Average Error: 58.8 → 0.5
Time: 6.3s
Precision: 64
\[-0.00017 \lt x\]
\[e^{x} - 1\]
↓
\[x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)\]
e^{x} - 1
↓
x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)
double f(double x) {
        double r5064794 = x;
        double r5064795 = exp(r5064794);
        double r5064796 = 1.0;
        double r5064797 = r5064795 - r5064796;
        return r5064797;
}
↓
double f(double x) {
        double r5064798 = x;
        double r5064799 = 0.16666666666666666;
        double r5064800 = r5064798 * r5064799;
        double r5064801 = 0.5;
        double r5064802 = r5064800 + r5064801;
        double r5064803 = r5064798 * r5064798;
        double r5064804 = r5064802 * r5064803;
        double r5064805 = r5064798 + r5064804;
        return r5064805;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.8
Target0.5
Herbie0.5
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right)\]

Derivation

  1. Initial program 58.8

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019120 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)

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

  (- (exp x) 1))