Average Error: 29.2 → 0.6
Time: 35.3s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[x \cdot x + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360} + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12}\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
x \cdot x + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360} + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12}\right)
double f(double x) {
        double r3664704 = x;
        double r3664705 = exp(r3664704);
        double r3664706 = 2.0;
        double r3664707 = r3664705 - r3664706;
        double r3664708 = -r3664704;
        double r3664709 = exp(r3664708);
        double r3664710 = r3664707 + r3664709;
        return r3664710;
}
↓
double f(double x) {
        double r3664711 = x;
        double r3664712 = r3664711 * r3664711;
        double r3664713 = r3664712 * r3664712;
        double r3664714 = r3664713 * r3664712;
        double r3664715 = 0.002777777777777778;
        double r3664716 = r3664714 * r3664715;
        double r3664717 = 0.08333333333333333;
        double r3664718 = r3664713 * r3664717;
        double r3664719 = r3664716 + r3664718;
        double r3664720 = r3664712 + r3664719;
        return r3664720;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 29.2

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

  2. Simplified29.2

    \[\leadsto \color{blue}{\left(e^{x} - 2\right) - \frac{-1}{e^{x}}}\]

  3. Taylor expanded around 0 0.6

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

  4. Simplified0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019139 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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