Average Error: 58.6 → 0.5
Time: 16.7s
Precision: 64
\[-1.7 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[{x}^{2} \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right) + x\]
e^{x} - 1
↓
{x}^{2} \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right) + x
double f(double x) {
        double r94780 = x;
        double r94781 = exp(r94780);
        double r94782 = 1.0;
        double r94783 = r94781 - r94782;
        return r94783;
}
↓
double f(double x) {
        double r94784 = x;
        double r94785 = 2.0;
        double r94786 = pow(r94784, r94785);
        double r94787 = 0.5;
        double r94788 = 0.16666666666666666;
        double r94789 = r94784 * r94788;
        double r94790 = r94787 + r94789;
        double r94791 = r94786 * r94790;
        double r94792 = r94791 + r94784;
        return r94792;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.6

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019199 
(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))