Average Error: 59.0 → 0.3
Time: 6.4s
Precision: 64
\[-0.00017 \lt x\]
\[e^{x} - 1\]
↓
\[x + \left(\left(\frac{1}{2} + x \cdot \frac{1}{6}\right) \cdot x\right) \cdot x\]
e^{x} - 1
↓
x + \left(\left(\frac{1}{2} + x \cdot \frac{1}{6}\right) \cdot x\right) \cdot x
double f(double x) {
        double r2532743 = x;
        double r2532744 = exp(r2532743);
        double r2532745 = 1.0;
        double r2532746 = r2532744 - r2532745;
        return r2532746;
}
↓
double f(double x) {
        double r2532747 = x;
        double r2532748 = 0.5;
        double r2532749 = 0.16666666666666666;
        double r2532750 = r2532747 * r2532749;
        double r2532751 = r2532748 + r2532750;
        double r2532752 = r2532751 * r2532747;
        double r2532753 = r2532752 * r2532747;
        double r2532754 = r2532747 + r2532753;
        return r2532754;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original59.0
Target0.4
Herbie0.3
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right)\]

Derivation

  1. Initial program 59.0

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019132 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)

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

  (- (exp x) 1))