Average Error: 0.0 → 0.8
Time: 28.6s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\frac{2}{\sqrt{e^{x} + e^{-x}}} \cdot \frac{1}{\sqrt{e^{x} + e^{-x}}}\]
\frac{2}{e^{x} + e^{-x}}
↓
\frac{2}{\sqrt{e^{x} + e^{-x}}} \cdot \frac{1}{\sqrt{e^{x} + e^{-x}}}
double f(double x) {
        double r2050175 = 2.0;
        double r2050176 = x;
        double r2050177 = exp(r2050176);
        double r2050178 = -r2050176;
        double r2050179 = exp(r2050178);
        double r2050180 = r2050177 + r2050179;
        double r2050181 = r2050175 / r2050180;
        return r2050181;
}
↓
double f(double x) {
        double r2050182 = 2.0;
        double r2050183 = x;
        double r2050184 = exp(r2050183);
        double r2050185 = -r2050183;
        double r2050186 = exp(r2050185);
        double r2050187 = r2050184 + r2050186;
        double r2050188 = sqrt(r2050187);
        double r2050189 = r2050182 / r2050188;
        double r2050190 = 1.0;
        double r2050191 = r2050190 / r2050188;
        double r2050192 = r2050189 * r2050191;
        return r2050192;
}

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 *-un-lft-identity0.8

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

  5. Applied times-frac0.8

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

  6. Final simplification0.8

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2.0 (+ (exp x) (exp (- x)))))