Average Error: 58.9 → 0.4
Time: 14.2s
Precision: 64
\[-1.700000000000000122124532708767219446599 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[x + \left(\frac{1}{2} + x \cdot \frac{1}{6}\right) \cdot \left(x \cdot x\right)\]
e^{x} - 1
↓
x + \left(\frac{1}{2} + x \cdot \frac{1}{6}\right) \cdot \left(x \cdot x\right)
double f(double x) {
        double r5065688 = x;
        double r5065689 = exp(r5065688);
        double r5065690 = 1.0;
        double r5065691 = r5065689 - r5065690;
        return r5065691;
}
↓
double f(double x) {
        double r5065692 = x;
        double r5065693 = 0.5;
        double r5065694 = 0.16666666666666666;
        double r5065695 = r5065692 * r5065694;
        double r5065696 = r5065693 + r5065695;
        double r5065697 = r5065692 * r5065692;
        double r5065698 = r5065696 * r5065697;
        double r5065699 = r5065692 + r5065698;
        return r5065699;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.9

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019192 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)

  :herbie-target
  (* x (+ (+ 1.0 (/ x 2.0)) (/ (* x x) 6.0)))

  (- (exp x) 1.0))