Average Error: 0.2 → 0.1
Time: 18.3s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{6}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{x - 1}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{6}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{x - 1}}
double f(double x) {
        double r519478 = 6.0;
        double r519479 = x;
        double r519480 = 1.0;
        double r519481 = r519479 - r519480;
        double r519482 = r519478 * r519481;
        double r519483 = r519479 + r519480;
        double r519484 = 4.0;
        double r519485 = sqrt(r519479);
        double r519486 = r519484 * r519485;
        double r519487 = r519483 + r519486;
        double r519488 = r519482 / r519487;
        return r519488;
}
↓
double f(double x) {
        double r519489 = 6.0;
        double r519490 = x;
        double r519491 = sqrt(r519490);
        double r519492 = 4.0;
        double r519493 = 1.0;
        double r519494 = r519490 + r519493;
        double r519495 = fma(r519491, r519492, r519494);
        double r519496 = r519490 - r519493;
        double r519497 = r519495 / r519496;
        double r519498 = r519489 / r519497;
        return r519498;
}

Error

Bits error versus x

Target

Original0.2
Target0.1
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. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019199 +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)))))