Average Error: 0.4 → 0.4
Time: 37.7s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(\mathsf{fma}\left(y, 3, \frac{0.3333333333333333148296162562473909929395}{x}\right) - 3\right) \cdot \sqrt{x}\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\left(\mathsf{fma}\left(y, 3, \frac{0.3333333333333333148296162562473909929395}{x}\right) - 3\right) \cdot \sqrt{x}
double f(double x, double y) {
        double r255981 = 3.0;
        double r255982 = x;
        double r255983 = sqrt(r255982);
        double r255984 = r255981 * r255983;
        double r255985 = y;
        double r255986 = 1.0;
        double r255987 = 9.0;
        double r255988 = r255982 * r255987;
        double r255989 = r255986 / r255988;
        double r255990 = r255985 + r255989;
        double r255991 = r255990 - r255986;
        double r255992 = r255984 * r255991;
        return r255992;
}
↓
double f(double x, double y) {
        double r255993 = y;
        double r255994 = 3.0;
        double r255995 = 0.3333333333333333;
        double r255996 = x;
        double r255997 = r255995 / r255996;
        double r255998 = fma(r255993, r255994, r255997);
        double r255999 = r255998 - r255994;
        double r256000 = sqrt(r255996);
        double r256001 = r255999 * r256000;
        return r256001;
}

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 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. Simplified0.4

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

  6. Applied associate-*r*0.4

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

  7. Taylor expanded around 0 0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019306 +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)))