Average Error: 0.3 → 0.4
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[(e^{\log_* (1 + \frac{1 - \tan x \cdot \tan x}{\tan x \cdot \tan x + 1})} - 1)^*\]

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. Using strategy rm

  3. Applied expm1-log1p-u0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019088 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))