Average Error: 0.0 → 0.1
Time: 4.5s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\sqrt[3]{\left(\frac{2}{e^{-x} + e^{x}} \cdot \frac{2}{e^{-x} + e^{x}}\right) \cdot \frac{2}{e^{-x} + e^{x}}}\]
\frac{2}{e^{x} + e^{-x}}
↓
\sqrt[3]{\left(\frac{2}{e^{-x} + e^{x}} \cdot \frac{2}{e^{-x} + e^{x}}\right) \cdot \frac{2}{e^{-x} + e^{x}}}
double f(double x) {
        double r827536 = 2.0;
        double r827537 = x;
        double r827538 = exp(r827537);
        double r827539 = -r827537;
        double r827540 = exp(r827539);
        double r827541 = r827538 + r827540;
        double r827542 = r827536 / r827541;
        return r827542;
}
↓
double f(double x) {
        double r827543 = 2.0;
        double r827544 = x;
        double r827545 = -r827544;
        double r827546 = exp(r827545);
        double r827547 = exp(r827544);
        double r827548 = r827546 + r827547;
        double r827549 = r827543 / r827548;
        double r827550 = r827549 * r827549;
        double r827551 = r827550 * r827549;
        double r827552 = cbrt(r827551);
        return r827552;
}

Error

Bits error versus x

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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

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019156 
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))