Average Error: 0.2 → 0.1
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 r655040 = 6.0;
        double r655041 = x;
        double r655042 = 1.0;
        double r655043 = r655041 - r655042;
        double r655044 = r655040 * r655043;
        double r655045 = r655041 + r655042;
        double r655046 = 4.0;
        double r655047 = sqrt(r655041);
        double r655048 = r655046 * r655047;
        double r655049 = r655045 + r655048;
        double r655050 = r655044 / r655049;
        return r655050;
}
↓
double f(double x) {
        double r655051 = 6.0;
        double r655052 = x;
        double r655053 = sqrt(r655052);
        double r655054 = 4.0;
        double r655055 = 1.0;
        double r655056 = r655052 + r655055;
        double r655057 = fma(r655053, r655054, r655056);
        double r655058 = r655052 - r655055;
        double r655059 = r655057 / r655058;
        double r655060 = r655051 / r655059;
        return r655060;
}

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 2019198 +o rules:numerics
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"

  :herbie-target
  (/ 6.0 (/ (+ (+ x 1.0) (* 4.0 (sqrt x))) (- x 1.0)))

  (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))