Average Error: 0.2 → 0.0
Time: 15.7s
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 r868602 = 6.0;
        double r868603 = x;
        double r868604 = 1.0;
        double r868605 = r868603 - r868604;
        double r868606 = r868602 * r868605;
        double r868607 = r868603 + r868604;
        double r868608 = 4.0;
        double r868609 = sqrt(r868603);
        double r868610 = r868608 * r868609;
        double r868611 = r868607 + r868610;
        double r868612 = r868606 / r868611;
        return r868612;
}
↓
double f(double x) {
        double r868613 = 6.0;
        double r868614 = x;
        double r868615 = sqrt(r868614);
        double r868616 = 4.0;
        double r868617 = 1.0;
        double r868618 = r868614 + r868617;
        double r868619 = fma(r868615, r868616, r868618);
        double r868620 = r868614 - r868617;
        double r868621 = r868619 / r868620;
        double r868622 = r868613 / r868621;
        return r868622;
}

Error

Bits error versus x

Target

Original0.2
Target0.0
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

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

  3. Final simplification0.0

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

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)))))