Average Error: 29.5 → 0.6
Time: 18.1s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\left(x \cdot x + \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \log \left(e^{\left(x \cdot x\right) \cdot \frac{1}{12}}\right) \cdot \left(x \cdot x\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
\left(x \cdot 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)\right) + \log \left(e^{\left(x \cdot x\right) \cdot \frac{1}{12}}\right) \cdot \left(x \cdot x\right)
double f(double x) {
        double r4598652 = x;
        double r4598653 = exp(r4598652);
        double r4598654 = 2.0;
        double r4598655 = r4598653 - r4598654;
        double r4598656 = -r4598652;
        double r4598657 = exp(r4598656);
        double r4598658 = r4598655 + r4598657;
        return r4598658;
}
↓
double f(double x) {
        double r4598659 = x;
        double r4598660 = r4598659 * r4598659;
        double r4598661 = r4598659 * r4598660;
        double r4598662 = 0.002777777777777778;
        double r4598663 = r4598661 * r4598662;
        double r4598664 = r4598663 * r4598661;
        double r4598665 = r4598660 + r4598664;
        double r4598666 = 0.08333333333333333;
        double r4598667 = r4598660 * r4598666;
        double r4598668 = exp(r4598667);
        double r4598669 = log(r4598668);
        double r4598670 = r4598669 * r4598660;
        double r4598671 = r4598665 + r4598670;
        return r4598671;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.5
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.5

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

  2. Simplified29.5

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

  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

    \[\leadsto \color{blue}{\left(x \cdot x + \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right)}\]
  5. Using strategy rm

  6. Applied add-log-exp0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2019174 
(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))))