Average Error: 0.2 → 0.0
Time: 51.2s
Precision: 64
\[\frac{6.0 \cdot \left(x - 1.0\right)}{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}}\]
↓
\[\left(\frac{x}{\mathsf{fma}\left(\sqrt{x}, 4.0, x + 1.0\right)} - \frac{1.0}{\mathsf{fma}\left(\sqrt{x}, 4.0, x + 1.0\right)}\right) \cdot 6.0\]
\frac{6.0 \cdot \left(x - 1.0\right)}{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}}
↓
\left(\frac{x}{\mathsf{fma}\left(\sqrt{x}, 4.0, x + 1.0\right)} - \frac{1.0}{\mathsf{fma}\left(\sqrt{x}, 4.0, x + 1.0\right)}\right) \cdot 6.0
double f(double x) {
        double r32036154 = 6.0;
        double r32036155 = x;
        double r32036156 = 1.0;
        double r32036157 = r32036155 - r32036156;
        double r32036158 = r32036154 * r32036157;
        double r32036159 = r32036155 + r32036156;
        double r32036160 = 4.0;
        double r32036161 = sqrt(r32036155);
        double r32036162 = r32036160 * r32036161;
        double r32036163 = r32036159 + r32036162;
        double r32036164 = r32036158 / r32036163;
        return r32036164;
}
↓
double f(double x) {
        double r32036165 = x;
        double r32036166 = sqrt(r32036165);
        double r32036167 = 4.0;
        double r32036168 = 1.0;
        double r32036169 = r32036165 + r32036168;
        double r32036170 = fma(r32036166, r32036167, r32036169);
        double r32036171 = r32036165 / r32036170;
        double r32036172 = r32036168 / r32036170;
        double r32036173 = r32036171 - r32036172;
        double r32036174 = 6.0;
        double r32036175 = r32036173 * r32036174;
        return r32036175;
}

Error

Bits error versus x

Target

Original0.2
Target0.0
Herbie0.0
\[\frac{6.0}{\frac{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}}{x - 1.0}}\]

Derivation

  1. Initial program 0.2

    \[\frac{6.0 \cdot \left(x - 1.0\right)}{\left(x + 1.0\right) + 4.0 \cdot \sqrt{x}}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{6.0}{\frac{\mathsf{fma}\left(\sqrt{x}, 4.0, x + 1.0\right)}{x - 1.0}}}\]
  3. Using strategy rm

  4. Applied div-inv0.1

    \[\leadsto \color{blue}{6.0 \cdot \frac{1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4.0, x + 1.0\right)}{x - 1.0}}}\]

  5. Simplified0.0

    \[\leadsto 6.0 \cdot \color{blue}{\frac{x - 1.0}{\mathsf{fma}\left(\sqrt{x}, 4.0, x + 1.0\right)}}\]
  6. Using strategy rm

  7. Applied div-sub0.0

    \[\leadsto 6.0 \cdot \color{blue}{\left(\frac{x}{\mathsf{fma}\left(\sqrt{x}, 4.0, x + 1.0\right)} - \frac{1.0}{\mathsf{fma}\left(\sqrt{x}, 4.0, x + 1.0\right)}\right)}\]

  8. Final simplification0.0

    \[\leadsto \left(\frac{x}{\mathsf{fma}\left(\sqrt{x}, 4.0, x + 1.0\right)} - \frac{1.0}{\mathsf{fma}\left(\sqrt{x}, 4.0, x + 1.0\right)}\right) \cdot 6.0\]

Reproduce

herbie shell --seed 2019165 +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)))))