Average Error: 58.0 → 0.7
Time: 4.1s
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 r32316 = x;
        double r32317 = exp(r32316);
        double r32318 = -r32316;
        double r32319 = exp(r32318);
        double r32320 = r32317 - r32319;
        double r32321 = 2.0;
        double r32322 = r32320 / r32321;
        return r32322;
}
↓
double f(double x) {
        double r32323 = 0.3333333333333333;
        double r32324 = x;
        double r32325 = 3.0;
        double r32326 = pow(r32324, r32325);
        double r32327 = r32323 * r32326;
        double r32328 = 0.016666666666666666;
        double r32329 = 5.0;
        double r32330 = pow(r32324, r32329);
        double r32331 = r32328 * r32330;
        double r32332 = 2.0;
        double r32333 = r32332 * r32324;
        double r32334 = r32331 + r32333;
        double r32335 = r32327 + r32334;
        double r32336 = 2.0;
        double r32337 = r32335 / r32336;
        return r32337;
}

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 2020047 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))