Average Error: 0.2 → 0.1
Time: 11.1s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{6}{\sqrt[3]{{\left(\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{x - 1}\right)}^{3}}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{6}{\sqrt[3]{{\left(\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{x - 1}\right)}^{3}}}
double f(double x) {
        double r614446 = 6.0;
        double r614447 = x;
        double r614448 = 1.0;
        double r614449 = r614447 - r614448;
        double r614450 = r614446 * r614449;
        double r614451 = r614447 + r614448;
        double r614452 = 4.0;
        double r614453 = sqrt(r614447);
        double r614454 = r614452 * r614453;
        double r614455 = r614451 + r614454;
        double r614456 = r614450 / r614455;
        return r614456;
}
↓
double f(double x) {
        double r614457 = 6.0;
        double r614458 = x;
        double r614459 = sqrt(r614458);
        double r614460 = 4.0;
        double r614461 = 1.0;
        double r614462 = r614458 + r614461;
        double r614463 = fma(r614459, r614460, r614462);
        double r614464 = r614458 - r614461;
        double r614465 = r614463 / r614464;
        double r614466 = 3.0;
        double r614467 = pow(r614465, r614466);
        double r614468 = cbrt(r614467);
        double r614469 = r614457 / r614468;
        return r614469;
}

Error

Bits error versus x

Target

Original0.2
Target0.0
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

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

  4. Applied add-cbrt-cube20.9

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

  5. Applied add-cbrt-cube20.9

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

  6. Applied cbrt-undiv20.9

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

  7. Simplified0.1

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

  8. Final simplification0.1

    \[\leadsto \frac{6}{\sqrt[3]{{\left(\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{x - 1}\right)}^{3}}}\]

Reproduce

herbie shell --seed 2019325 +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)))))