Average Error: 29.4 → 0.6
Time: 16.7s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{360} + \frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + x \cdot x\]
\left(e^{x} - 2\right) + e^{-x}
↓
\left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{360} + \frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + x \cdot x
double f(double x) {
        double r1511440 = x;
        double r1511441 = exp(r1511440);
        double r1511442 = 2.0;
        double r1511443 = r1511441 - r1511442;
        double r1511444 = -r1511440;
        double r1511445 = exp(r1511444);
        double r1511446 = r1511443 + r1511445;
        return r1511446;
}
↓
double f(double x) {
        double r1511447 = x;
        double r1511448 = r1511447 * r1511447;
        double r1511449 = r1511447 * r1511448;
        double r1511450 = r1511449 * r1511449;
        double r1511451 = 0.002777777777777778;
        double r1511452 = r1511450 * r1511451;
        double r1511453 = 0.08333333333333333;
        double r1511454 = r1511448 * r1511448;
        double r1511455 = r1511453 * r1511454;
        double r1511456 = r1511452 + r1511455;
        double r1511457 = r1511456 + r1511448;
        return r1511457;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 29.4

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019128 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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