Average Error: 29.9 → 0.7
Time: 15.6s
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 r94015 = x;
        double r94016 = exp(r94015);
        double r94017 = 2.0;
        double r94018 = r94016 - r94017;
        double r94019 = -r94015;
        double r94020 = exp(r94019);
        double r94021 = r94018 + r94020;
        return r94021;
}
↓
double f(double x) {
        double r94022 = x;
        double r94023 = 2.0;
        double r94024 = pow(r94022, r94023);
        double r94025 = 0.002777777777777778;
        double r94026 = 6.0;
        double r94027 = pow(r94022, r94026);
        double r94028 = r94025 * r94027;
        double r94029 = 0.08333333333333333;
        double r94030 = 4.0;
        double r94031 = pow(r94022, r94030);
        double r94032 = r94029 * r94031;
        double r94033 = r94028 + r94032;
        double r94034 = r94024 + r94033;
        return r94034;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.9

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

  2. Taylor expanded around 0 0.7

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

  3. Final simplification0.7

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

Reproduce

herbie shell --seed 2019305 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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