Average Error: 58.8 → 0.4
Time: 6.8s
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 r2047434 = x;
        double r2047435 = exp(r2047434);
        double r2047436 = 1.0;
        double r2047437 = r2047435 - r2047436;
        return r2047437;
}
↓
double f(double x) {
        double r2047438 = x;
        double r2047439 = 0.5;
        double r2047440 = 0.16666666666666666;
        double r2047441 = r2047438 * r2047440;
        double r2047442 = r2047439 + r2047441;
        double r2047443 = r2047442 * r2047438;
        double r2047444 = r2047443 * r2047438;
        double r2047445 = r2047438 + r2047444;
        return r2047445;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.8
Target0.4
Herbie0.4
\[x \cdot \left(\left(1 + \frac{x}{2}\right) + \frac{x \cdot x}{6}\right)\]

Derivation

  1. Initial program 58.8

    \[e^{x} - 1\]

  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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019130 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)

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

  (- (exp x) 1))