Average Error: 0.2 → 0.1
Time: 8.2s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[6 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}\right)\right)\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
6 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}\right)\right)
double f(double x) {
        double r875935 = 6.0;
        double r875936 = x;
        double r875937 = 1.0;
        double r875938 = r875936 - r875937;
        double r875939 = r875935 * r875938;
        double r875940 = r875936 + r875937;
        double r875941 = 4.0;
        double r875942 = sqrt(r875936);
        double r875943 = r875941 * r875942;
        double r875944 = r875940 + r875943;
        double r875945 = r875939 / r875944;
        return r875945;
}
↓
double f(double x) {
        double r875946 = 6.0;
        double r875947 = x;
        double r875948 = 1.0;
        double r875949 = r875947 - r875948;
        double r875950 = sqrt(r875947);
        double r875951 = 4.0;
        double r875952 = r875947 + r875948;
        double r875953 = fma(r875950, r875951, r875952);
        double r875954 = r875949 / r875953;
        double r875955 = expm1(r875954);
        double r875956 = log1p(r875955);
        double r875957 = r875946 * r875956;
        return r875957;
}

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 div-inv0.1

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

  5. Simplified0.0

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

  7. Applied log1p-expm1-u0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2020043 +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)))))