Average Error: 58.7 → 0.4
Time: 3.2s
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 r69161 = x;
        double r69162 = exp(r69161);
        double r69163 = 1.0;
        double r69164 = r69162 - r69163;
        return r69164;
}
↓
double f(double x) {
        double r69165 = 0.5;
        double r69166 = x;
        double r69167 = 2.0;
        double r69168 = pow(r69166, r69167);
        double r69169 = 0.16666666666666666;
        double r69170 = 3.0;
        double r69171 = pow(r69166, r69170);
        double r69172 = fma(r69169, r69171, r69166);
        double r69173 = fma(r69165, r69168, r69172);
        return r69173;
}

Error

Bits error versus x

Target

Original58.7
Target0.4
Herbie0.4
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right)\]

Derivation

  1. Initial program 58.7

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2020035 +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))