Average Error: 0.2 → 0.1
Time: 11.2s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{6}{\sqrt[3]{{\left(\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{x - 1}\right)}^{3}}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{6}{\sqrt[3]{{\left(\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{x - 1}\right)}^{3}}}
double f(double x) {
        double r550556 = 6.0;
        double r550557 = x;
        double r550558 = 1.0;
        double r550559 = r550557 - r550558;
        double r550560 = r550556 * r550559;
        double r550561 = r550557 + r550558;
        double r550562 = 4.0;
        double r550563 = sqrt(r550557);
        double r550564 = r550562 * r550563;
        double r550565 = r550561 + r550564;
        double r550566 = r550560 / r550565;
        return r550566;
}
↓
double f(double x) {
        double r550567 = 6.0;
        double r550568 = x;
        double r550569 = sqrt(r550568);
        double r550570 = 4.0;
        double r550571 = 1.0;
        double r550572 = r550568 + r550571;
        double r550573 = fma(r550569, r550570, r550572);
        double r550574 = r550568 - r550571;
        double r550575 = r550573 / r550574;
        double r550576 = 3.0;
        double r550577 = pow(r550575, r550576);
        double r550578 = cbrt(r550577);
        double r550579 = r550567 / r550578;
        return r550579;
}

Error

Bits error versus x

Target

Original0.2
Target0.0
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. Simplified0.0

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

  4. Applied add-cbrt-cube20.9

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

  5. Applied add-cbrt-cube20.9

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

  6. Applied cbrt-undiv20.9

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

  7. Simplified0.1

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

  8. Final simplification0.1

    \[\leadsto \frac{6}{\sqrt[3]{{\left(\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{x - 1}\right)}^{3}}}\]

Reproduce

herbie shell --seed 2019325 +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)))))