Average Error: 58.0 → 0.7
Time: 15.6s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, x \cdot 2 + x \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right)\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, x \cdot 2 + x \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right)\right)}{2}
double f(double x) {
        double r3088123 = x;
        double r3088124 = exp(r3088123);
        double r3088125 = -r3088123;
        double r3088126 = exp(r3088125);
        double r3088127 = r3088124 - r3088126;
        double r3088128 = 2.0;
        double r3088129 = r3088127 / r3088128;
        return r3088129;
}
↓
double f(double x) {
        double r3088130 = x;
        double r3088131 = 5.0;
        double r3088132 = pow(r3088130, r3088131);
        double r3088133 = 0.016666666666666666;
        double r3088134 = 2.0;
        double r3088135 = r3088130 * r3088134;
        double r3088136 = 0.3333333333333333;
        double r3088137 = r3088130 * r3088130;
        double r3088138 = r3088136 * r3088137;
        double r3088139 = r3088130 * r3088138;
        double r3088140 = r3088135 + r3088139;
        double r3088141 = fma(r3088132, r3088133, r3088140);
        double r3088142 = 2.0;
        double r3088143 = r3088141 / r3088142;
        return r3088143;
}

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

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

  5. Applied fma-udef0.7

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

  6. Applied distribute-lft-in0.7

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

  7. Final simplification0.7

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2.0))