Average Error: 30.1 → 0.7
Time: 27.1s
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 r129810 = x;
        double r129811 = exp(r129810);
        double r129812 = 2.0;
        double r129813 = r129811 - r129812;
        double r129814 = -r129810;
        double r129815 = exp(r129814);
        double r129816 = r129813 + r129815;
        return r129816;
}
↓
double f(double x) {
        double r129817 = x;
        double r129818 = 4.0;
        double r129819 = pow(r129817, r129818);
        double r129820 = 12.0;
        double r129821 = r129819 / r129820;
        double r129822 = 2.0;
        double r129823 = pow(r129817, r129822);
        double r129824 = 1.0;
        double r129825 = 360.0;
        double r129826 = r129824 / r129825;
        double r129827 = 6.0;
        double r129828 = pow(r129817, r129827);
        double r129829 = r129826 * r129828;
        double r129830 = r129823 + r129829;
        double r129831 = r129821 + r129830;
        return r129831;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.1
Target0.0
Herbie0.7
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 30.1

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

  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 2019303 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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