Average Error: 58.1 → 0.6
Time: 6.7s
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 r53096 = x;
        double r53097 = exp(r53096);
        double r53098 = -r53096;
        double r53099 = exp(r53098);
        double r53100 = r53097 - r53099;
        double r53101 = 2.0;
        double r53102 = r53100 / r53101;
        return r53102;
}
↓
double f(double x) {
        double r53103 = 0.3333333333333333;
        double r53104 = x;
        double r53105 = 3.0;
        double r53106 = pow(r53104, r53105);
        double r53107 = r53103 * r53106;
        double r53108 = 0.016666666666666666;
        double r53109 = 5.0;
        double r53110 = pow(r53104, r53109);
        double r53111 = r53108 * r53110;
        double r53112 = r53107 + r53111;
        double r53113 = 2.0;
        double r53114 = r53113 * r53104;
        double r53115 = r53112 + r53114;
        double r53116 = 2.0;
        double r53117 = r53115 / r53116;
        return r53117;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.1

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

  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 2019308 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))