Average Error: 58.0 → 0.7
Time: 1.1m
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{(\frac{1}{60} \cdot \left({x}^{5}\right) + \left(2 \cdot x + \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right))_*}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{(\frac{1}{60} \cdot \left({x}^{5}\right) + \left(2 \cdot x + \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right))_*}{2}
double f(double x) {
        double r5933026 = x;
        double r5933027 = exp(r5933026);
        double r5933028 = -r5933026;
        double r5933029 = exp(r5933028);
        double r5933030 = r5933027 - r5933029;
        double r5933031 = 2.0;
        double r5933032 = r5933030 / r5933031;
        return r5933032;
}
↓
double f(double x) {
        double r5933033 = 0.016666666666666666;
        double r5933034 = x;
        double r5933035 = 5.0;
        double r5933036 = pow(r5933034, r5933035);
        double r5933037 = 2.0;
        double r5933038 = r5933037 * r5933034;
        double r5933039 = 0.3333333333333333;
        double r5933040 = r5933034 * r5933034;
        double r5933041 = r5933039 * r5933040;
        double r5933042 = r5933041 * r5933034;
        double r5933043 = r5933038 + r5933042;
        double r5933044 = fma(r5933033, r5933036, r5933043);
        double r5933045 = r5933044 / r5933037;
        return r5933045;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.0

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

  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

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

  5. Applied fma-udef0.7

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

  6. Applied distribute-rgt-in0.7

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

  7. Final simplification0.7

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

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))