Average Error: 58.7 → 0.4
Time: 8.8s
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 r2965807 = x;
        double r2965808 = exp(r2965807);
        double r2965809 = 1.0;
        double r2965810 = r2965808 - r2965809;
        return r2965810;
}
↓
double f(double x) {
        double r2965811 = x;
        double r2965812 = 0.5;
        double r2965813 = 0.16666666666666666;
        double r2965814 = r2965811 * r2965813;
        double r2965815 = r2965812 + r2965814;
        double r2965816 = r2965815 * r2965811;
        double r2965817 = r2965816 * r2965811;
        double r2965818 = r2965811 + r2965817;
        return r2965818;
}

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

Derivation

  1. Initial program 58.7

    \[e^{x} - 1\]

  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

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

  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 2019134 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)

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

  (- (exp x) 1))