Average Error: 0.2 → 0.0
Time: 7.1s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{x - 1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{x - 1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}}
double f(double x) {
        double r911004 = 6.0;
        double r911005 = x;
        double r911006 = 1.0;
        double r911007 = r911005 - r911006;
        double r911008 = r911004 * r911007;
        double r911009 = r911005 + r911006;
        double r911010 = 4.0;
        double r911011 = sqrt(r911005);
        double r911012 = r911010 * r911011;
        double r911013 = r911009 + r911012;
        double r911014 = r911008 / r911013;
        return r911014;
}
↓
double f(double x) {
        double r911015 = x;
        double r911016 = 1.0;
        double r911017 = r911015 - r911016;
        double r911018 = sqrt(r911015);
        double r911019 = 4.0;
        double r911020 = r911015 + r911016;
        double r911021 = fma(r911018, r911019, r911020);
        double r911022 = 6.0;
        double r911023 = r911021 / r911022;
        double r911024 = r911017 / r911023;
        return r911024;
}

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. Simplified0.1

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

  4. Applied associate-/r/0.1

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

  6. Applied *-un-lft-identity0.1

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

  7. Applied *-un-lft-identity0.1

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

  8. Applied times-frac0.1

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

  9. Applied associate-*l*0.1

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2020046 +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)))))