Average Error: 0.3 → 0.4
Time: 15.9min
Precision: binary64
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1}{1 + {\left(\tan x\right)}^{2}} + \frac{-{\left(\tan x\right)}^{2}}{1 + {\left(\tan x\right)}^{2}}\]

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. Using strategy rm

  3. Applied flip-+0.4

    \[\leadsto \frac{1 - \tan x \cdot \tan x}{\color{blue}{\frac{1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)}{1 - \tan x \cdot \tan x}}}\]

  4. Applied associate-/r/0.4

    \[\leadsto \color{blue}{\frac{1 - \tan x \cdot \tan x}{1 \cdot 1 - \left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)} \cdot \left(1 - \tan x \cdot \tan x\right)}\]

  5. Simplified0.4

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

  7. Applied sub-neg0.4

    \[\leadsto \frac{1}{1 + {\left(\tan x\right)}^{2}} \cdot \color{blue}{\left(1 + \left(-\tan x \cdot \tan x\right)\right)}\]

  8. Applied distribute-lft-in0.4

    \[\leadsto \color{blue}{\frac{1}{1 + {\left(\tan x\right)}^{2}} \cdot 1 + \frac{1}{1 + {\left(\tan x\right)}^{2}} \cdot \left(-\tan x \cdot \tan x\right)}\]

  9. Simplified0.4

    \[\leadsto \color{blue}{\frac{1}{1 + {\left(\tan x\right)}^{2}}} + \frac{1}{1 + {\left(\tan x\right)}^{2}} \cdot \left(-\tan x \cdot \tan x\right)\]

  10. Simplified0.4

    \[\leadsto \frac{1}{1 + {\left(\tan x\right)}^{2}} + \color{blue}{\frac{-{\left(\tan x\right)}^{2}}{1 + {\left(\tan x\right)}^{2}}}\]

  11. Final simplification0.4

    \[\leadsto \frac{1}{1 + {\left(\tan x\right)}^{2}} + \frac{-{\left(\tan x\right)}^{2}}{1 + {\left(\tan x\right)}^{2}}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))