Average Error: 0.0 → 0.4
Time: 16.3s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{2}{{x}^{2} + \left(\frac{1}{12} \cdot {x}^{4} + 2\right)}}\]
\frac{2}{e^{x} + e^{-x}}
↓
\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{2}{{x}^{2} + \left(\frac{1}{12} \cdot {x}^{4} + 2\right)}}
double f(double x) {
        double r67512 = 2.0;
        double r67513 = x;
        double r67514 = exp(r67513);
        double r67515 = -r67513;
        double r67516 = exp(r67515);
        double r67517 = r67514 + r67516;
        double r67518 = r67512 / r67517;
        return r67518;
}
↓
double f(double x) {
        double r67519 = 2.0;
        double r67520 = x;
        double r67521 = exp(r67520);
        double r67522 = -r67520;
        double r67523 = exp(r67522);
        double r67524 = r67521 + r67523;
        double r67525 = r67519 / r67524;
        double r67526 = sqrt(r67525);
        double r67527 = 2.0;
        double r67528 = pow(r67520, r67527);
        double r67529 = 0.08333333333333333;
        double r67530 = 4.0;
        double r67531 = pow(r67520, r67530);
        double r67532 = r67529 * r67531;
        double r67533 = r67532 + r67527;
        double r67534 = r67528 + r67533;
        double r67535 = r67519 / r67534;
        double r67536 = sqrt(r67535);
        double r67537 = r67526 * r67536;
        return r67537;
}

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-sqr-sqrt0.0

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

  4. Taylor expanded around 0 0.4

    \[\leadsto \sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{2}{\color{blue}{{x}^{2} + \left(\frac{1}{12} \cdot {x}^{4} + 2\right)}}}\]

  5. Final simplification0.4

    \[\leadsto \sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{2}{{x}^{2} + \left(\frac{1}{12} \cdot {x}^{4} + 2\right)}}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2 (+ (exp x) (exp (- x)))))