Average Error: 29.9 → 0.6
Time: 26.2s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x + \frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) + \left(x \cdot x + \frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)
double f(double x) {
        double r3888428 = x;
        double r3888429 = exp(r3888428);
        double r3888430 = 2.0;
        double r3888431 = r3888429 - r3888430;
        double r3888432 = -r3888428;
        double r3888433 = exp(r3888432);
        double r3888434 = r3888431 + r3888433;
        return r3888434;
}
↓
double f(double x) {
        double r3888435 = x;
        double r3888436 = r3888435 * r3888435;
        double r3888437 = r3888435 * r3888436;
        double r3888438 = 0.002777777777777778;
        double r3888439 = r3888437 * r3888438;
        double r3888440 = r3888439 * r3888437;
        double r3888441 = 0.08333333333333333;
        double r3888442 = r3888436 * r3888436;
        double r3888443 = r3888441 * r3888442;
        double r3888444 = r3888436 + r3888443;
        double r3888445 = r3888440 + r3888444;
        return r3888445;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.9

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

  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

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019164 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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