Average Error: 58.8 → 0.3
Time: 9.1s
Precision: 64
\[-1.700000000000000122124532708767219446599 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[{x}^{2} \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right) + x\]
e^{x} - 1
↓
{x}^{2} \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right) + x
double f(double x) {
        double r102730 = x;
        double r102731 = exp(r102730);
        double r102732 = 1.0;
        double r102733 = r102731 - r102732;
        return r102733;
}
↓
double f(double x) {
        double r102734 = x;
        double r102735 = 2.0;
        double r102736 = pow(r102734, r102735);
        double r102737 = 0.16666666666666666;
        double r102738 = r102737 * r102734;
        double r102739 = 0.5;
        double r102740 = r102738 + r102739;
        double r102741 = r102736 * r102740;
        double r102742 = r102741 + r102734;
        return r102742;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.8
Target0.3
Herbie0.3
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right)\]

Derivation

  1. Initial program 58.8

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019351 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :precision binary64
  :pre (< -0.00017 x)

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

  (- (exp x) 1))