Average Error: 58.7 → 0.5
Time: 6.6s
Precision: 64
\[-1.700000000000000122124532708767219446599 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[\mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right), x\right)\]
e^{x} - 1
↓
\mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right), x\right)
double f(double x) {
        double r40890 = x;
        double r40891 = exp(r40890);
        double r40892 = 1.0;
        double r40893 = r40891 - r40892;
        return r40893;
}
↓
double f(double x) {
        double r40894 = x;
        double r40895 = 2.0;
        double r40896 = pow(r40894, r40895);
        double r40897 = 0.16666666666666666;
        double r40898 = 0.5;
        double r40899 = fma(r40894, r40897, r40898);
        double r40900 = fma(r40896, r40899, r40894);
        return r40900;
}

Error

Bits error versus x

Target

Original58.7
Target0.5
Herbie0.5
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right)\]

Derivation

  1. Initial program 58.7

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (x)
  :name "expm1 (example 3.7)"
  :precision binary64
  :pre (< -1.7e-4 x)

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

  (- (exp x) 1))