Average Error: 0.2 → 0.0
Time: 9.1s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, 1 + x\right)} \cdot 6\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, 1 + x\right)} \cdot 6
double f(double x) {
        double r710423 = 6.0;
        double r710424 = x;
        double r710425 = 1.0;
        double r710426 = r710424 - r710425;
        double r710427 = r710423 * r710426;
        double r710428 = r710424 + r710425;
        double r710429 = 4.0;
        double r710430 = sqrt(r710424);
        double r710431 = r710429 * r710430;
        double r710432 = r710428 + r710431;
        double r710433 = r710427 / r710432;
        return r710433;
}
↓
double f(double x) {
        double r710434 = x;
        double r710435 = 1.0;
        double r710436 = r710434 - r710435;
        double r710437 = sqrt(r710434);
        double r710438 = 4.0;
        double r710439 = r710435 + r710434;
        double r710440 = fma(r710437, r710438, r710439);
        double r710441 = r710436 / r710440;
        double r710442 = 6.0;
        double r710443 = r710441 * r710442;
        return r710443;
}

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

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

  4. Applied *-commutative0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"

  :herbie-target
  (/ 6.0 (/ (+ (+ x 1.0) (* 4.0 (sqrt x))) (- x 1.0)))

  (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))