Average Error: 0.2 → 0.0
Time: 3.4s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)} \cdot 6\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)} \cdot 6
double f(double x) {
        double r799384 = 6.0;
        double r799385 = x;
        double r799386 = 1.0;
        double r799387 = r799385 - r799386;
        double r799388 = r799384 * r799387;
        double r799389 = r799385 + r799386;
        double r799390 = 4.0;
        double r799391 = sqrt(r799385);
        double r799392 = r799390 * r799391;
        double r799393 = r799389 + r799392;
        double r799394 = r799388 / r799393;
        return r799394;
}
↓
double f(double x) {
        double r799395 = x;
        double r799396 = 1.0;
        double r799397 = r799395 - r799396;
        double r799398 = sqrt(r799395);
        double r799399 = 4.0;
        double r799400 = r799395 + r799396;
        double r799401 = fma(r799398, r799399, r799400);
        double r799402 = r799397 / r799401;
        double r799403 = 6.0;
        double r799404 = r799402 * r799403;
        return r799404;
}

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. Using strategy rm

  4. Applied associate-/r/0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020001 +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)))))