Average Error: 0.2 → 0.1
Time: 5.4s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{\frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{\frac{1}{6}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{\frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{\frac{1}{6}}
double f(double x) {
        double r989534 = 6.0;
        double r989535 = x;
        double r989536 = 1.0;
        double r989537 = r989535 - r989536;
        double r989538 = r989534 * r989537;
        double r989539 = r989535 + r989536;
        double r989540 = 4.0;
        double r989541 = sqrt(r989535);
        double r989542 = r989540 * r989541;
        double r989543 = r989539 + r989542;
        double r989544 = r989538 / r989543;
        return r989544;
}
↓
double f(double x) {
        double r989545 = x;
        double r989546 = 1.0;
        double r989547 = r989545 - r989546;
        double r989548 = sqrt(r989545);
        double r989549 = 4.0;
        double r989550 = r989545 + r989546;
        double r989551 = fma(r989548, r989549, r989550);
        double r989552 = r989547 / r989551;
        double r989553 = 1.0;
        double r989554 = 6.0;
        double r989555 = r989553 / r989554;
        double r989556 = r989552 / r989555;
        return r989556;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 0.2

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

  2. Simplified0.0

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

  4. Applied div-inv0.2

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

  5. Applied associate-/r*0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 +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)))))