Average Error: 29.5 → 0.6
Time: 40.9s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[x \cdot x + \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360} + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12}\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
x \cdot x + \left(\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} + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12}\right)
double f(double x) {
        double r2826409 = x;
        double r2826410 = exp(r2826409);
        double r2826411 = 2.0;
        double r2826412 = r2826410 - r2826411;
        double r2826413 = -r2826409;
        double r2826414 = exp(r2826413);
        double r2826415 = r2826412 + r2826414;
        return r2826415;
}
↓
double f(double x) {
        double r2826416 = x;
        double r2826417 = r2826416 * r2826416;
        double r2826418 = r2826417 * r2826417;
        double r2826419 = r2826418 * r2826417;
        double r2826420 = 0.002777777777777778;
        double r2826421 = r2826419 * r2826420;
        double r2826422 = 0.08333333333333333;
        double r2826423 = r2826418 * r2826422;
        double r2826424 = r2826421 + r2826423;
        double r2826425 = r2826417 + r2826424;
        return r2826425;
}

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

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019138 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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