Average Error: 58.7 → 0.5
Time: 9.0s
Precision: 64
\[-1.700000000000000122124532708767219446599 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[{x}^{2} \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right) + x\]
e^{x} - 1
↓
{x}^{2} \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right) + x
double f(double x) {
        double r80216 = x;
        double r80217 = exp(r80216);
        double r80218 = 1.0;
        double r80219 = r80217 - r80218;
        return r80219;
}
↓
double f(double x) {
        double r80220 = x;
        double r80221 = 2.0;
        double r80222 = pow(r80220, r80221);
        double r80223 = 0.16666666666666666;
        double r80224 = r80223 * r80220;
        double r80225 = 0.5;
        double r80226 = r80224 + r80225;
        double r80227 = r80222 * r80226;
        double r80228 = r80227 + r80220;
        return r80228;
}

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

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019306 
(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))