Average Error: 0.2 → 0.1
Time: 7.0s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{x - 1}} \cdot 6\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{x - 1}} \cdot 6
double f(double x) {
        double r920188 = 6.0;
        double r920189 = x;
        double r920190 = 1.0;
        double r920191 = r920189 - r920190;
        double r920192 = r920188 * r920191;
        double r920193 = r920189 + r920190;
        double r920194 = 4.0;
        double r920195 = sqrt(r920189);
        double r920196 = r920194 * r920195;
        double r920197 = r920193 + r920196;
        double r920198 = r920192 / r920197;
        return r920198;
}
↓
double f(double x) {
        double r920199 = 1.0;
        double r920200 = x;
        double r920201 = sqrt(r920200);
        double r920202 = 4.0;
        double r920203 = 1.0;
        double r920204 = r920200 + r920203;
        double r920205 = fma(r920201, r920202, r920204);
        double r920206 = r920200 - r920203;
        double r920207 = r920205 / r920206;
        double r920208 = r920199 / r920207;
        double r920209 = 6.0;
        double r920210 = r920208 * r920209;
        return r920210;
}

Error

Bits error versus x

Target

Original0.2
Target0.1
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

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

  4. Applied associate-/r/0.0

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

  6. Applied clear-num0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2020100 +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)))))