Hyperbolic secant

Average Error: 0.0 → 0.0

Time: 16.3s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r2662256 = 2.0;
        double r2662257 = x;
        double r2662258 = exp(r2662257);
        double r2662259 = -r2662257;
        double r2662260 = exp(r2662259);
        double r2662261 = r2662258 + r2662260;
        double r2662262 = r2662256 / r2662261;
        return r2662262;
}

↓

double f(double x) {
        double r2662263 = 2.0;
        double r2662264 = sqrt(r2662263);
        double r2662265 = x;
        double r2662266 = exp(r2662265);
        double r2662267 = -r2662265;
        double r2662268 = exp(r2662267);
        double r2662269 = r2662266 + r2662268;
        double r2662270 = r2662264 / r2662269;
        double r2662271 = r2662264 * r2662270;
        double r2662272 = cbrt(r2662271);
        double r2662273 = r2662263 / r2662269;
        double r2662274 = cbrt(r2662273);
        double r2662275 = sqrt(r2662274);
        double r2662276 = r2662274 * r2662275;
        double r2662277 = r2662276 * r2662275;
        double r2662278 = r2662272 * r2662277;
        return r2662278;
}

Error

Bits error versus x

Derivation

1. Initial program 0.0

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

3. Applied add-cube-cbrt 0.0

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

5. Applied add-sqr-sqrt 0.0

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

6. Applied associate-*l* 0.0

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

8. Applied *-un-lft-identity 0.0

   \[\leadsto \left(\sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}} \cdot \left(\sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}} \cdot \sqrt[3]{\frac{2}{e^{x} + e^{-x}}}\right)\right) \cdot \sqrt[3]{\frac{2}{\color{blue}{1 \cdot \left(e^{x} + e^{-x}\right)}}}\]

9. Applied add-sqr-sqrt 0.0

   \[\leadsto \left(\sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}} \cdot \left(\sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}} \cdot \sqrt[3]{\frac{2}{e^{x} + e^{-x}}}\right)\right) \cdot \sqrt[3]{\frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{1 \cdot \left(e^{x} + e^{-x}\right)}}\]

10. Applied times-frac 0.0

    \[\leadsto \left(\sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}} \cdot \left(\sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}} \cdot \sqrt[3]{\frac{2}{e^{x} + e^{-x}}}\right)\right) \cdot \sqrt[3]{\color{blue}{\frac{\sqrt{2}}{1} \cdot \frac{\sqrt{2}}{e^{x} + e^{-x}}}}\]

11. Simplified 0.0

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

12. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019174 
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2.0 (+ (exp x) (exp (- x)))))