Average Error: 58.7 → 0.4
Time: 2.6s
Precision: 64
\[-1.700000000000000122124532708767219446599 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), {x}^{2}, x\right)\]
e^{x} - 1
↓
\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), {x}^{2}, x\right)
double f(double x) {
        double r91855 = x;
        double r91856 = exp(r91855);
        double r91857 = 1.0;
        double r91858 = r91856 - r91857;
        return r91858;
}
↓
double f(double x) {
        double r91859 = 0.16666666666666666;
        double r91860 = x;
        double r91861 = 0.5;
        double r91862 = fma(r91859, r91860, r91861);
        double r91863 = 2.0;
        double r91864 = pow(r91860, r91863);
        double r91865 = fma(r91862, r91864, r91860);
        return r91865;
}

Error

Bits error versus x

Target

Original58.7
Target0.4
Herbie0.4
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right)\]

Derivation

  1. Initial program 58.7

    \[e^{x} - 1\]
  2. Using strategy rm

  3. Applied add-cube-cbrt58.9

    \[\leadsto \color{blue}{\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right) \cdot \sqrt[3]{e^{x}}} - 1\]

  4. Applied fma-neg58.9

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}, \sqrt[3]{e^{x}}, -1\right)}\]

  5. Taylor expanded around 0 0.4

    \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2} + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}\]

  6. Simplified0.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), {x}^{2}, x\right)}\]

  7. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), {x}^{2}, x\right)\]

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (x)
  :name "expm1 (example 3.7)"
  :precision binary64
  :pre (< -0.00017 x)

  :herbie-target
  (* x (+ (+ 1 (/ x 2)) (/ (* x x) 6)))

  (- (exp x) 1))