Average Error: 58.5 → 0.4
Time: 1.4m
Precision: 64
\[-0.00017 \lt x\]
\[e^{x} - 1\]
↓
\[x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)\]
e^{x} - 1
↓
x + \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) \cdot \left(x \cdot x\right)
double f(double x) {
        double r8900694 = x;
        double r8900695 = exp(r8900694);
        double r8900696 = 1.0;
        double r8900697 = r8900695 - r8900696;
        return r8900697;
}
↓
double f(double x) {
        double r8900698 = x;
        double r8900699 = 0.16666666666666666;
        double r8900700 = r8900698 * r8900699;
        double r8900701 = 0.5;
        double r8900702 = r8900700 + r8900701;
        double r8900703 = r8900698 * r8900698;
        double r8900704 = r8900702 * r8900703;
        double r8900705 = r8900698 + r8900704;
        return r8900705;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.5

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019124 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)

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

  (- (exp x) 1))