Average Error: 0.4 → 0.5
Time: 4.6s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\mathsf{fma}\left(0.1111111111111111, \frac{1}{x}, y\right) - 1\right)\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\left(3 \cdot \sqrt{x}\right) \cdot \left(\mathsf{fma}\left(0.1111111111111111, \frac{1}{x}, y\right) - 1\right)
double f(double x, double y) {
        double r409550 = 3.0;
        double r409551 = x;
        double r409552 = sqrt(r409551);
        double r409553 = r409550 * r409552;
        double r409554 = y;
        double r409555 = 1.0;
        double r409556 = 9.0;
        double r409557 = r409551 * r409556;
        double r409558 = r409555 / r409557;
        double r409559 = r409554 + r409558;
        double r409560 = r409559 - r409555;
        double r409561 = r409553 * r409560;
        return r409561;
}
↓
double f(double x, double y) {
        double r409562 = 3.0;
        double r409563 = x;
        double r409564 = sqrt(r409563);
        double r409565 = r409562 * r409564;
        double r409566 = 0.1111111111111111;
        double r409567 = 1.0;
        double r409568 = r409567 / r409563;
        double r409569 = y;
        double r409570 = fma(r409566, r409568, r409569);
        double r409571 = 1.0;
        double r409572 = r409570 - r409571;
        double r409573 = r409565 * r409572;
        return r409573;
}

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-/r*0.4

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

  4. Taylor expanded around 0 0.5

    \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\left(y + 0.1111111111111111 \cdot \frac{1}{x}\right)} - 1\right)\]

  5. Simplified0.5

    \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\mathsf{fma}\left(0.1111111111111111, \frac{1}{x}, y\right)} - 1\right)\]

  6. Final simplification0.5

    \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\mathsf{fma}\left(0.1111111111111111, \frac{1}{x}, y\right) - 1\right)\]

Reproduce

herbie shell --seed 2020089 +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)))