Average Error: 0.2 → 0.0
Time: 12.7s
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 r875981 = 6.0;
        double r875982 = x;
        double r875983 = 1.0;
        double r875984 = r875982 - r875983;
        double r875985 = r875981 * r875984;
        double r875986 = r875982 + r875983;
        double r875987 = 4.0;
        double r875988 = sqrt(r875982);
        double r875989 = r875987 * r875988;
        double r875990 = r875986 + r875989;
        double r875991 = r875985 / r875990;
        return r875991;
}
↓
double f(double x) {
        double r875992 = 6.0;
        double r875993 = x;
        double r875994 = sqrt(r875993);
        double r875995 = 4.0;
        double r875996 = 1.0;
        double r875997 = r875993 + r875996;
        double r875998 = fma(r875994, r875995, r875997);
        double r875999 = r875993 - r875996;
        double r876000 = r875998 / r875999;
        double r876001 = r875992 / r876000;
        return r876001;
}

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)))))