Average Error: 58.7 → 0.4
Time: 30.4s
Precision: 64
\[-0.00017 \lt x\]
\[e^{x} - 1.0\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x \cdot x, x\right)\]
e^{x} - 1.0
↓
\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), x \cdot x, x\right)
double f(double x) {
        double r5245856 = x;
        double r5245857 = exp(r5245856);
        double r5245858 = 1.0;
        double r5245859 = r5245857 - r5245858;
        return r5245859;
}
↓
double f(double x) {
        double r5245860 = 0.16666666666666666;
        double r5245861 = x;
        double r5245862 = 0.5;
        double r5245863 = fma(r5245860, r5245861, r5245862);
        double r5245864 = r5245861 * r5245861;
        double r5245865 = fma(r5245863, r5245864, r5245861);
        return r5245865;
}

Error

Bits error versus x

Target

Original58.7
Target0.4
Herbie0.4
\[x \cdot \left(\left(1.0 + \frac{x}{2.0}\right) + \frac{x \cdot x}{6.0}\right)\]

Derivation

  1. Initial program 58.7

    \[e^{x} - 1.0\]

  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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019165 +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))