Average Error: 58.6 → 0.4
Time: 9.6s
Precision: 64
\[-1.700000000000000122124532708767219446599 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x\right)\]
e^{x} - 1
↓
\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x\right)
double f(double x) {
        double r83617 = x;
        double r83618 = exp(r83617);
        double r83619 = 1.0;
        double r83620 = r83618 - r83619;
        return r83620;
}
↓
double f(double x) {
        double r83621 = x;
        double r83622 = r83621 * r83621;
        double r83623 = 0.16666666666666666;
        double r83624 = 0.5;
        double r83625 = fma(r83623, r83621, r83624);
        double r83626 = fma(r83622, r83625, r83621);
        return r83626;
}

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

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)

  :herbie-target
  (* x (+ (+ 1.0 (/ x 2.0)) (/ (* x x) 6.0)))

  (- (exp x) 1.0))