Average Error: 29.8 → 0.5
Time: 13.1s
Precision: 64
\[\left(e^{x} - 2\right) + e^{-x}\]
↓
\[{x}^{4} \cdot \frac{1}{12} + \left(x \cdot x + {x}^{6} \cdot \frac{1}{360}\right)\]
\left(e^{x} - 2\right) + e^{-x}
↓
{x}^{4} \cdot \frac{1}{12} + \left(x \cdot x + {x}^{6} \cdot \frac{1}{360}\right)
double f(double x) {
        double r127935 = x;
        double r127936 = exp(r127935);
        double r127937 = 2.0;
        double r127938 = r127936 - r127937;
        double r127939 = -r127935;
        double r127940 = exp(r127939);
        double r127941 = r127938 + r127940;
        return r127941;
}
↓
double f(double x) {
        double r127942 = x;
        double r127943 = 4.0;
        double r127944 = pow(r127942, r127943);
        double r127945 = 0.08333333333333333;
        double r127946 = r127944 * r127945;
        double r127947 = r127942 * r127942;
        double r127948 = 6.0;
        double r127949 = pow(r127942, r127948);
        double r127950 = 0.002777777777777778;
        double r127951 = r127949 * r127950;
        double r127952 = r127947 + r127951;
        double r127953 = r127946 + r127952;
        return r127953;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.8
Target0.0
Herbie0.5
\[4 \cdot {\left(\sinh \left(\frac{x}{2}\right)\right)}^{2}\]

Derivation

  1. Initial program 29.8

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

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019196 
(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))))