Average Error: 58.6 → 0.4
Time: 20.5s
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 r56737 = x;
        double r56738 = exp(r56737);
        double r56739 = 1.0;
        double r56740 = r56738 - r56739;
        return r56740;
}
↓
double f(double x) {
        double r56741 = x;
        double r56742 = 2.0;
        double r56743 = pow(r56741, r56742);
        double r56744 = 0.16666666666666666;
        double r56745 = 0.5;
        double r56746 = fma(r56741, r56744, r56745);
        double r56747 = fma(r56743, r56746, r56741);
        return r56747;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 58.6

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019304 +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))