Average Error: 58.6 → 0.5
Time: 8.8s
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 r64391 = x;
        double r64392 = exp(r64391);
        double r64393 = 1.0;
        double r64394 = r64392 - r64393;
        return r64394;
}
↓
double f(double x) {
        double r64395 = x;
        double r64396 = 2.0;
        double r64397 = pow(r64395, r64396);
        double r64398 = 0.16666666666666666;
        double r64399 = 0.5;
        double r64400 = fma(r64395, r64398, r64399);
        double r64401 = fma(r64397, r64400, r64395);
        return r64401;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 58.6

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.5

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

  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 2019350 +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))