Average Error: 29.4 → 0.7
Time: 43.2s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\left(x \cdot x + \frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right)\]
\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) + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right)
double f(double x) {
        double r5240424 = x;
        double r5240425 = exp(r5240424);
        double r5240426 = 2.0;
        double r5240427 = r5240425 - r5240426;
        double r5240428 = -r5240424;
        double r5240429 = exp(r5240428);
        double r5240430 = r5240427 + r5240429;
        return r5240430;
}
↓
double f(double x) {
        double r5240431 = x;
        double r5240432 = r5240431 * r5240431;
        double r5240433 = 0.08333333333333333;
        double r5240434 = r5240432 * r5240432;
        double r5240435 = r5240433 * r5240434;
        double r5240436 = r5240432 + r5240435;
        double r5240437 = r5240431 * r5240432;
        double r5240438 = 0.002777777777777778;
        double r5240439 = r5240437 * r5240438;
        double r5240440 = r5240437 * r5240439;
        double r5240441 = r5240436 + r5240440;
        return r5240441;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.4
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.4

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

  2. Simplified29.4

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

  3. Taylor expanded around 0 0.7

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

  4. Simplified0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2019168 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

  :herbie-target
  (* 4.0 (pow (sinh (/ x 2.0)) 2.0))

  (+ (- (exp x) 2.0) (exp (- x))))