Average Error: 0.0 → 0.0
Time: 7.7s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\sqrt{-\frac{2}{\frac{-1}{e^{x}} - e^{x}}} \cdot \sqrt{-\frac{2}{\frac{-1}{e^{x}} - e^{x}}}\]
\frac{2}{e^{x} + e^{-x}}
↓
\sqrt{-\frac{2}{\frac{-1}{e^{x}} - e^{x}}} \cdot \sqrt{-\frac{2}{\frac{-1}{e^{x}} - e^{x}}}
double f(double x) {
        double r67229 = 2.0;
        double r67230 = x;
        double r67231 = exp(r67230);
        double r67232 = -r67230;
        double r67233 = exp(r67232);
        double r67234 = r67231 + r67233;
        double r67235 = r67229 / r67234;
        return r67235;
}
↓
double f(double x) {
        double r67236 = 2.0;
        double r67237 = -1.0;
        double r67238 = x;
        double r67239 = exp(r67238);
        double r67240 = r67237 / r67239;
        double r67241 = r67240 - r67239;
        double r67242 = r67236 / r67241;
        double r67243 = -r67242;
        double r67244 = sqrt(r67243);
        double r67245 = r67244 * r67244;
        return r67245;
}

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 frac-2neg0.0

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

  4. Simplified0.0

    \[\leadsto \frac{-2}{\color{blue}{\frac{-1}{e^{x}} - e^{x}}}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.0

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

  7. Simplified0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2.0 (+ (exp x) (exp (- x)))))