Average Error: 29.7 → 0.7
Time: 13.8s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[\frac{{x}^{4}}{12} + \left({x}^{2} + \frac{1}{360} \cdot {x}^{6}\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
\frac{{x}^{4}}{12} + \left({x}^{2} + \frac{1}{360} \cdot {x}^{6}\right)
double f(double x) {
        double r404834 = x;
        double r404835 = exp(r404834);
        double r404836 = 2.0;
        double r404837 = r404835 - r404836;
        double r404838 = -r404834;
        double r404839 = exp(r404838);
        double r404840 = r404837 + r404839;
        return r404840;
}
↓
double f(double x) {
        double r404841 = x;
        double r404842 = 4.0;
        double r404843 = pow(r404841, r404842);
        double r404844 = 12.0;
        double r404845 = r404843 / r404844;
        double r404846 = 2.0;
        double r404847 = pow(r404841, r404846);
        double r404848 = 1.0;
        double r404849 = 360.0;
        double r404850 = r404848 / r404849;
        double r404851 = 6.0;
        double r404852 = pow(r404841, r404851);
        double r404853 = r404850 * r404852;
        double r404854 = r404847 + r404853;
        double r404855 = r404845 + r404854;
        return r404855;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 29.7

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

  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. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 350497007 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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