Average Error: 0.2 → 0.1
Time: 12.0s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{x}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}} - \frac{1}{\sqrt{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}} \cdot \sqrt{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{x}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}} - \frac{1}{\sqrt{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}} \cdot \sqrt{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}}}
double f(double x) {
        double r5328 = 6.0;
        double r5329 = x;
        double r5330 = 1.0;
        double r5331 = r5329 - r5330;
        double r5332 = r5328 * r5331;
        double r5333 = r5329 + r5330;
        double r5334 = 4.0;
        double r5335 = sqrt(r5329);
        double r5336 = r5334 * r5335;
        double r5337 = r5333 + r5336;
        double r5338 = r5332 / r5337;
        return r5338;
}
↓
double f(double x) {
        double r5339 = x;
        double r5340 = sqrt(r5339);
        double r5341 = 4.0;
        double r5342 = 1.0;
        double r5343 = r5339 + r5342;
        double r5344 = fma(r5340, r5341, r5343);
        double r5345 = 6.0;
        double r5346 = r5344 / r5345;
        double r5347 = r5339 / r5346;
        double r5348 = sqrt(r5346);
        double r5349 = r5348 * r5348;
        double r5350 = r5342 / r5349;
        double r5351 = r5347 - r5350;
        return r5351;
}

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

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

  4. Applied div-sub0.0

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

  6. Applied add-sqr-sqrt0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2020025 +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)))))