Average Error: 0.0 → 0.1
Time: 37.2s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\log \left(e^{\frac{\frac{2}{\sqrt{e^{x} + e^{-x}}}}{\sqrt{e^{x} + e^{-x}}}}\right)\]
\frac{2}{e^{x} + e^{-x}}
↓
\log \left(e^{\frac{\frac{2}{\sqrt{e^{x} + e^{-x}}}}{\sqrt{e^{x} + e^{-x}}}}\right)
double f(double x) {
        double r12312625 = 2.0;
        double r12312626 = x;
        double r12312627 = exp(r12312626);
        double r12312628 = -r12312626;
        double r12312629 = exp(r12312628);
        double r12312630 = r12312627 + r12312629;
        double r12312631 = r12312625 / r12312630;
        return r12312631;
}
↓
double f(double x) {
        double r12312632 = 2.0;
        double r12312633 = x;
        double r12312634 = exp(r12312633);
        double r12312635 = -r12312633;
        double r12312636 = exp(r12312635);
        double r12312637 = r12312634 + r12312636;
        double r12312638 = sqrt(r12312637);
        double r12312639 = r12312632 / r12312638;
        double r12312640 = r12312639 / r12312638;
        double r12312641 = exp(r12312640);
        double r12312642 = log(r12312641);
        return r12312642;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

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

  3. Applied add-log-exp0.1

    \[\leadsto \color{blue}{\log \left(e^{\frac{2}{e^{x} + e^{-x}}}\right)}\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt0.9

    \[\leadsto \log \left(e^{\frac{2}{\color{blue}{\sqrt{e^{x} + e^{-x}} \cdot \sqrt{e^{x} + e^{-x}}}}}\right)\]

  6. Applied associate-/r*0.1

    \[\leadsto \log \left(e^{\color{blue}{\frac{\frac{2}{\sqrt{e^{x} + e^{-x}}}}{\sqrt{e^{x} + e^{-x}}}}}\right)\]

  7. Final simplification0.1

    \[\leadsto \log \left(e^{\frac{\frac{2}{\sqrt{e^{x} + e^{-x}}}}{\sqrt{e^{x} + e^{-x}}}}\right)\]

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))