Average Error: 58.5 → 0.4
Time: 9.6s
Precision: 64
\[-1.7 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[\mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x\right)\]
e^{x} - 1
↓
\mathsf{fma}\left({x}^{2}, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x\right)
double f(double x) {
        double r77258 = x;
        double r77259 = exp(r77258);
        double r77260 = 1.0;
        double r77261 = r77259 - r77260;
        return r77261;
}
↓
double f(double x) {
        double r77262 = x;
        double r77263 = 2.0;
        double r77264 = pow(r77262, r77263);
        double r77265 = 0.16666666666666666;
        double r77266 = 0.5;
        double r77267 = fma(r77265, r77262, r77266);
        double r77268 = fma(r77264, r77267, r77262);
        return r77268;
}

Error

Bits error versus x

Target

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

Derivation

  1. Initial program 58.5

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2020045 +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))