Average Error: 30.3 → 0.5
Time: 19.2s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[x \cdot x + \left({x}^{6} \cdot \frac{1}{360} + \frac{1}{12} \cdot {x}^{4}\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
x \cdot x + \left({x}^{6} \cdot \frac{1}{360} + \frac{1}{12} \cdot {x}^{4}\right)
double f(double x) {
        double r98003 = x;
        double r98004 = exp(r98003);
        double r98005 = 2.0;
        double r98006 = r98004 - r98005;
        double r98007 = -r98003;
        double r98008 = exp(r98007);
        double r98009 = r98006 + r98008;
        return r98009;
}
↓
double f(double x) {
        double r98010 = x;
        double r98011 = r98010 * r98010;
        double r98012 = 6.0;
        double r98013 = pow(r98010, r98012);
        double r98014 = 0.002777777777777778;
        double r98015 = r98013 * r98014;
        double r98016 = 0.08333333333333333;
        double r98017 = 4.0;
        double r98018 = pow(r98010, r98017);
        double r98019 = r98016 * r98018;
        double r98020 = r98015 + r98019;
        double r98021 = r98011 + r98020;
        return r98021;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.3
Target0.0
Herbie0.5
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 30.3

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

  2. Simplified30.3

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

  3. Taylor expanded around 0 0.5

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

  4. Simplified0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2019194 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

  :herbie-target
  (* 4.0 (pow (sinh (/ x 2.0)) 2.0))

  (+ (- (exp x) 2.0) (exp (- x))))