Average Error: 58.7 → 0.5
Time: 7.9s
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 r6189666 = x;
        double r6189667 = exp(r6189666);
        double r6189668 = 1.0;
        double r6189669 = r6189667 - r6189668;
        return r6189669;
}
↓
double f(double x) {
        double r6189670 = x;
        double r6189671 = 0.16666666666666666;
        double r6189672 = r6189670 * r6189671;
        double r6189673 = 0.5;
        double r6189674 = r6189672 + r6189673;
        double r6189675 = r6189670 * r6189670;
        double r6189676 = r6189674 * r6189675;
        double r6189677 = r6189670 + r6189676;
        return r6189677;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.7

    \[e^{x} - 1\]

  2. 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 2019107 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)

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

  (- (exp x) 1))