Average Error: 58.0 → 0.6
Time: 6.1s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right) + 2 \cdot x}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right) + 2 \cdot x}{2}
double f(double x) {
        double r52154 = x;
        double r52155 = exp(r52154);
        double r52156 = -r52154;
        double r52157 = exp(r52156);
        double r52158 = r52155 - r52157;
        double r52159 = 2.0;
        double r52160 = r52158 / r52159;
        return r52160;
}
↓
double f(double x) {
        double r52161 = 0.3333333333333333;
        double r52162 = x;
        double r52163 = 3.0;
        double r52164 = pow(r52162, r52163);
        double r52165 = r52161 * r52164;
        double r52166 = 0.016666666666666666;
        double r52167 = 5.0;
        double r52168 = pow(r52162, r52167);
        double r52169 = r52166 * r52168;
        double r52170 = r52165 + r52169;
        double r52171 = 2.0;
        double r52172 = r52171 * r52162;
        double r52173 = r52170 + r52172;
        double r52174 = 2.0;
        double r52175 = r52173 / r52174;
        return r52175;
}

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 0.6

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

  4. Applied associate-+r+0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019304 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))