Average Error: 58.0 → 0.6
Time: 18.5s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\frac{1}{60} \cdot {x}^{5} + \left(2 \cdot x + \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\frac{1}{60} \cdot {x}^{5} + \left(2 \cdot x + \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x\right)}{2}
double f(double x) {
        double r2979196 = x;
        double r2979197 = exp(r2979196);
        double r2979198 = -r2979196;
        double r2979199 = exp(r2979198);
        double r2979200 = r2979197 - r2979199;
        double r2979201 = 2.0;
        double r2979202 = r2979200 / r2979201;
        return r2979202;
}
↓
double f(double x) {
        double r2979203 = 0.016666666666666666;
        double r2979204 = x;
        double r2979205 = 5.0;
        double r2979206 = pow(r2979204, r2979205);
        double r2979207 = r2979203 * r2979206;
        double r2979208 = 2.0;
        double r2979209 = r2979208 * r2979204;
        double r2979210 = r2979204 * r2979204;
        double r2979211 = 0.3333333333333333;
        double r2979212 = r2979210 * r2979211;
        double r2979213 = r2979212 * r2979204;
        double r2979214 = r2979209 + r2979213;
        double r2979215 = r2979207 + r2979214;
        double r2979216 = r2979215 / r2979208;
        return r2979216;
}

Error

Bits error versus x

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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

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

  3. Simplified0.6

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

  5. Applied distribute-lft-in0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2019158 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))