Average Error: 0.2 → 0.0
Time: 5.4s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{x}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)} \cdot 6 - \frac{1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{x}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)} \cdot 6 - \frac{1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}}
double f(double x) {
        double r834519 = 6.0;
        double r834520 = x;
        double r834521 = 1.0;
        double r834522 = r834520 - r834521;
        double r834523 = r834519 * r834522;
        double r834524 = r834520 + r834521;
        double r834525 = 4.0;
        double r834526 = sqrt(r834520);
        double r834527 = r834525 * r834526;
        double r834528 = r834524 + r834527;
        double r834529 = r834523 / r834528;
        return r834529;
}
↓
double f(double x) {
        double r834530 = x;
        double r834531 = sqrt(r834530);
        double r834532 = 4.0;
        double r834533 = 1.0;
        double r834534 = r834530 + r834533;
        double r834535 = fma(r834531, r834532, r834534);
        double r834536 = r834530 / r834535;
        double r834537 = 6.0;
        double r834538 = r834536 * r834537;
        double r834539 = r834535 / r834537;
        double r834540 = r834533 / r834539;
        double r834541 = r834538 - r834540;
        return r834541;
}

Error

Bits error versus x

Target

Original0.2
Target0.1
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-sub0.1

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

  6. Applied associate-/r/0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020089 +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)))))