Average Error: 58.0 → 0.7
Time: 10.0s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{{x}^{5} \cdot \frac{1}{60} + \left(2 + \frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot x}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{{x}^{5} \cdot \frac{1}{60} + \left(2 + \frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot x}{2}
double f(double x) {
        double r2265074 = x;
        double r2265075 = exp(r2265074);
        double r2265076 = -r2265074;
        double r2265077 = exp(r2265076);
        double r2265078 = r2265075 - r2265077;
        double r2265079 = 2.0;
        double r2265080 = r2265078 / r2265079;
        return r2265080;
}
↓
double f(double x) {
        double r2265081 = x;
        double r2265082 = 5.0;
        double r2265083 = pow(r2265081, r2265082);
        double r2265084 = 0.016666666666666666;
        double r2265085 = r2265083 * r2265084;
        double r2265086 = 2.0;
        double r2265087 = 0.3333333333333333;
        double r2265088 = r2265081 * r2265081;
        double r2265089 = r2265087 * r2265088;
        double r2265090 = r2265086 + r2265089;
        double r2265091 = r2265090 * r2265081;
        double r2265092 = r2265085 + r2265091;
        double r2265093 = r2265092 / r2265086;
        return r2265093;
}

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

    \[\frac{e^{x} - e^{-x}}{2}\]

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019133 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))