Average Error: 0.3 → 0.5
Time: 1.2m
Precision: 64
Internal Precision: 128
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}{1 + \sqrt[3]{\left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)}}\]

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. Taylor expanded around -inf 0.4

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

  3. Simplified0.4

    \[\leadsto \color{blue}{\frac{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}{1 + \frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}}\]
  4. Using strategy rm

  5. Applied add-cbrt-cube0.5

    \[\leadsto \frac{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}{1 + \color{blue}{\sqrt[3]{\left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)}}}\]

  6. Final simplification0.5

    \[\leadsto \frac{1 - \frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}}{1 + \sqrt[3]{\left(\left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)\right) \cdot \left(\frac{\sin x}{\cos x} \cdot \frac{\sin x}{\cos x}\right)}}\]

Reproduce

herbie shell --seed 2019089 
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))