Average Error: 58.7 → 0.4
Time: 11.4s
Precision: 64
\[-1.700000000000000122124532708767219446599 \cdot 10^{-4} \lt x\]
\[e^{x} - 1\]
↓
\[x + {x}^{2} \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)\]
e^{x} - 1
↓
x + {x}^{2} \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right)
double f(double x) {
        double r92737 = x;
        double r92738 = exp(r92737);
        double r92739 = 1.0;
        double r92740 = r92738 - r92739;
        return r92740;
}
↓
double f(double x) {
        double r92741 = x;
        double r92742 = 2.0;
        double r92743 = pow(r92741, r92742);
        double r92744 = 0.5;
        double r92745 = 0.16666666666666666;
        double r92746 = r92741 * r92745;
        double r92747 = r92744 + r92746;
        double r92748 = r92743 * r92747;
        double r92749 = r92741 + r92748;
        return r92749;
}

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

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019179 
(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))