Average Error: 58.0 → 0.7
Time: 4.3s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2}
double f(double x) {
        double r50250 = x;
        double r50251 = exp(r50250);
        double r50252 = -r50250;
        double r50253 = exp(r50252);
        double r50254 = r50251 - r50253;
        double r50255 = 2.0;
        double r50256 = r50254 / r50255;
        return r50256;
}
↓
double f(double x) {
        double r50257 = 0.3333333333333333;
        double r50258 = x;
        double r50259 = 3.0;
        double r50260 = pow(r50258, r50259);
        double r50261 = r50257 * r50260;
        double r50262 = 0.016666666666666666;
        double r50263 = 5.0;
        double r50264 = pow(r50258, r50263);
        double r50265 = r50262 * r50264;
        double r50266 = 2.0;
        double r50267 = r50266 * r50258;
        double r50268 = r50265 + r50267;
        double r50269 = r50261 + r50268;
        double r50270 = 2.0;
        double r50271 = r50269 / r50270;
        return r50271;
}

Error

Bits error versus x

Try it out

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 0.7

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

  3. Final simplification0.7

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

Reproduce

herbie shell --seed 2020064 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))