Average Error: 0.0 → 0.1
Time: 14.1s
Precision: 64
Internal Precision: 320
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\sqrt[3]{\log_* (1 + (e^{\frac{\frac{2}{\sqrt{e^{x} + e^{-x}}}}{\sqrt{e^{x} + e^{-x}}}} - 1)^*) \cdot \left(\log_* (1 + (e^{\frac{2}{e^{x} + e^{-x}}} - 1)^*) \cdot \log_* (1 + (e^{\frac{2}{e^{x} + e^{-x}}} - 1)^*)\right)}\]

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 log1p-expm1-u0.0

    \[\leadsto \color{blue}{\log_* (1 + (e^{\frac{2}{e^{x} + e^{-x}}} - 1)^*)}\]
  4. Using strategy rm

  5. Applied add-cbrt-cube0.1

    \[\leadsto \color{blue}{\sqrt[3]{\left(\log_* (1 + (e^{\frac{2}{e^{x} + e^{-x}}} - 1)^*) \cdot \log_* (1 + (e^{\frac{2}{e^{x} + e^{-x}}} - 1)^*)\right) \cdot \log_* (1 + (e^{\frac{2}{e^{x} + e^{-x}}} - 1)^*)}}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt0.5

    \[\leadsto \sqrt[3]{\left(\log_* (1 + (e^{\frac{2}{e^{x} + e^{-x}}} - 1)^*) \cdot \log_* (1 + (e^{\frac{2}{e^{x} + e^{-x}}} - 1)^*)\right) \cdot \log_* (1 + (e^{\frac{2}{\color{blue}{\sqrt{e^{x} + e^{-x}} \cdot \sqrt{e^{x} + e^{-x}}}}} - 1)^*)}\]

  8. Applied associate-/r*0.1

    \[\leadsto \sqrt[3]{\left(\log_* (1 + (e^{\frac{2}{e^{x} + e^{-x}}} - 1)^*) \cdot \log_* (1 + (e^{\frac{2}{e^{x} + e^{-x}}} - 1)^*)\right) \cdot \log_* (1 + (e^{\color{blue}{\frac{\frac{2}{\sqrt{e^{x} + e^{-x}}}}{\sqrt{e^{x} + e^{-x}}}}} - 1)^*)}\]

  9. Final simplification0.1

    \[\leadsto \sqrt[3]{\log_* (1 + (e^{\frac{\frac{2}{\sqrt{e^{x} + e^{-x}}}}{\sqrt{e^{x} + e^{-x}}}} - 1)^*) \cdot \left(\log_* (1 + (e^{\frac{2}{e^{x} + e^{-x}}} - 1)^*) \cdot \log_* (1 + (e^{\frac{2}{e^{x} + e^{-x}}} - 1)^*)\right)}\]

Runtime

Time bar (total: 14.1s)Debug logProfile

herbie shell --seed 2018250 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))