Average Error: 0.2 → 0.0
Time: 14.9s
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 r872230 = 6.0;
        double r872231 = x;
        double r872232 = 1.0;
        double r872233 = r872231 - r872232;
        double r872234 = r872230 * r872233;
        double r872235 = r872231 + r872232;
        double r872236 = 4.0;
        double r872237 = sqrt(r872231);
        double r872238 = r872236 * r872237;
        double r872239 = r872235 + r872238;
        double r872240 = r872234 / r872239;
        return r872240;
}
↓
double f(double x) {
        double r872241 = 6.0;
        double r872242 = x;
        double r872243 = sqrt(r872242);
        double r872244 = 4.0;
        double r872245 = 1.0;
        double r872246 = r872242 + r872245;
        double r872247 = fma(r872243, r872244, r872246);
        double r872248 = r872242 - r872245;
        double r872249 = r872247 / r872248;
        double r872250 = r872241 / r872249;
        return r872250;
}

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)))))