Average Error: 0.2 → 0.1
Time: 49.0s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{6}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{x - 1}\right)\right)}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{6}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{x - 1}\right)\right)}
double f(double x) {
        double r29985224 = 6.0;
        double r29985225 = x;
        double r29985226 = 1.0;
        double r29985227 = r29985225 - r29985226;
        double r29985228 = r29985224 * r29985227;
        double r29985229 = r29985225 + r29985226;
        double r29985230 = 4.0;
        double r29985231 = sqrt(r29985225);
        double r29985232 = r29985230 * r29985231;
        double r29985233 = r29985229 + r29985232;
        double r29985234 = r29985228 / r29985233;
        return r29985234;
}
↓
double f(double x) {
        double r29985235 = 6.0;
        double r29985236 = x;
        double r29985237 = sqrt(r29985236);
        double r29985238 = 4.0;
        double r29985239 = 1.0;
        double r29985240 = r29985236 + r29985239;
        double r29985241 = fma(r29985237, r29985238, r29985240);
        double r29985242 = r29985236 - r29985239;
        double r29985243 = r29985241 / r29985242;
        double r29985244 = expm1(r29985243);
        double r29985245 = log1p(r29985244);
        double r29985246 = r29985235 / r29985245;
        return r29985246;
}

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. Simplified0.1

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

  4. Applied log1p-expm1-u0.1

    \[\leadsto \frac{6}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{x - 1}\right)\right)}}\]

  5. Final simplification0.1

    \[\leadsto \frac{6}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{x - 1}\right)\right)}\]

Reproduce

herbie shell --seed 2019169 +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)))))