Average Error: 0.2 → 0.0
Time: 1.3m
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{\frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{\frac{1}{6}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{\frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{\frac{1}{6}}
double f(double x) {
        double r1158451 = 6.0;
        double r1158452 = x;
        double r1158453 = 1.0;
        double r1158454 = r1158452 - r1158453;
        double r1158455 = r1158451 * r1158454;
        double r1158456 = r1158452 + r1158453;
        double r1158457 = 4.0;
        double r1158458 = sqrt(r1158452);
        double r1158459 = r1158457 * r1158458;
        double r1158460 = r1158456 + r1158459;
        double r1158461 = r1158455 / r1158460;
        return r1158461;
}
↓
double f(double x) {
        double r1158462 = x;
        double r1158463 = 1.0;
        double r1158464 = r1158462 - r1158463;
        double r1158465 = sqrt(r1158462);
        double r1158466 = 4.0;
        double r1158467 = r1158462 + r1158463;
        double r1158468 = fma(r1158465, r1158466, r1158467);
        double r1158469 = r1158464 / r1158468;
        double r1158470 = 1.0;
        double r1158471 = 6.0;
        double r1158472 = r1158470 / r1158471;
        double r1158473 = r1158469 / r1158472;
        return r1158473;
}

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-inv0.2

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

  5. Applied associate-/r*0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020036 +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)))))