Average Error: 0.2 → 0.1
Time: 13.9s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{6}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{x - 1}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{6}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{x - 1}}
double f(double x) {
        double r535105 = 6.0;
        double r535106 = x;
        double r535107 = 1.0;
        double r535108 = r535106 - r535107;
        double r535109 = r535105 * r535108;
        double r535110 = r535106 + r535107;
        double r535111 = 4.0;
        double r535112 = sqrt(r535106);
        double r535113 = r535111 * r535112;
        double r535114 = r535110 + r535113;
        double r535115 = r535109 / r535114;
        return r535115;
}
↓
double f(double x) {
        double r535116 = 6.0;
        double r535117 = x;
        double r535118 = sqrt(r535117);
        double r535119 = 4.0;
        double r535120 = 1.0;
        double r535121 = r535117 + r535120;
        double r535122 = fma(r535118, r535119, r535121);
        double r535123 = r535117 - r535120;
        double r535124 = r535122 / r535123;
        double r535125 = r535116 / r535124;
        return r535125;
}

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. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019347 +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)))))