Average Error: 0.2 → 0.0
Time: 22.2s
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 r987200 = 6.0;
        double r987201 = x;
        double r987202 = 1.0;
        double r987203 = r987201 - r987202;
        double r987204 = r987200 * r987203;
        double r987205 = r987201 + r987202;
        double r987206 = 4.0;
        double r987207 = sqrt(r987201);
        double r987208 = r987206 * r987207;
        double r987209 = r987205 + r987208;
        double r987210 = r987204 / r987209;
        return r987210;
}
↓
double f(double x) {
        double r987211 = x;
        double r987212 = 1.0;
        double r987213 = r987211 - r987212;
        double r987214 = sqrt(r987211);
        double r987215 = 4.0;
        double r987216 = r987211 + r987212;
        double r987217 = fma(r987214, r987215, r987216);
        double r987218 = 6.0;
        double r987219 = r987217 / r987218;
        double r987220 = r987213 / r987219;
        return r987220;
}

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 2020046 +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)))))