Average Error: 0.2 → 0.1
Time: 12.5s
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 r969328 = 6.0;
        double r969329 = x;
        double r969330 = 1.0;
        double r969331 = r969329 - r969330;
        double r969332 = r969328 * r969331;
        double r969333 = r969329 + r969330;
        double r969334 = 4.0;
        double r969335 = sqrt(r969329);
        double r969336 = r969334 * r969335;
        double r969337 = r969333 + r969336;
        double r969338 = r969332 / r969337;
        return r969338;
}
↓
double f(double x) {
        double r969339 = 6.0;
        double r969340 = x;
        double r969341 = sqrt(r969340);
        double r969342 = 4.0;
        double r969343 = 1.0;
        double r969344 = r969340 + r969343;
        double r969345 = fma(r969341, r969342, r969344);
        double r969346 = r969340 - r969343;
        double r969347 = r969345 / r969346;
        double r969348 = r969339 / r969347;
        return r969348;
}

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)))))