Average Error: 58.0 → 0.6
Time: 35.2s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2.0}\]
↓
\[\frac{x \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right) + 2\right) + {x}^{5} \cdot \frac{1}{60}}{2.0}\]
\frac{e^{x} - e^{-x}}{2.0}
↓
\frac{x \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right) + 2\right) + {x}^{5} \cdot \frac{1}{60}}{2.0}
double f(double x) {
        double r2246968 = x;
        double r2246969 = exp(r2246968);
        double r2246970 = -r2246968;
        double r2246971 = exp(r2246970);
        double r2246972 = r2246969 - r2246971;
        double r2246973 = 2.0;
        double r2246974 = r2246972 / r2246973;
        return r2246974;
}
↓
double f(double x) {
        double r2246975 = x;
        double r2246976 = 0.3333333333333333;
        double r2246977 = r2246975 * r2246975;
        double r2246978 = r2246976 * r2246977;
        double r2246979 = 2.0;
        double r2246980 = r2246978 + r2246979;
        double r2246981 = r2246975 * r2246980;
        double r2246982 = 5.0;
        double r2246983 = pow(r2246975, r2246982);
        double r2246984 = 0.016666666666666666;
        double r2246985 = r2246983 * r2246984;
        double r2246986 = r2246981 + r2246985;
        double r2246987 = 2.0;
        double r2246988 = r2246986 / r2246987;
        return r2246988;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.0

    \[\frac{e^{x} - e^{-x}}{2.0}\]

  2. Taylor expanded around 0 0.6

    \[\leadsto \frac{\color{blue}{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right)}}{2.0}\]

  3. Simplified0.6

    \[\leadsto \frac{\color{blue}{\frac{1}{60} \cdot {x}^{5} + x \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right) + 2\right)}}{2.0}\]

  4. Final simplification0.6

    \[\leadsto \frac{x \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right) + 2\right) + {x}^{5} \cdot \frac{1}{60}}{2.0}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2.0))