Average Error: 58.6 → 0.5
Time: 2.9s
Precision: 64
\[-1.7 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[\mathsf{fma}\left(\frac{1}{2}, {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {x}^{3}, x\right)\right)\]
e^{x} - 1
↓
\mathsf{fma}\left(\frac{1}{2}, {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {x}^{3}, x\right)\right)
double f(double x) {
        double r85339 = x;
        double r85340 = exp(r85339);
        double r85341 = 1.0;
        double r85342 = r85340 - r85341;
        return r85342;
}
↓
double f(double x) {
        double r85343 = 0.5;
        double r85344 = x;
        double r85345 = 2.0;
        double r85346 = pow(r85344, r85345);
        double r85347 = 0.16666666666666666;
        double r85348 = 3.0;
        double r85349 = pow(r85344, r85348);
        double r85350 = fma(r85347, r85349, r85344);
        double r85351 = fma(r85343, r85346, r85350);
        return r85351;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {x}^{3}, x\right)\right)}\]

  4. Final simplification0.5

    \[\leadsto \mathsf{fma}\left(\frac{1}{2}, {x}^{2}, \mathsf{fma}\left(\frac{1}{6}, {x}^{3}, x\right)\right)\]

Reproduce

herbie shell --seed 2020027 +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))