Hyperbolic secant

Average Error: 0.0 → 0.0

Time: 3.8s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r55268 = 2.0;
        double r55269 = x;
        double r55270 = exp(r55269);
        double r55271 = -r55269;
        double r55272 = exp(r55271);
        double r55273 = r55270 + r55272;
        double r55274 = r55268 / r55273;
        return r55274;
}

↓

double f(double x) {
        double r55275 = 2.0;
        double r55276 = x;
        double r55277 = exp(r55276);
        double r55278 = -r55276;
        double r55279 = exp(r55278);
        double r55280 = r55277 + r55279;
        double r55281 = r55275 / r55280;
        return r55281;
}

Error

Bits error versus x

Derivation

1. Initial program 0.0

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

3. Applied add-sqr-sqrt 0.0

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

5. Applied *-un-lft-identity 0.0

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

6. Applied add-sqr-sqrt 0.0

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

7. Applied times-frac 0.0

   \[\leadsto \sqrt{\color{blue}{\frac{\sqrt{2}}{1} \cdot \frac{\sqrt{2}}{e^{x} + e^{-x}}}} \cdot \sqrt{\frac{2}{e^{x} + e^{-x}}}\]

8. Simplified 0.0

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

9. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019308 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2 (+ (exp x) (exp (- x)))))