Average Error: 57.8 → 0.8
Time: 38.8s
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 r7376004 = x;
        double r7376005 = exp(r7376004);
        double r7376006 = -r7376004;
        double r7376007 = exp(r7376006);
        double r7376008 = r7376005 - r7376007;
        double r7376009 = 2.0;
        double r7376010 = r7376008 / r7376009;
        return r7376010;
}
↓
double f(double x) {
        double r7376011 = 0.016666666666666666;
        double r7376012 = x;
        double r7376013 = 5.0;
        double r7376014 = pow(r7376012, r7376013);
        double r7376015 = 2.0;
        double r7376016 = r7376015 * r7376012;
        double r7376017 = 0.3333333333333333;
        double r7376018 = r7376012 * r7376012;
        double r7376019 = r7376017 * r7376018;
        double r7376020 = r7376019 * r7376012;
        double r7376021 = r7376016 + r7376020;
        double r7376022 = fma(r7376011, r7376014, r7376021);
        double r7376023 = r7376022 / r7376015;
        return r7376023;
}

Error

Bits error versus x

Derivation

  1. Initial program 57.8

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  5. Applied fma-udef0.8

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

  6. Applied distribute-lft-in0.8

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

  7. Final simplification0.8

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

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))