Average Error: 0.2 → 0.0
Time: 2.9s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{x}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}} - \frac{1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}}\]
\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}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}}
double f(double x) {
        double r877404 = 6.0;
        double r877405 = x;
        double r877406 = 1.0;
        double r877407 = r877405 - r877406;
        double r877408 = r877404 * r877407;
        double r877409 = r877405 + r877406;
        double r877410 = 4.0;
        double r877411 = sqrt(r877405);
        double r877412 = r877410 * r877411;
        double r877413 = r877409 + r877412;
        double r877414 = r877408 / r877413;
        return r877414;
}
↓
double f(double x) {
        double r877415 = x;
        double r877416 = sqrt(r877415);
        double r877417 = 4.0;
        double r877418 = 1.0;
        double r877419 = r877415 + r877418;
        double r877420 = fma(r877416, r877417, r877419);
        double r877421 = 6.0;
        double r877422 = r877420 / r877421;
        double r877423 = r877415 / r877422;
        double r877424 = r877418 / r877422;
        double r877425 = r877423 - r877424;
        return r877425;
}

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 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. Final simplification0.0

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

Reproduce

herbie shell --seed 2020021 +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)))))