Average Error: 0.0 → 0.0
Time: 15.2s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\sqrt{\frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}}} \cdot \left(\sqrt{\frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}}} \cdot \frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}}\right)\]
\frac{2}{e^{x} + e^{-x}}
↓
\sqrt{\frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}}} \cdot \left(\sqrt{\frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}}} \cdot \frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}}\right)
double f(double x) {
        double r1293070 = 2.0;
        double r1293071 = x;
        double r1293072 = exp(r1293071);
        double r1293073 = -r1293071;
        double r1293074 = exp(r1293073);
        double r1293075 = r1293072 + r1293074;
        double r1293076 = r1293070 / r1293075;
        return r1293076;
}
↓
double f(double x) {
        double r1293077 = 2.0;
        double r1293078 = sqrt(r1293077);
        double r1293079 = x;
        double r1293080 = exp(r1293079);
        double r1293081 = -r1293079;
        double r1293082 = exp(r1293081);
        double r1293083 = r1293080 + r1293082;
        double r1293084 = sqrt(r1293083);
        double r1293085 = r1293078 / r1293084;
        double r1293086 = sqrt(r1293085);
        double r1293087 = r1293086 * r1293085;
        double r1293088 = r1293086 * r1293087;
        return r1293088;
}

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-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 add-sqr-sqrt0.0

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

  8. Applied associate-*r*0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))