Average Error: 58.7 → 0.0
Time: 3.4s
Precision: 64
\[e^{x} - 1\]
↓
\[(e^{x} - 1)^*\]
double f(double x) {
        double r6607874 = x;
        double r6607875 = exp(r6607874);
        double r6607876 = 1.0;
        double r6607877 = r6607875 - r6607876;
        return r6607877;
}
↓
double f(double x) {
        double r6607878 = x;
        double r6607879 = expm1(r6607878);
        return r6607879;
}


e^{x} - 1
↓
(e^{x} - 1)^*

Error

Bits error versus x

Target

Original58.7
Target0.5
Herbie0.0
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right)\]

Derivation

  1. Initial program 58.7

    \[e^{x} - 1\]

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019101 +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))