Average Error: 29.4 → 0.5
Time: 13.9s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\left(x \cdot x + \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{360}\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12}\]
\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) + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{12}
double f(double x) {
        double r6355404 = x;
        double r6355405 = exp(r6355404);
        double r6355406 = 2.0;
        double r6355407 = r6355405 - r6355406;
        double r6355408 = -r6355404;
        double r6355409 = exp(r6355408);
        double r6355410 = r6355407 + r6355409;
        return r6355410;
}
↓
double f(double x) {
        double r6355411 = x;
        double r6355412 = r6355411 * r6355411;
        double r6355413 = r6355411 * r6355412;
        double r6355414 = 0.002777777777777778;
        double r6355415 = r6355413 * r6355414;
        double r6355416 = r6355415 * r6355413;
        double r6355417 = r6355412 + r6355416;
        double r6355418 = r6355412 * r6355412;
        double r6355419 = 0.08333333333333333;
        double r6355420 = r6355418 * r6355419;
        double r6355421 = r6355417 + r6355420;
        return r6355421;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 29.4

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

  2. Simplified29.4

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

  3. Taylor expanded around 0 0.5

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

  4. Simplified0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2019171 
(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))))