Average Error: 0.2 → 0.0
Time: 4.6s
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 r1007522 = 6.0;
        double r1007523 = x;
        double r1007524 = 1.0;
        double r1007525 = r1007523 - r1007524;
        double r1007526 = r1007522 * r1007525;
        double r1007527 = r1007523 + r1007524;
        double r1007528 = 4.0;
        double r1007529 = sqrt(r1007523);
        double r1007530 = r1007528 * r1007529;
        double r1007531 = r1007527 + r1007530;
        double r1007532 = r1007526 / r1007531;
        return r1007532;
}
↓
double f(double x) {
        double r1007533 = x;
        double r1007534 = 1.0;
        double r1007535 = r1007533 - r1007534;
        double r1007536 = sqrt(r1007533);
        double r1007537 = 4.0;
        double r1007538 = r1007533 + r1007534;
        double r1007539 = fma(r1007536, r1007537, r1007538);
        double r1007540 = r1007535 / r1007539;
        double r1007541 = 6.0;
        double r1007542 = r1007540 * r1007541;
        return r1007542;
}

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

    \[\leadsto \color{blue}{\frac{x - 1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}}}\]
  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 2020083 +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)))))