Average Error: 0.2 → 0.0
Time: 10.4s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[6 \cdot \frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
6 \cdot \frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}
double f(double x) {
        double r700238 = 6.0;
        double r700239 = x;
        double r700240 = 1.0;
        double r700241 = r700239 - r700240;
        double r700242 = r700238 * r700241;
        double r700243 = r700239 + r700240;
        double r700244 = 4.0;
        double r700245 = sqrt(r700239);
        double r700246 = r700244 * r700245;
        double r700247 = r700243 + r700246;
        double r700248 = r700242 / r700247;
        return r700248;
}
↓
double f(double x) {
        double r700249 = 6.0;
        double r700250 = x;
        double r700251 = 1.0;
        double r700252 = r700250 - r700251;
        double r700253 = sqrt(r700250);
        double r700254 = 4.0;
        double r700255 = r700250 + r700251;
        double r700256 = fma(r700253, r700254, r700255);
        double r700257 = r700252 / r700256;
        double r700258 = r700249 * r700257;
        return r700258;
}

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

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

  6. Applied div-inv0.1

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019209 +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)))))