Average Error: 0.4 → 0.4
Time: 3.3s
Precision: 64
\[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\]
↓
\[\left(\mathsf{fma}\left(0.1111111111111111, \frac{1}{x}, y\right) - 1\right) \cdot \left(3 \cdot \sqrt{x}\right)\]
\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)
↓
\left(\mathsf{fma}\left(0.1111111111111111, \frac{1}{x}, y\right) - 1\right) \cdot \left(3 \cdot \sqrt{x}\right)
double f(double x, double y) {
        double r428400 = 3.0;
        double r428401 = x;
        double r428402 = sqrt(r428401);
        double r428403 = r428400 * r428402;
        double r428404 = y;
        double r428405 = 1.0;
        double r428406 = 9.0;
        double r428407 = r428401 * r428406;
        double r428408 = r428405 / r428407;
        double r428409 = r428404 + r428408;
        double r428410 = r428409 - r428405;
        double r428411 = r428403 * r428410;
        return r428411;
}
↓
double f(double x, double y) {
        double r428412 = 0.1111111111111111;
        double r428413 = 1.0;
        double r428414 = x;
        double r428415 = r428413 / r428414;
        double r428416 = y;
        double r428417 = fma(r428412, r428415, r428416);
        double r428418 = 1.0;
        double r428419 = r428417 - r428418;
        double r428420 = 3.0;
        double r428421 = sqrt(r428414);
        double r428422 = r428420 * r428421;
        double r428423 = r428419 * r428422;
        return r428423;
}

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. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  5. Applied associate-*l*0.4

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

  7. Applied *-un-lft-identity0.4

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

  8. Applied associate-*l*0.4

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2020025 +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)))