Average Error: 58.5 → 0.5
Time: 11.4s
Precision: 64
\[-1.700000000000000122124532708767219446599 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[x + \left(\frac{1}{2} + x \cdot \frac{1}{6}\right) \cdot \left(x \cdot x\right)\]
e^{x} - 1
↓
x + \left(\frac{1}{2} + x \cdot \frac{1}{6}\right) \cdot \left(x \cdot x\right)
double f(double x) {
        double r5772462 = x;
        double r5772463 = exp(r5772462);
        double r5772464 = 1.0;
        double r5772465 = r5772463 - r5772464;
        return r5772465;
}
↓
double f(double x) {
        double r5772466 = x;
        double r5772467 = 0.5;
        double r5772468 = 0.16666666666666666;
        double r5772469 = r5772466 * r5772468;
        double r5772470 = r5772467 + r5772469;
        double r5772471 = r5772466 * r5772466;
        double r5772472 = r5772470 * r5772471;
        double r5772473 = r5772466 + r5772472;
        return r5772473;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.5

    \[e^{x} - 1\]

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019174 
(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))