Average Error: 58.5 → 0.5
Time: 7.7s
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 r75990 = x;
        double r75991 = exp(r75990);
        double r75992 = 1.0;
        double r75993 = r75991 - r75992;
        return r75993;
}
↓
double f(double x) {
        double r75994 = x;
        double r75995 = r75994 * r75994;
        double r75996 = 0.16666666666666666;
        double r75997 = 0.5;
        double r75998 = fma(r75996, r75994, r75997);
        double r75999 = fma(r75995, r75998, r75994);
        return r75999;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 58.5

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019174 +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))