Average Error: 58.6 → 0.4
Time: 9.1s
Precision: 64
\[-1.700000000000000122124532708767219446599 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[\left(x + \frac{{x}^{3}}{6}\right) - \frac{{x}^{2}}{-2}\]
e^{x} - 1
↓
\left(x + \frac{{x}^{3}}{6}\right) - \frac{{x}^{2}}{-2}
double f(double x) {
        double r387833 = x;
        double r387834 = exp(r387833);
        double r387835 = 1.0;
        double r387836 = r387834 - r387835;
        return r387836;
}
↓
double f(double x) {
        double r387837 = x;
        double r387838 = 3.0;
        double r387839 = pow(r387837, r387838);
        double r387840 = 6.0;
        double r387841 = r387839 / r387840;
        double r387842 = r387837 + r387841;
        double r387843 = 2.0;
        double r387844 = pow(r387837, r387843);
        double r387845 = -r387843;
        double r387846 = r387844 / r387845;
        double r387847 = r387842 - r387846;
        return r387847;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.6

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.4

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

  3. Simplified0.4

    \[\leadsto \color{blue}{\left(\frac{{x}^{2}}{2} + x\right) + \frac{{x}^{3}}{6}}\]
  4. Using strategy rm

  5. Applied frac-2neg0.4

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

  6. Final simplification0.4

    \[\leadsto \left(x + \frac{{x}^{3}}{6}\right) - \frac{{x}^{2}}{-2}\]

Reproduce

herbie shell --seed 350497007 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :precision binary64
  :pre (< -1.7e-4 x)

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

  (- (exp x) 1))