Average Error: 0.0 → 0.0
Time: 3.9s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{\sqrt{2}}{\frac{e^{x} + e^{-x}}{\sqrt{2}}}}\]
\frac{2}{e^{x} + e^{-x}}
↓
\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{\sqrt{2}}{\frac{e^{x} + e^{-x}}{\sqrt{2}}}}
double f(double x) {
        double r82174 = 2.0;
        double r82175 = x;
        double r82176 = exp(r82175);
        double r82177 = -r82175;
        double r82178 = exp(r82177);
        double r82179 = r82176 + r82178;
        double r82180 = r82174 / r82179;
        return r82180;
}
↓
double f(double x) {
        double r82181 = 2.0;
        double r82182 = x;
        double r82183 = exp(r82182);
        double r82184 = -r82182;
        double r82185 = exp(r82184);
        double r82186 = r82183 + r82185;
        double r82187 = r82181 / r82186;
        double r82188 = sqrt(r82187);
        double r82189 = sqrt(r82181);
        double r82190 = r82186 / r82189;
        double r82191 = r82189 / r82190;
        double r82192 = sqrt(r82191);
        double r82193 = r82188 * r82192;
        return r82193;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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Derivation

  1. Initial program 0.0

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

  3. Applied add-sqr-sqrt0.0

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

  5. Applied add-sqr-sqrt0.0

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

  6. Applied associate-/l*0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020034 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2 (+ (exp x) (exp (- x)))))