Average Error: 58.8 → 0.4
Time: 8.7s
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 r87348 = x;
        double r87349 = exp(r87348);
        double r87350 = 1.0;
        double r87351 = r87349 - r87350;
        return r87351;
}
↓
double f(double x) {
        double r87352 = x;
        double r87353 = r87352 * r87352;
        double r87354 = 0.16666666666666666;
        double r87355 = 0.5;
        double r87356 = fma(r87354, r87352, r87355);
        double r87357 = fma(r87353, r87356, r87352);
        return r87357;
}

Error

Bits error versus x

Target

Original58.8
Target0.5
Herbie0.4
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right)\]

Derivation

  1. Initial program 58.8

    \[e^{x} - 1\]

  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 2019195 +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))