Average Error: 29.9 → 0.6
Time: 6.0s
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 r121068 = x;
        double r121069 = exp(r121068);
        double r121070 = 2.0;
        double r121071 = r121069 - r121070;
        double r121072 = -r121068;
        double r121073 = exp(r121072);
        double r121074 = r121071 + r121073;
        return r121074;
}
↓
double f(double x) {
        double r121075 = x;
        double r121076 = 2.0;
        double r121077 = pow(r121075, r121076);
        double r121078 = 0.002777777777777778;
        double r121079 = 6.0;
        double r121080 = pow(r121075, r121079);
        double r121081 = r121078 * r121080;
        double r121082 = 0.08333333333333333;
        double r121083 = 4.0;
        double r121084 = pow(r121075, r121083);
        double r121085 = r121082 * r121084;
        double r121086 = r121081 + r121085;
        double r121087 = r121077 + r121086;
        return r121087;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.0
Herbie0.6
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.9

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

  2. Taylor expanded around 0 0.6

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

  3. Final simplification0.6

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

Reproduce

herbie shell --seed 2019346 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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