Average Error: 58.5 → 0.5
Time: 8.7s
Precision: 64
\[-1.700000000000000122124532708767219446599 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right), x\right)\]
e^{x} - 1
↓
\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right), x\right)
double f(double x) {
        double r3396232 = x;
        double r3396233 = exp(r3396232);
        double r3396234 = 1.0;
        double r3396235 = r3396233 - r3396234;
        return r3396235;
}
↓
double f(double x) {
        double r3396236 = x;
        double r3396237 = r3396236 * r3396236;
        double r3396238 = 0.16666666666666666;
        double r3396239 = 0.5;
        double r3396240 = fma(r3396236, r3396238, r3396239);
        double r3396241 = fma(r3396237, r3396240, r3396236);
        return r3396241;
}

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

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019200 +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))