Average Error: 58.5 → 0.5
Time: 12.5s
Precision: 64
\[-1.700000000000000122124532708767219446599 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[x + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\]
e^{x} - 1
↓
x + \left(x \cdot x\right) \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)
double f(double x) {
        double r7310884 = x;
        double r7310885 = exp(r7310884);
        double r7310886 = 1.0;
        double r7310887 = r7310885 - r7310886;
        return r7310887;
}
↓
double f(double x) {
        double r7310888 = x;
        double r7310889 = r7310888 * r7310888;
        double r7310890 = 0.5;
        double r7310891 = 0.16666666666666666;
        double r7310892 = r7310891 * r7310888;
        double r7310893 = r7310890 + r7310892;
        double r7310894 = r7310889 * r7310893;
        double r7310895 = r7310888 + r7310894;
        return r7310895;
}

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

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

  4. Final simplification0.5

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

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))