Average Error: 0.0 → 0.0
Time: 2.5s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\frac{2}{\mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, e^{-x}\right)}\]
\frac{2}{e^{x} + e^{-x}}
↓
\frac{2}{\mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, e^{-x}\right)}
double f(double x) {
        double r66787 = 2.0;
        double r66788 = x;
        double r66789 = exp(r66788);
        double r66790 = -r66788;
        double r66791 = exp(r66790);
        double r66792 = r66789 + r66791;
        double r66793 = r66787 / r66792;
        return r66793;
}
↓
double f(double x) {
        double r66794 = 2.0;
        double r66795 = x;
        double r66796 = exp(r66795);
        double r66797 = sqrt(r66796);
        double r66798 = -r66795;
        double r66799 = exp(r66798);
        double r66800 = fma(r66797, r66797, r66799);
        double r66801 = r66794 / r66800;
        return r66801;
}

Error

Bits error versus x

Derivation

  1. Initial program 0.0

    \[\frac{2}{e^{x} + e^{-x}}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.0

    \[\leadsto \frac{2}{\color{blue}{\sqrt{e^{x}} \cdot \sqrt{e^{x}}} + e^{-x}}\]

  4. Applied fma-def0.0

    \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, e^{-x}\right)}}\]

  5. Final simplification0.0

    \[\leadsto \frac{2}{\mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, e^{-x}\right)}\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2 (+ (exp x) (exp (- x)))))