Average Error: 0.3 → 0.0
Time: 5.3s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{x}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}} - \frac{1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{x}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}} - \frac{1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}}
double f(double x) {
        double r764090 = 6.0;
        double r764091 = x;
        double r764092 = 1.0;
        double r764093 = r764091 - r764092;
        double r764094 = r764090 * r764093;
        double r764095 = r764091 + r764092;
        double r764096 = 4.0;
        double r764097 = sqrt(r764091);
        double r764098 = r764096 * r764097;
        double r764099 = r764095 + r764098;
        double r764100 = r764094 / r764099;
        return r764100;
}
↓
double f(double x) {
        double r764101 = x;
        double r764102 = sqrt(r764101);
        double r764103 = 4.0;
        double r764104 = 1.0;
        double r764105 = r764101 + r764104;
        double r764106 = fma(r764102, r764103, r764105);
        double r764107 = 6.0;
        double r764108 = r764106 / r764107;
        double r764109 = r764101 / r764108;
        double r764110 = r764104 / r764108;
        double r764111 = r764109 - r764110;
        return r764111;
}

Error

Bits error versus x

Target

Original0.3
Target0.1
Herbie0.0
\[\frac{6}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}}\]

Derivation

  1. Initial program 0.3

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

  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-sub0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020002 +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)))))