Average Error: 58.7 → 0.5
Time: 3.5s
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 r89225 = x;
        double r89226 = exp(r89225);
        double r89227 = 1.0;
        double r89228 = r89226 - r89227;
        return r89228;
}
↓
double f(double x) {
        double r89229 = 0.5;
        double r89230 = x;
        double r89231 = 2.0;
        double r89232 = pow(r89230, r89231);
        double r89233 = 0.16666666666666666;
        double r89234 = 3.0;
        double r89235 = pow(r89230, r89234);
        double r89236 = fma(r89233, r89235, r89230);
        double r89237 = fma(r89229, r89232, r89236);
        return r89237;
}

Error

Bits error versus x

Target

Original58.7
Target0.5
Herbie0.5
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right)\]

Derivation

  1. Initial program 58.7

    \[e^{x} - 1\]

  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 2020060 +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))