Average Error: 0.2 → 0.0
Time: 13.0s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[6 \cdot \frac{x - 1}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
6 \cdot \frac{x - 1}{\mathsf{fma}\left(4, \sqrt{x}, x + 1\right)}
double f(double x) {
        double r500087 = 6.0;
        double r500088 = x;
        double r500089 = 1.0;
        double r500090 = r500088 - r500089;
        double r500091 = r500087 * r500090;
        double r500092 = r500088 + r500089;
        double r500093 = 4.0;
        double r500094 = sqrt(r500088);
        double r500095 = r500093 * r500094;
        double r500096 = r500092 + r500095;
        double r500097 = r500091 / r500096;
        return r500097;
}
↓
double f(double x) {
        double r500098 = 6.0;
        double r500099 = x;
        double r500100 = 1.0;
        double r500101 = r500099 - r500100;
        double r500102 = 4.0;
        double r500103 = sqrt(r500099);
        double r500104 = r500099 + r500100;
        double r500105 = fma(r500102, r500103, r500104);
        double r500106 = r500101 / r500105;
        double r500107 = r500098 * r500106;
        return r500107;
}

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

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019235 +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)))))