Average Error: 57.9 → 0.6
Time: 3.8s
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 r53926 = x;
        double r53927 = exp(r53926);
        double r53928 = -r53926;
        double r53929 = exp(r53928);
        double r53930 = r53927 - r53929;
        double r53931 = 2.0;
        double r53932 = r53930 / r53931;
        return r53932;
}
↓
double f(double x) {
        double r53933 = 0.3333333333333333;
        double r53934 = x;
        double r53935 = 3.0;
        double r53936 = pow(r53934, r53935);
        double r53937 = r53933 * r53936;
        double r53938 = 0.016666666666666666;
        double r53939 = 5.0;
        double r53940 = pow(r53934, r53939);
        double r53941 = r53938 * r53940;
        double r53942 = r53937 + r53941;
        double r53943 = 2.0;
        double r53944 = r53943 * r53934;
        double r53945 = r53942 + r53944;
        double r53946 = 2.0;
        double r53947 = r53945 / r53946;
        return r53947;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 57.9

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

  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 2020046 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))