Average Error: 0.2 → 0.0
Time: 14.9s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{6}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, 1\right) + x}{x - 1}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{6}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, 1\right) + x}{x - 1}}
double f(double x) {
        double r705165 = 6.0;
        double r705166 = x;
        double r705167 = 1.0;
        double r705168 = r705166 - r705167;
        double r705169 = r705165 * r705168;
        double r705170 = r705166 + r705167;
        double r705171 = 4.0;
        double r705172 = sqrt(r705166);
        double r705173 = r705171 * r705172;
        double r705174 = r705170 + r705173;
        double r705175 = r705169 / r705174;
        return r705175;
}
↓
double f(double x) {
        double r705176 = 6.0;
        double r705177 = x;
        double r705178 = sqrt(r705177);
        double r705179 = 4.0;
        double r705180 = 1.0;
        double r705181 = fma(r705178, r705179, r705180);
        double r705182 = r705181 + r705177;
        double r705183 = r705177 - r705180;
        double r705184 = r705182 / r705183;
        double r705185 = r705176 / r705184;
        return r705185;
}

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

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

  4. Applied *-un-lft-identity0.0

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

  5. Applied *-un-lft-identity0.0

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

  6. Applied times-frac0.0

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

  7. Simplified0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019326 +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)))))