Average Error: 0.2 → 0.1
Time: 3.8s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}\right)\right) \cdot 6\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}\right)\right) \cdot 6
double f(double x) {
        double r888773 = 6.0;
        double r888774 = x;
        double r888775 = 1.0;
        double r888776 = r888774 - r888775;
        double r888777 = r888773 * r888776;
        double r888778 = r888774 + r888775;
        double r888779 = 4.0;
        double r888780 = sqrt(r888774);
        double r888781 = r888779 * r888780;
        double r888782 = r888778 + r888781;
        double r888783 = r888777 / r888782;
        return r888783;
}
↓
double f(double x) {
        double r888784 = x;
        double r888785 = 1.0;
        double r888786 = r888784 - r888785;
        double r888787 = sqrt(r888784);
        double r888788 = 4.0;
        double r888789 = r888784 + r888785;
        double r888790 = fma(r888787, r888788, r888789);
        double r888791 = r888786 / r888790;
        double r888792 = expm1(r888791);
        double r888793 = log1p(r888792);
        double r888794 = 6.0;
        double r888795 = r888793 * r888794;
        return r888795;
}

Error

Bits error versus x

Target

Original0.2
Target0.0
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 log1p-expm1-u0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2020057 +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)))))