Average Error: 0.2 → 0.0
Time: 3.9s
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 r989169 = 6.0;
        double r989170 = x;
        double r989171 = 1.0;
        double r989172 = r989170 - r989171;
        double r989173 = r989169 * r989172;
        double r989174 = r989170 + r989171;
        double r989175 = 4.0;
        double r989176 = sqrt(r989170);
        double r989177 = r989175 * r989176;
        double r989178 = r989174 + r989177;
        double r989179 = r989173 / r989178;
        return r989179;
}
↓
double f(double x) {
        double r989180 = x;
        double r989181 = 1.0;
        double r989182 = r989180 - r989181;
        double r989183 = 1.0;
        double r989184 = sqrt(r989180);
        double r989185 = 4.0;
        double r989186 = r989180 + r989181;
        double r989187 = fma(r989184, r989185, r989186);
        double r989188 = r989183 * r989187;
        double r989189 = 6.0;
        double r989190 = r989188 / r989189;
        double r989191 = r989182 / r989190;
        return r989191;
}

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)))))