Average Error: 0.3 → 0.4
Time: 28.4s
Precision: 64
Internal Precision: 576
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[(e^{\log \left(e^{\log_* (1 + \frac{1 - \tan x \cdot \tan x}{(\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*})}\right)} - 1)^*\]

Error

Bits error versus x

Derivation

  1. Initial program 0.3

    \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]

  2. Initial simplification0.3

    \[\leadsto \frac{1 - \tan x \cdot \tan x}{(\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*}\]
  3. Using strategy rm

  4. Applied expm1-log1p-u0.4

    \[\leadsto \color{blue}{(e^{\log_* (1 + \frac{1 - \tan x \cdot \tan x}{(\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*})} - 1)^*}\]
  5. Using strategy rm

  6. Applied add-log-exp0.4

    \[\leadsto (e^{\color{blue}{\log \left(e^{\log_* (1 + \frac{1 - \tan x \cdot \tan x}{(\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*})}\right)}} - 1)^*\]

  7. Final simplification0.4

    \[\leadsto (e^{\log \left(e^{\log_* (1 + \frac{1 - \tan x \cdot \tan x}{(\left(\tan x\right) \cdot \left(\tan x\right) + 1)_*})}\right)} - 1)^*\]

Runtime

Time bar (total: 28.4s)Debug logProfile

herbie shell --seed 2018235 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))