Average Error: 57.9 → 0.7
Time: 17.8s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\frac{\left(8 + \left(\left(x \cdot x\right) \cdot \frac{1}{27}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot x}{\left(\left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) - 2 \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right)\right) + 4} + {x}^{5} \cdot \frac{1}{60}}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\frac{\left(8 + \left(\left(x \cdot x\right) \cdot \frac{1}{27}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot x}{\left(\left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) - 2 \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right)\right) + 4} + {x}^{5} \cdot \frac{1}{60}}{2}
double f(double x) {
        double r1917802 = x;
        double r1917803 = exp(r1917802);
        double r1917804 = -r1917802;
        double r1917805 = exp(r1917804);
        double r1917806 = r1917803 - r1917805;
        double r1917807 = 2.0;
        double r1917808 = r1917806 / r1917807;
        return r1917808;
}
↓
double f(double x) {
        double r1917809 = 8.0;
        double r1917810 = x;
        double r1917811 = r1917810 * r1917810;
        double r1917812 = 0.037037037037037035;
        double r1917813 = r1917811 * r1917812;
        double r1917814 = r1917811 * r1917811;
        double r1917815 = r1917813 * r1917814;
        double r1917816 = r1917809 + r1917815;
        double r1917817 = r1917816 * r1917810;
        double r1917818 = 0.3333333333333333;
        double r1917819 = r1917811 * r1917818;
        double r1917820 = r1917819 * r1917819;
        double r1917821 = 2.0;
        double r1917822 = r1917821 * r1917819;
        double r1917823 = r1917820 - r1917822;
        double r1917824 = 4.0;
        double r1917825 = r1917823 + r1917824;
        double r1917826 = r1917817 / r1917825;
        double r1917827 = 5.0;
        double r1917828 = pow(r1917810, r1917827);
        double r1917829 = 0.016666666666666666;
        double r1917830 = r1917828 * r1917829;
        double r1917831 = r1917826 + r1917830;
        double r1917832 = r1917831 / r1917821;
        return r1917832;
}

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 0.7

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

  3. Simplified0.7

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

  5. Applied flip3-+0.7

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

  6. Applied associate-*l/0.7

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

  7. Simplified0.7

    \[\leadsto \frac{\frac{\color{blue}{\left(8 + \left(\frac{1}{27} \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot x}}{2 \cdot 2 + \left(\left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) - 2 \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right)\right)} + \frac{1}{60} \cdot {x}^{5}}{2}\]

  8. Final simplification0.7

    \[\leadsto \frac{\frac{\left(8 + \left(\left(x \cdot x\right) \cdot \frac{1}{27}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot x}{\left(\left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) - 2 \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right)\right) + 4} + {x}^{5} \cdot \frac{1}{60}}{2}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))