Average Error: 0.2 → 0.1
Time: 18.1s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[6 \cdot \frac{1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, 1 + x\right)}{x - 1}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
6 \cdot \frac{1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, 1 + x\right)}{x - 1}}
double f(double x) {
        double r36880739 = 6.0;
        double r36880740 = x;
        double r36880741 = 1.0;
        double r36880742 = r36880740 - r36880741;
        double r36880743 = r36880739 * r36880742;
        double r36880744 = r36880740 + r36880741;
        double r36880745 = 4.0;
        double r36880746 = sqrt(r36880740);
        double r36880747 = r36880745 * r36880746;
        double r36880748 = r36880744 + r36880747;
        double r36880749 = r36880743 / r36880748;
        return r36880749;
}
↓
double f(double x) {
        double r36880750 = 6.0;
        double r36880751 = 1.0;
        double r36880752 = x;
        double r36880753 = sqrt(r36880752);
        double r36880754 = 4.0;
        double r36880755 = 1.0;
        double r36880756 = r36880755 + r36880752;
        double r36880757 = fma(r36880753, r36880754, r36880756);
        double r36880758 = r36880752 - r36880755;
        double r36880759 = r36880757 / r36880758;
        double r36880760 = r36880751 / r36880759;
        double r36880761 = r36880750 * r36880760;
        return r36880761;
}

Error

Bits error versus x

Target

Original0.2
Target0.1
Herbie0.1
\[\frac{6}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}}\]

Derivation

  1. Initial program 0.2

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

  2. Simplified0.0

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

  4. Applied clear-num0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"

  :herbie-target
  (/ 6.0 (/ (+ (+ x 1.0) (* 4.0 (sqrt x))) (- x 1.0)))

  (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))