Average Error: 58.6 → 0.5
Time: 10.5s
Precision: 64
\[-0.00017 \lt x\]
\[e^{x} - 1\]
↓
\[x + \left(\left(\frac{1}{2} + x \cdot \frac{1}{6}\right) \cdot x\right) \cdot x\]
e^{x} - 1
↓
x + \left(\left(\frac{1}{2} + x \cdot \frac{1}{6}\right) \cdot x\right) \cdot x
double f(double x) {
        double r2794421 = x;
        double r2794422 = exp(r2794421);
        double r2794423 = 1.0;
        double r2794424 = r2794422 - r2794423;
        return r2794424;
}
↓
double f(double x) {
        double r2794425 = x;
        double r2794426 = 0.5;
        double r2794427 = 0.16666666666666666;
        double r2794428 = r2794425 * r2794427;
        double r2794429 = r2794426 + r2794428;
        double r2794430 = r2794429 * r2794425;
        double r2794431 = r2794430 * r2794425;
        double r2794432 = r2794425 + r2794431;
        return r2794432;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 58.6

    \[e^{x} - 1\]

  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

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019133 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)

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

  (- (exp x) 1))