Average Error: 0.2 → 0.0
Time: 15.2s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[6 \cdot \frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
6 \cdot \frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}
double f(double x) {
        double r646631 = 6.0;
        double r646632 = x;
        double r646633 = 1.0;
        double r646634 = r646632 - r646633;
        double r646635 = r646631 * r646634;
        double r646636 = r646632 + r646633;
        double r646637 = 4.0;
        double r646638 = sqrt(r646632);
        double r646639 = r646637 * r646638;
        double r646640 = r646636 + r646639;
        double r646641 = r646635 / r646640;
        return r646641;
}
↓
double f(double x) {
        double r646642 = 6.0;
        double r646643 = x;
        double r646644 = 1.0;
        double r646645 = r646643 - r646644;
        double r646646 = sqrt(r646643);
        double r646647 = 4.0;
        double r646648 = r646643 + r646644;
        double r646649 = fma(r646646, r646647, r646648);
        double r646650 = r646645 / r646649;
        double r646651 = r646642 * r646650;
        return r646651;
}

Error

Bits error versus x

Target

Original0.2
Target0.1
Herbie0.0
\[\frac{6}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}}\]

Derivation

  1. Initial program 0.2

    \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\frac{6}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{x - 1}}}\]
  3. Using strategy rm

  4. Applied div-inv0.1

    \[\leadsto \color{blue}{6 \cdot \frac{1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{x - 1}}}\]

  5. Simplified0.0

    \[\leadsto 6 \cdot \color{blue}{\frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}\]

  6. Final simplification0.0

    \[\leadsto 6 \cdot \frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"
  :precision binary64

  :herbie-target
  (/ 6 (/ (+ (+ x 1) (* 4 (sqrt x))) (- x 1)))

  (/ (* 6 (- x 1)) (+ (+ x 1) (* 4 (sqrt x)))))