Average Error: 0.2 → 0.1
Time: 14.2s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[6 \cdot \frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
6 \cdot \frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}
double f(double x) {
        double r1008470 = 6.0;
        double r1008471 = x;
        double r1008472 = 1.0;
        double r1008473 = r1008471 - r1008472;
        double r1008474 = r1008470 * r1008473;
        double r1008475 = r1008471 + r1008472;
        double r1008476 = 4.0;
        double r1008477 = sqrt(r1008471);
        double r1008478 = r1008476 * r1008477;
        double r1008479 = r1008475 + r1008478;
        double r1008480 = r1008474 / r1008479;
        return r1008480;
}
↓
double f(double x) {
        double r1008481 = 6.0;
        double r1008482 = x;
        double r1008483 = 1.0;
        double r1008484 = r1008482 - r1008483;
        double r1008485 = sqrt(r1008482);
        double r1008486 = 4.0;
        double r1008487 = r1008482 + r1008483;
        double r1008488 = fma(r1008485, r1008486, r1008487);
        double r1008489 = r1008484 / r1008488;
        double r1008490 = r1008481 * r1008489;
        return r1008490;
}

Error

Bits error versus x

Target

Original0.2
Target0.1
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. Simplified0.1

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

  4. Applied div-inv0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2020042 +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)))))