.4
Time: 40.8s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(-1\right) \cdot \left(3 \cdot \sqrt{x}\right) + \mathsf{fma}\left(3, y, \frac{0.3333333333333333148296162562473909929395}{x}\right) \cdot \sqrt{x}\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\left(-1\right) \cdot \left(3 \cdot \sqrt{x}\right) + \mathsf{fma}\left(3, y, \frac{0.3333333333333333148296162562473909929395}{x}\right) \cdot \sqrt{x}
double f(double x, double y) {
        double r22038532 = 3.0;
        double r22038533 = x;
        double r22038534 = sqrt(r22038533);
        double r22038535 = r22038532 * r22038534;
        double r22038536 = y;
        double r22038537 = 1.0;
        double r22038538 = 9.0;
        double r22038539 = r22038533 * r22038538;
        double r22038540 = r22038537 / r22038539;
        double r22038541 = r22038536 + r22038540;
        double r22038542 = r22038541 - r22038537;
        double r22038543 = r22038535 * r22038542;
        return r22038543;
}
↓
double f(double x, double y) {
        double r22038544 = 1.0;
        double r22038545 = -r22038544;
        double r22038546 = 3.0;
        double r22038547 = x;
        double r22038548 = sqrt(r22038547);
        double r22038549 = r22038546 * r22038548;
        double r22038550 = r22038545 * r22038549;
        double r22038551 = y;
        double r22038552 = 0.3333333333333333;
        double r22038553 = r22038552 / r22038547;
        double r22038554 = fma(r22038546, r22038551, r22038553);
        double r22038555 = r22038554 * r22038548;
        double r22038556 = r22038550 + r22038555;
        return r22038556;
}

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 sub-neg0.4

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

  4. Applied distribute-rgt-in0.4

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

  6. Applied associate-*r*0.4

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

  8. Applied associate-/r*0.4

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

  9. Taylor expanded around 0 0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019168 +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)))