Average Error: 0.2 → 0.0
Time: 3.6s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{x - 1}{\frac{1 \cdot \mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{x - 1}{\frac{1 \cdot \mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}}
double f(double x) {
        double r1053023 = 6.0;
        double r1053024 = x;
        double r1053025 = 1.0;
        double r1053026 = r1053024 - r1053025;
        double r1053027 = r1053023 * r1053026;
        double r1053028 = r1053024 + r1053025;
        double r1053029 = 4.0;
        double r1053030 = sqrt(r1053024);
        double r1053031 = r1053029 * r1053030;
        double r1053032 = r1053028 + r1053031;
        double r1053033 = r1053027 / r1053032;
        return r1053033;
}
↓
double f(double x) {
        double r1053034 = x;
        double r1053035 = 1.0;
        double r1053036 = r1053034 - r1053035;
        double r1053037 = 1.0;
        double r1053038 = sqrt(r1053034);
        double r1053039 = 4.0;
        double r1053040 = r1053034 + r1053035;
        double r1053041 = fma(r1053038, r1053039, r1053040);
        double r1053042 = r1053037 * r1053041;
        double r1053043 = 6.0;
        double r1053044 = r1053042 / r1053043;
        double r1053045 = r1053036 / r1053044;
        return r1053045;
}

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 *-un-lft-identity0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019353 +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)))))