Average Error: 58.0 → 0.7
Time: 34.7s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x + \left(\left(x \cdot \frac{1}{3}\right) \cdot x\right) \cdot x\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x + \left(\left(x \cdot \frac{1}{3}\right) \cdot x\right) \cdot x\right)}{2}
double f(double x) {
        double r3772122 = x;
        double r3772123 = exp(r3772122);
        double r3772124 = -r3772122;
        double r3772125 = exp(r3772124);
        double r3772126 = r3772123 - r3772125;
        double r3772127 = 2.0;
        double r3772128 = r3772126 / r3772127;
        return r3772128;
}
↓
double f(double x) {
        double r3772129 = 0.016666666666666666;
        double r3772130 = x;
        double r3772131 = 5.0;
        double r3772132 = pow(r3772130, r3772131);
        double r3772133 = 2.0;
        double r3772134 = r3772133 * r3772130;
        double r3772135 = 0.3333333333333333;
        double r3772136 = r3772130 * r3772135;
        double r3772137 = r3772136 * r3772130;
        double r3772138 = r3772137 * r3772130;
        double r3772139 = r3772134 + r3772138;
        double r3772140 = fma(r3772129, r3772132, r3772139);
        double r3772141 = 2.0;
        double r3772142 = r3772140 / r3772141;
        return r3772142;
}

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

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

  5. Applied fma-udef0.7

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

  6. Applied distribute-lft-in0.7

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

  7. Final simplification0.7

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2.0))