Average Error: 58.7 → 0.4
Time: 14.0s
Precision: 64
\[-1.700000000000000122124532708767219446599 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[x + \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot {x}^{2}\]
e^{x} - 1
↓
x + \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot {x}^{2}
double f(double x) {
        double r116736 = x;
        double r116737 = exp(r116736);
        double r116738 = 1.0;
        double r116739 = r116737 - r116738;
        return r116739;
}
↓
double f(double x) {
        double r116740 = x;
        double r116741 = 0.5;
        double r116742 = 0.16666666666666666;
        double r116743 = r116742 * r116740;
        double r116744 = r116741 + r116743;
        double r116745 = 2.0;
        double r116746 = pow(r116740, r116745);
        double r116747 = r116744 * r116746;
        double r116748 = r116740 + r116747;
        return r116748;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.7
Target0.5
Herbie0.4
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right)\]

Derivation

  1. Initial program 58.7

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019323 
(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))