Hyperbolic secant

Average Error: 0.0 → 0.1

Time: 15.3s

Precision: 64

Math

\[\frac{2}{e^{x} + e^{-x}}\]

↓

\[\sqrt[3]{{\left(\frac{2}{e^{x} + e^{-x}}\right)}^{3}}\]

C

double f(double x) {
        double r53120 = 2.0;
        double r53121 = x;
        double r53122 = exp(r53121);
        double r53123 = -r53121;
        double r53124 = exp(r53123);
        double r53125 = r53122 + r53124;
        double r53126 = r53120 / r53125;
        return r53126;
}

↓

double f(double x) {
        double r53127 = 2.0;
        double r53128 = x;
        double r53129 = exp(r53128);
        double r53130 = -r53128;
        double r53131 = exp(r53130);
        double r53132 = r53129 + r53131;
        double r53133 = r53127 / r53132;
        double r53134 = 3.0;
        double r53135 = pow(r53133, r53134);
        double r53136 = cbrt(r53135);
        return r53136;
}

Error

Bits error versus x

Derivation

1. Initial program 0.0

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

3. Applied add-cbrt-cube 0.1

   \[\leadsto \frac{2}{\color{blue}{\sqrt[3]{\left(\left(e^{x} + e^{-x}\right) \cdot \left(e^{x} + e^{-x}\right)\right) \cdot \left(e^{x} + e^{-x}\right)}}}\]

4. Applied add-cbrt-cube 0.1

   \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(2 \cdot 2\right) \cdot 2}}}{\sqrt[3]{\left(\left(e^{x} + e^{-x}\right) \cdot \left(e^{x} + e^{-x}\right)\right) \cdot \left(e^{x} + e^{-x}\right)}}\]

5. Applied cbrt-undiv 0.1

   \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(2 \cdot 2\right) \cdot 2}{\left(\left(e^{x} + e^{-x}\right) \cdot \left(e^{x} + e^{-x}\right)\right) \cdot \left(e^{x} + e^{-x}\right)}}}\]

6. Simplified 0.1

   \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{2}{e^{x} + e^{-x}}\right)}^{3}}}\]

7. Final simplification 0.1

   \[\leadsto \sqrt[3]{{\left(\frac{2}{e^{x} + e^{-x}}\right)}^{3}}\]

Reproduce

herbie shell --seed 2019199 
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2.0 (+ (exp x) (exp (- x)))))