Average Error: 29.4 → 0.8
Time: 8.5s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
{x}^{2} + \left(\frac{1}{360} \cdot {x}^{6} + \frac{1}{12} \cdot {x}^{4}\right)
double f(double x) {
        double r94465 = x;
        double r94466 = exp(r94465);
        double r94467 = 2.0;
        double r94468 = r94466 - r94467;
        double r94469 = -r94465;
        double r94470 = exp(r94469);
        double r94471 = r94468 + r94470;
        return r94471;
}
↓
double f(double x) {
        double r94472 = x;
        double r94473 = 2.0;
        double r94474 = pow(r94472, r94473);
        double r94475 = 0.002777777777777778;
        double r94476 = 6.0;
        double r94477 = pow(r94472, r94476);
        double r94478 = r94475 * r94477;
        double r94479 = 0.08333333333333333;
        double r94480 = 4.0;
        double r94481 = pow(r94472, r94480);
        double r94482 = r94479 * r94481;
        double r94483 = r94478 + r94482;
        double r94484 = r94474 + r94483;
        return r94484;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.4
Target0.0
Herbie0.8
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.4

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

  2. Taylor expanded around 0 0.8

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

  3. Final simplification0.8

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

Reproduce

herbie shell --seed 197574269 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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