Average Error: 0.0 → 0.0
Time: 18.6s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}} \cdot \sqrt{\frac{2}{e^{x} + e^{-x}}}\]
\frac{2}{e^{x} + e^{-x}}
↓
\frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}} \cdot \sqrt{\frac{2}{e^{x} + e^{-x}}}
double f(double x) {
        double r2110050 = 2.0;
        double r2110051 = x;
        double r2110052 = exp(r2110051);
        double r2110053 = -r2110051;
        double r2110054 = exp(r2110053);
        double r2110055 = r2110052 + r2110054;
        double r2110056 = r2110050 / r2110055;
        return r2110056;
}
↓
double f(double x) {
        double r2110057 = 2.0;
        double r2110058 = sqrt(r2110057);
        double r2110059 = x;
        double r2110060 = exp(r2110059);
        double r2110061 = -r2110059;
        double r2110062 = exp(r2110061);
        double r2110063 = r2110060 + r2110062;
        double r2110064 = sqrt(r2110063);
        double r2110065 = r2110058 / r2110064;
        double r2110066 = r2110057 / r2110063;
        double r2110067 = sqrt(r2110066);
        double r2110068 = r2110065 * r2110067;
        return r2110068;
}

Error

Bits error versus x

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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-sqrt0.8

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

  4. Applied add-sqr-sqrt0.0

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

  5. Applied times-frac0.0

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

  7. Applied sqrt-undiv0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019130 
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))