Average Error: 0.2 → 0.0
Time: 12.7s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{6}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, 1\right) + x}{x - 1}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{6}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, 1\right) + x}{x - 1}}
double f(double x) {
        double r537270 = 6.0;
        double r537271 = x;
        double r537272 = 1.0;
        double r537273 = r537271 - r537272;
        double r537274 = r537270 * r537273;
        double r537275 = r537271 + r537272;
        double r537276 = 4.0;
        double r537277 = sqrt(r537271);
        double r537278 = r537276 * r537277;
        double r537279 = r537275 + r537278;
        double r537280 = r537274 / r537279;
        return r537280;
}
↓
double f(double x) {
        double r537281 = 6.0;
        double r537282 = x;
        double r537283 = sqrt(r537282);
        double r537284 = 4.0;
        double r537285 = 1.0;
        double r537286 = fma(r537283, r537284, r537285);
        double r537287 = r537286 + r537282;
        double r537288 = r537282 - r537285;
        double r537289 = r537287 / r537288;
        double r537290 = r537281 / r537289;
        return r537290;
}

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. Using strategy rm

  4. Applied *-un-lft-identity0.0

    \[\leadsto \frac{6}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{\color{blue}{1 \cdot \left(x - 1\right)}}}\]

  5. Applied associate-/r*0.0

    \[\leadsto \frac{6}{\color{blue}{\frac{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{1}}{x - 1}}}\]

  6. Simplified0.0

    \[\leadsto \frac{6}{\frac{\color{blue}{\mathsf{fma}\left(\sqrt{x}, 4, 1\right) + x}}{x - 1}}\]

  7. Final simplification0.0

    \[\leadsto \frac{6}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, 1\right) + x}{x - 1}}\]

Reproduce

herbie shell --seed 2019326 +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)))))