Average Error: 0.2 → 0.0
Time: 13.0s
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 r822510 = 6.0;
        double r822511 = x;
        double r822512 = 1.0;
        double r822513 = r822511 - r822512;
        double r822514 = r822510 * r822513;
        double r822515 = r822511 + r822512;
        double r822516 = 4.0;
        double r822517 = sqrt(r822511);
        double r822518 = r822516 * r822517;
        double r822519 = r822515 + r822518;
        double r822520 = r822514 / r822519;
        return r822520;
}
↓
double f(double x) {
        double r822521 = 6.0;
        double r822522 = x;
        double r822523 = 1.0;
        double r822524 = r822522 - r822523;
        double r822525 = sqrt(r822522);
        double r822526 = 4.0;
        double r822527 = r822522 + r822523;
        double r822528 = fma(r822525, r822526, r822527);
        double r822529 = r822524 / r822528;
        double r822530 = r822521 * r822529;
        return r822530;
}

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. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"

  :herbie-target
  (/ 6.0 (/ (+ (+ x 1.0) (* 4.0 (sqrt x))) (- x 1.0)))

  (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))