Average Error: 0.0 → 0.1
Time: 14.5s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\sqrt[3]{\frac{\frac{8}{e^{x} + \frac{1}{e^{x}}}}{\left(e^{x} + \frac{1}{e^{x}}\right) \cdot \left(e^{x} + \frac{1}{e^{x}}\right)}}\]
\frac{2}{e^{x} + e^{-x}}
↓
\sqrt[3]{\frac{\frac{8}{e^{x} + \frac{1}{e^{x}}}}{\left(e^{x} + \frac{1}{e^{x}}\right) \cdot \left(e^{x} + \frac{1}{e^{x}}\right)}}
double f(double x) {
        double r2729139 = 2.0;
        double r2729140 = x;
        double r2729141 = exp(r2729140);
        double r2729142 = -r2729140;
        double r2729143 = exp(r2729142);
        double r2729144 = r2729141 + r2729143;
        double r2729145 = r2729139 / r2729144;
        return r2729145;
}
↓
double f(double x) {
        double r2729146 = 8.0;
        double r2729147 = x;
        double r2729148 = exp(r2729147);
        double r2729149 = 1.0;
        double r2729150 = r2729149 / r2729148;
        double r2729151 = r2729148 + r2729150;
        double r2729152 = r2729146 / r2729151;
        double r2729153 = r2729151 * r2729151;
        double r2729154 = r2729152 / r2729153;
        double r2729155 = cbrt(r2729154);
        return r2729155;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

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

  3. Applied add-cbrt-cube0.1

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

  4. Applied add-cbrt-cube0.1

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

  5. Applied cbrt-undiv0.1

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

  6. Simplified0.1

    \[\leadsto \sqrt[3]{\color{blue}{\frac{\frac{8}{e^{x} + \frac{1}{e^{x}}}}{\left(e^{x} + \frac{1}{e^{x}}\right) \cdot \left(e^{x} + \frac{1}{e^{x}}\right)}}}\]

  7. Final simplification0.1

    \[\leadsto \sqrt[3]{\frac{\frac{8}{e^{x} + \frac{1}{e^{x}}}}{\left(e^{x} + \frac{1}{e^{x}}\right) \cdot \left(e^{x} + \frac{1}{e^{x}}\right)}}\]

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))