Average Error: 0.3 → 0.4
Time: 22.4s
Precision: 64
Internal Precision: 384
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1 - \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\cos x \cdot \cos x}}{1 + \frac{\sin x \cdot \sin x}{\cos x \cdot \cos x}}\]

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 tan-quot0.4

    \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \color{blue}{\frac{\sin x}{\cos x}}}\]

  4. Applied tan-quot0.4

    \[\leadsto \frac{1 - \tan x \cdot \tan x}{1 + \color{blue}{\frac{\sin x}{\cos x}} \cdot \frac{\sin x}{\cos x}}\]

  5. Applied frac-times0.4

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

  7. Applied tan-quot0.4

    \[\leadsto \frac{1 - \tan x \cdot \color{blue}{\frac{\sin x}{\cos x}}}{1 + \frac{\sin x \cdot \sin x}{\cos x \cdot \cos x}}\]

  8. Applied tan-quot0.5

    \[\leadsto \frac{1 - \color{blue}{\frac{\sin x}{\cos x}} \cdot \frac{\sin x}{\cos x}}{1 + \frac{\sin x \cdot \sin x}{\cos x \cdot \cos x}}\]

  9. Applied frac-times0.4

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

  11. Applied sqr-sin0.4

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

Runtime

Time bar (total: 22.4s)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))