Average Error: 0.2 → 0.0
Time: 9.0s
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 r1099994 = 6.0;
        double r1099995 = x;
        double r1099996 = 1.0;
        double r1099997 = r1099995 - r1099996;
        double r1099998 = r1099994 * r1099997;
        double r1099999 = r1099995 + r1099996;
        double r1100000 = 4.0;
        double r1100001 = sqrt(r1099995);
        double r1100002 = r1100000 * r1100001;
        double r1100003 = r1099999 + r1100002;
        double r1100004 = r1099998 / r1100003;
        return r1100004;
}
↓
double f(double x) {
        double r1100005 = 6.0;
        double r1100006 = x;
        double r1100007 = 1.0;
        double r1100008 = r1100006 - r1100007;
        double r1100009 = sqrt(r1100006);
        double r1100010 = 4.0;
        double r1100011 = r1100006 + r1100007;
        double r1100012 = fma(r1100009, r1100010, r1100011);
        double r1100013 = r1100008 / r1100012;
        double r1100014 = r1100005 * r1100013;
        return r1100014;
}

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. Simplified0.1

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

  4. Applied div-inv0.1

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019351 +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)))))