Average Error: 58.8 → 0.4
Time: 11.4s
Precision: 64
\[-0.00017 \lt x\]
\[e^{x} - 1\]
↓
\[x + \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot \left(x \cdot x\right)\]
e^{x} - 1
↓
x + \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot \left(x \cdot x\right)
double f(double x) {
        double r2875814 = x;
        double r2875815 = exp(r2875814);
        double r2875816 = 1.0;
        double r2875817 = r2875815 - r2875816;
        return r2875817;
}
↓
double f(double x) {
        double r2875818 = x;
        double r2875819 = 0.5;
        double r2875820 = 0.16666666666666666;
        double r2875821 = r2875820 * r2875818;
        double r2875822 = r2875819 + r2875821;
        double r2875823 = r2875818 * r2875818;
        double r2875824 = r2875822 * r2875823;
        double r2875825 = r2875818 + r2875824;
        return r2875825;
}

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

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

  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 2019144 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :pre (< -0.00017 x)

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

  (- (exp x) 1))