Average Error: 58.7 → 0.5
Time: 12.0s
Precision: 64
\[-1.700000000000000122124532708767219446599 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[x + \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot \left(x \cdot x\right)\]
e^{x} - 1
↓
x + \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot \left(x \cdot x\right)
double f(double x) {
        double r5351825 = x;
        double r5351826 = exp(r5351825);
        double r5351827 = 1.0;
        double r5351828 = r5351826 - r5351827;
        return r5351828;
}
↓
double f(double x) {
        double r5351829 = x;
        double r5351830 = 0.5;
        double r5351831 = 0.16666666666666666;
        double r5351832 = r5351831 * r5351829;
        double r5351833 = r5351830 + r5351832;
        double r5351834 = r5351829 * r5351829;
        double r5351835 = r5351833 * r5351834;
        double r5351836 = r5351829 + r5351835;
        return r5351836;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.7

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019170 
(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))