Average Error: 58.5 → 0.4
Time: 5.7s
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 r61344 = x;
        double r61345 = exp(r61344);
        double r61346 = 1.0;
        double r61347 = r61345 - r61346;
        return r61347;
}
↓
double f(double x) {
        double r61348 = 0.5;
        double r61349 = x;
        double r61350 = 2.0;
        double r61351 = pow(r61349, r61350);
        double r61352 = 0.16666666666666666;
        double r61353 = 3.0;
        double r61354 = pow(r61349, r61353);
        double r61355 = fma(r61352, r61354, r61349);
        double r61356 = fma(r61348, r61351, r61355);
        return r61356;
}

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

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

  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 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))