Average Error: 29.2 → 0.8
Time: 13.0s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[x \cdot x + \left({x}^{6} \cdot \frac{1}{360} + {x}^{4} \cdot \frac{1}{12}\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
x \cdot x + \left({x}^{6} \cdot \frac{1}{360} + {x}^{4} \cdot \frac{1}{12}\right)
double f(double x) {
        double r3954884 = x;
        double r3954885 = exp(r3954884);
        double r3954886 = 2.0;
        double r3954887 = r3954885 - r3954886;
        double r3954888 = -r3954884;
        double r3954889 = exp(r3954888);
        double r3954890 = r3954887 + r3954889;
        return r3954890;
}
↓
double f(double x) {
        double r3954891 = x;
        double r3954892 = r3954891 * r3954891;
        double r3954893 = 6.0;
        double r3954894 = pow(r3954891, r3954893);
        double r3954895 = 0.002777777777777778;
        double r3954896 = r3954894 * r3954895;
        double r3954897 = 4.0;
        double r3954898 = pow(r3954891, r3954897);
        double r3954899 = 0.08333333333333333;
        double r3954900 = r3954898 * r3954899;
        double r3954901 = r3954896 + r3954900;
        double r3954902 = r3954892 + r3954901;
        return r3954902;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.2
Target0.0
Herbie0.8
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.2

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019134 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"

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

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