Average Error: 0.0 → 0.4
Time: 7.1s
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 r69088 = 2.0;
        double r69089 = x;
        double r69090 = exp(r69089);
        double r69091 = -r69089;
        double r69092 = exp(r69091);
        double r69093 = r69090 + r69092;
        double r69094 = r69088 / r69093;
        return r69094;
}
↓
double f(double x) {
        double r69095 = 2.0;
        double r69096 = x;
        double r69097 = exp(r69096);
        double r69098 = -r69096;
        double r69099 = exp(r69098);
        double r69100 = r69097 + r69099;
        double r69101 = r69095 / r69100;
        double r69102 = sqrt(r69101);
        double r69103 = 2.0;
        double r69104 = pow(r69096, r69103);
        double r69105 = 0.08333333333333333;
        double r69106 = 4.0;
        double r69107 = pow(r69096, r69106);
        double r69108 = r69105 * r69107;
        double r69109 = r69108 + r69103;
        double r69110 = r69104 + r69109;
        double r69111 = r69095 / r69110;
        double r69112 = sqrt(r69111);
        double r69113 = r69102 * r69112;
        return r69113;
}

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)))))