Average Error: 0.3 → 0.5
Time: 22.1s
Precision: 64
Internal Precision: 576
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1 - \sqrt[3]{\left(\tan x \cdot \tan x\right) \cdot \left(\left(\tan x \cdot \tan x\right) \cdot \left(\tan x \cdot \tan x\right)\right)}}{\tan x \cdot \tan x + 1}\]

Error

Bits error versus x

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Results

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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 add-cbrt-cube0.5

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

  4. Final simplification0.5

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

Runtime

Time bar (total: 22.1s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes0.50.50.10.40%
herbie shell --seed 2018340 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))