Average Error: 58.7 → 0.4
Time: 31.7s
Precision: 64
\[-0.00017 \lt x\]
\[e^{x} - 1.0\]
↓
\[x + \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot \left(x \cdot x\right)\]
e^{x} - 1.0
↓
x + \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot \left(x \cdot x\right)
double f(double x) {
        double r4993359 = x;
        double r4993360 = exp(r4993359);
        double r4993361 = 1.0;
        double r4993362 = r4993360 - r4993361;
        return r4993362;
}
↓
double f(double x) {
        double r4993363 = x;
        double r4993364 = 0.5;
        double r4993365 = 0.16666666666666666;
        double r4993366 = r4993365 * r4993363;
        double r4993367 = r4993364 + r4993366;
        double r4993368 = r4993363 * r4993363;
        double r4993369 = r4993367 * r4993368;
        double r4993370 = r4993363 + r4993369;
        return r4993370;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.7
Target0.4
Herbie0.4
\[x \cdot \left(\left(1.0 + \frac{x}{2.0}\right) + \frac{x \cdot x}{6.0}\right)\]

Derivation

  1. Initial program 58.7

    \[e^{x} - 1.0\]

  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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019165 
(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))