Average Error: 0.2 → 0.1
Time: 10.8s
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 r1091932 = 6.0;
        double r1091933 = x;
        double r1091934 = 1.0;
        double r1091935 = r1091933 - r1091934;
        double r1091936 = r1091932 * r1091935;
        double r1091937 = r1091933 + r1091934;
        double r1091938 = 4.0;
        double r1091939 = sqrt(r1091933);
        double r1091940 = r1091938 * r1091939;
        double r1091941 = r1091937 + r1091940;
        double r1091942 = r1091936 / r1091941;
        return r1091942;
}
↓
double f(double x) {
        double r1091943 = 6.0;
        double r1091944 = x;
        double r1091945 = sqrt(r1091944);
        double r1091946 = 4.0;
        double r1091947 = 1.0;
        double r1091948 = r1091944 + r1091947;
        double r1091949 = fma(r1091945, r1091946, r1091948);
        double r1091950 = r1091944 - r1091947;
        double r1091951 = r1091949 / r1091950;
        double r1091952 = r1091943 / r1091951;
        return r1091952;
}

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 2020045 +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)))))