Average Error: 30.5 → 0.6
Time: 42.3s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\frac{1}{360} \cdot \log \left(e^{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right) + \left(x \cdot x + \frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
\frac{1}{360} \cdot \log \left(e^{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \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 r6444624 = x;
        double r6444625 = exp(r6444624);
        double r6444626 = 2.0;
        double r6444627 = r6444625 - r6444626;
        double r6444628 = -r6444624;
        double r6444629 = exp(r6444628);
        double r6444630 = r6444627 + r6444629;
        return r6444630;
}
↓
double f(double x) {
        double r6444631 = 0.002777777777777778;
        double r6444632 = x;
        double r6444633 = r6444632 * r6444632;
        double r6444634 = r6444633 * r6444633;
        double r6444635 = r6444634 * r6444633;
        double r6444636 = exp(r6444635);
        double r6444637 = log(r6444636);
        double r6444638 = r6444631 * r6444637;
        double r6444639 = 0.08333333333333333;
        double r6444640 = r6444639 * r6444634;
        double r6444641 = r6444633 + r6444640;
        double r6444642 = r6444638 + r6444641;
        return r6444642;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.5
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 30.5

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

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  5. Applied add-log-exp0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2019104 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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