Average Error: 58.2 → 0.6
Time: 22.7s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{{x}^{5} \cdot \frac{1}{60} + x \cdot \left(2 + \left(x \cdot x\right) \cdot \frac{1}{3}\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{{x}^{5} \cdot \frac{1}{60} + x \cdot \left(2 + \left(x \cdot x\right) \cdot \frac{1}{3}\right)}{2}
double f(double x) {
        double r1888091 = x;
        double r1888092 = exp(r1888091);
        double r1888093 = -r1888091;
        double r1888094 = exp(r1888093);
        double r1888095 = r1888092 - r1888094;
        double r1888096 = 2.0;
        double r1888097 = r1888095 / r1888096;
        return r1888097;
}
↓
double f(double x) {
        double r1888098 = x;
        double r1888099 = 5.0;
        double r1888100 = pow(r1888098, r1888099);
        double r1888101 = 0.016666666666666666;
        double r1888102 = r1888100 * r1888101;
        double r1888103 = 2.0;
        double r1888104 = r1888098 * r1888098;
        double r1888105 = 0.3333333333333333;
        double r1888106 = r1888104 * r1888105;
        double r1888107 = r1888103 + r1888106;
        double r1888108 = r1888098 * r1888107;
        double r1888109 = r1888102 + r1888108;
        double r1888110 = r1888109 / r1888103;
        return r1888110;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.2

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

  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. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019142 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))