Hyperbolic secant

Average Error: 0.0 → 0.1

Time: 1.7s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r57075 = 2.0;
        double r57076 = x;
        double r57077 = exp(r57076);
        double r57078 = -r57076;
        double r57079 = exp(r57078);
        double r57080 = r57077 + r57079;
        double r57081 = r57075 / r57080;
        return r57081;
}

↓

double f(double x) {
        double r57082 = 2.0;
        double r57083 = -1.0;
        double r57084 = x;
        double r57085 = r57083 * r57084;
        double r57086 = exp(r57085);
        double r57087 = exp(r57084);
        double r57088 = r57086 + r57087;
        double r57089 = r57082 / r57088;
        double r57090 = 3.0;
        double r57091 = pow(r57089, r57090);
        double r57092 = cbrt(r57091);
        return r57092;
}

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

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

7. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2 (+ (exp x) (exp (- x)))))