Average Error: 29.4 → 29.5
Time: 10.4s
Precision: 64
Internal Precision: 576
\[\tan \left(\tan^{-1} x - {10}^{\left(-50\right)}\right)\]
\[\frac{\sin \left(\tan^{-1} x - {10}^{-50}\right)}{\left(\sqrt[3]{\cos \left(\tan^{-1} x - {10}^{-50}\right)} \cdot \sqrt[3]{\cos \left(\tan^{-1} x - {10}^{-50}\right)}\right) \cdot \sqrt[3]{\cos \left(\tan^{-1} x - {10}^{-50}\right)}}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.4

    \[\tan \left(\tan^{-1} x - {10}^{\left(-50\right)}\right)\]
  2. Initial simplification29.4

    \[\leadsto \tan \left(\tan^{-1} x - {10}^{\left(-50\right)}\right)\]
  3. Taylor expanded around 0 29.4

    \[\leadsto \color{blue}{\frac{\sin \left(\tan^{-1} x - {10}^{-50}\right)}{\cos \left(\tan^{-1} x - {10}^{-50}\right)}}\]
  4. Using strategy rm
  5. Applied add-cube-cbrt29.5

    \[\leadsto \frac{\sin \left(\tan^{-1} x - {10}^{-50}\right)}{\color{blue}{\left(\sqrt[3]{\cos \left(\tan^{-1} x - {10}^{-50}\right)} \cdot \sqrt[3]{\cos \left(\tan^{-1} x - {10}^{-50}\right)}\right) \cdot \sqrt[3]{\cos \left(\tan^{-1} x - {10}^{-50}\right)}}}\]
  6. Final simplification29.5

    \[\leadsto \frac{\sin \left(\tan^{-1} x - {10}^{-50}\right)}{\left(\sqrt[3]{\cos \left(\tan^{-1} x - {10}^{-50}\right)} \cdot \sqrt[3]{\cos \left(\tan^{-1} x - {10}^{-50}\right)}\right) \cdot \sqrt[3]{\cos \left(\tan^{-1} x - {10}^{-50}\right)}}\]

Runtime

Time bar (total: 10.4s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (x)
  :name "tan(atan(x)-10^-50)"
  (tan (- (atan x) (pow 10 (- 50)))))