Average Error: 0.0 → 0.0
Time: 1.9s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{e^{x} + e^{-x}} \cdot \sqrt[3]{e^{x} + e^{-x}}} \cdot \frac{\sqrt[3]{2}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{e^{x} + e^{-x}}\right)\right)}\]
\frac{2}{e^{x} + e^{-x}}
↓
\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{e^{x} + e^{-x}} \cdot \sqrt[3]{e^{x} + e^{-x}}} \cdot \frac{\sqrt[3]{2}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{e^{x} + e^{-x}}\right)\right)}
double f(double x) {
        double r65948 = 2.0;
        double r65949 = x;
        double r65950 = exp(r65949);
        double r65951 = -r65949;
        double r65952 = exp(r65951);
        double r65953 = r65950 + r65952;
        double r65954 = r65948 / r65953;
        return r65954;
}
↓
double f(double x) {
        double r65955 = 2.0;
        double r65956 = cbrt(r65955);
        double r65957 = r65956 * r65956;
        double r65958 = x;
        double r65959 = exp(r65958);
        double r65960 = -r65958;
        double r65961 = exp(r65960);
        double r65962 = r65959 + r65961;
        double r65963 = cbrt(r65962);
        double r65964 = r65963 * r65963;
        double r65965 = r65957 / r65964;
        double r65966 = log1p(r65963);
        double r65967 = expm1(r65966);
        double r65968 = r65956 / r65967;
        double r65969 = r65965 * r65968;
        return r65969;
}

Error

Bits error versus x

Try it out

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-cube-cbrt1.2

    \[\leadsto \frac{2}{\color{blue}{\left(\sqrt[3]{e^{x} + e^{-x}} \cdot \sqrt[3]{e^{x} + e^{-x}}\right) \cdot \sqrt[3]{e^{x} + e^{-x}}}}\]

  4. Applied add-cube-cbrt0.0

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{2}}}{\left(\sqrt[3]{e^{x} + e^{-x}} \cdot \sqrt[3]{e^{x} + e^{-x}}\right) \cdot \sqrt[3]{e^{x} + e^{-x}}}\]

  5. Applied times-frac0.0

    \[\leadsto \color{blue}{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{e^{x} + e^{-x}} \cdot \sqrt[3]{e^{x} + e^{-x}}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{e^{x} + e^{-x}}}}\]
  6. Using strategy rm

  7. Applied expm1-log1p-u0.0

    \[\leadsto \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{e^{x} + e^{-x}} \cdot \sqrt[3]{e^{x} + e^{-x}}} \cdot \frac{\sqrt[3]{2}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{e^{x} + e^{-x}}\right)\right)}}\]

  8. Final simplification0.0

    \[\leadsto \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{e^{x} + e^{-x}} \cdot \sqrt[3]{e^{x} + e^{-x}}} \cdot \frac{\sqrt[3]{2}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{e^{x} + e^{-x}}\right)\right)}\]

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2 (+ (exp x) (exp (- x)))))