Average Error: 0.2 → 0.0
Time: 5.3s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{x - 1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{x - 1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}}
double f(double x) {
        double r901932 = 6.0;
        double r901933 = x;
        double r901934 = 1.0;
        double r901935 = r901933 - r901934;
        double r901936 = r901932 * r901935;
        double r901937 = r901933 + r901934;
        double r901938 = 4.0;
        double r901939 = sqrt(r901933);
        double r901940 = r901938 * r901939;
        double r901941 = r901937 + r901940;
        double r901942 = r901936 / r901941;
        return r901942;
}
↓
double f(double x) {
        double r901943 = x;
        double r901944 = 1.0;
        double r901945 = r901943 - r901944;
        double r901946 = sqrt(r901943);
        double r901947 = 4.0;
        double r901948 = r901943 + r901944;
        double r901949 = fma(r901946, r901947, r901948);
        double r901950 = 6.0;
        double r901951 = r901949 / r901950;
        double r901952 = r901945 / r901951;
        return r901952;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2020056 +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)))))