Average Error: 29.3 → 0.6
Time: 56.8s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\left(x \cdot x + \frac{1}{12} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \frac{1}{360} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\]
\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) + \frac{1}{360} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)
double f(double x) {
        double r4751393 = x;
        double r4751394 = exp(r4751393);
        double r4751395 = 2.0;
        double r4751396 = r4751394 - r4751395;
        double r4751397 = -r4751393;
        double r4751398 = exp(r4751397);
        double r4751399 = r4751396 + r4751398;
        return r4751399;
}
↓
double f(double x) {
        double r4751400 = x;
        double r4751401 = r4751400 * r4751400;
        double r4751402 = 0.08333333333333333;
        double r4751403 = r4751401 * r4751401;
        double r4751404 = r4751402 * r4751403;
        double r4751405 = r4751401 + r4751404;
        double r4751406 = 0.002777777777777778;
        double r4751407 = r4751403 * r4751401;
        double r4751408 = r4751406 * r4751407;
        double r4751409 = r4751405 + r4751408;
        return r4751409;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.3
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.3

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

  2. Simplified29.3

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

  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

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019121 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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