Hyperbolic secant

Average Error: 0.0 → 0.1

Time: 14.1s

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 r49509 = 2.0;
        double r49510 = x;
        double r49511 = exp(r49510);
        double r49512 = -r49510;
        double r49513 = exp(r49512);
        double r49514 = r49511 + r49513;
        double r49515 = r49509 / r49514;
        return r49515;
}

↓

double f(double x) {
        double r49516 = 2.0;
        double r49517 = x;
        double r49518 = -r49517;
        double r49519 = exp(r49518);
        double r49520 = exp(r49517);
        double r49521 = r49519 + r49520;
        double r49522 = r49516 / r49521;
        double r49523 = 3.0;
        double r49524 = pow(r49522, r49523);
        double r49525 = cbrt(r49524);
        return r49525;
}

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 2019325 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2 (+ (exp x) (exp (- x)))))