Average Error: 58.7 → 0.3
Time: 2.7s
Precision: 64
\[-1.7 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x\]
e^{x} - 1
↓
{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x
double f(double x) {
        double r121853 = x;
        double r121854 = exp(r121853);
        double r121855 = 1.0;
        double r121856 = r121854 - r121855;
        return r121856;
}
↓
double f(double x) {
        double r121857 = x;
        double r121858 = 2.0;
        double r121859 = pow(r121857, r121858);
        double r121860 = 0.16666666666666666;
        double r121861 = r121857 * r121860;
        double r121862 = 0.5;
        double r121863 = r121861 + r121862;
        double r121864 = r121859 * r121863;
        double r121865 = r121864 + r121857;
        return r121865;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.7
Target0.4
Herbie0.3
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right)\]

Derivation

  1. Initial program 58.7

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2020056 
(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))