Average Error: 58.9 → 0.0
Time: 2.9s
Precision: 64
\[e^{x} - 1\]
↓
\[(e^{x} - 1)^*\]
double f(double x) {
        double r4553139 = x;
        double r4553140 = exp(r4553139);
        double r4553141 = 1.0;
        double r4553142 = r4553140 - r4553141;
        return r4553142;
}
↓
double f(double x) {
        double r4553143 = x;
        double r4553144 = expm1(r4553143);
        return r4553144;
}


e^{x} - 1
↓
(e^{x} - 1)^*

Error

Bits error versus x

Target

Original58.9
Target0.4
Herbie0.0
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right)\]

Derivation

  1. Initial program 58.9

    \[e^{x} - 1\]

  2. Simplified0.0

    \[\leadsto \color{blue}{(e^{x} - 1)^*}\]

  3. Final simplification0.0

    \[\leadsto (e^{x} - 1)^*\]

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)

  :herbie-target
  (* x (+ (+ 1 (/ x 2)) (/ (* x x) 6)))

  (- (exp x) 1))