Average Error: 29.6 → 0.8
Time: 47.2s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\left(x \cdot x + \frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1}{360} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
\left(x \cdot x + \frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1}{360} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)
double f(double x) {
        double r6160705 = x;
        double r6160706 = exp(r6160705);
        double r6160707 = 2.0;
        double r6160708 = r6160706 - r6160707;
        double r6160709 = -r6160705;
        double r6160710 = exp(r6160709);
        double r6160711 = r6160708 + r6160710;
        return r6160711;
}
↓
double f(double x) {
        double r6160712 = x;
        double r6160713 = r6160712 * r6160712;
        double r6160714 = 0.08333333333333333;
        double r6160715 = r6160713 * r6160713;
        double r6160716 = r6160714 * r6160715;
        double r6160717 = r6160713 + r6160716;
        double r6160718 = 0.002777777777777778;
        double r6160719 = r6160715 * r6160713;
        double r6160720 = r6160718 * r6160719;
        double r6160721 = r6160717 + r6160720;
        return r6160721;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.6
Target0.0
Herbie0.8
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.6

    \[\left(e^{x} - 2\right) + e^{-x}\]

  2. Taylor expanded around 0 0.8

    \[\leadsto \color{blue}{{x}^{2} + \left(\frac{1}{12} \cdot {x}^{4} + \frac{1}{360} \cdot {x}^{6}\right)}\]

  3. Simplified0.8

    \[\leadsto \color{blue}{\left(x \cdot x + \frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{360}}\]

  4. Final simplification0.8

    \[\leadsto \left(x \cdot x + \frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1}{360} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\]

Reproduce

herbie shell --seed 2019124 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

  :herbie-target
  (* 4 (pow (sinh (/ x 2)) 2))

  (+ (- (exp x) 2) (exp (- x))))