Average Error: 0.3 → 0.4
Time: 25.8s
Precision: 64
Internal Precision: 576
\[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x}\]
↓
\[\frac{1 - \frac{\log_* (1 + (e^{{\left(\sin x\right)}^{2}} - 1)^*)}{{\left(\cos x\right)}^{2}}}{1 + \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}\]

Error

Bits error versus x

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Your Program's Arguments

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. 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. Using strategy rm

  4. Applied log1p-expm1-u0.4

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

  5. Final simplification0.4

    \[\leadsto \frac{1 - \frac{\log_* (1 + (e^{{\left(\sin x\right)}^{2}} - 1)^*)}{{\left(\cos x\right)}^{2}}}{1 + \frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}}}\]

Runtime

Time bar (total: 25.8s)Debug logProfile

herbie shell --seed 2018225 +o rules:numerics
(FPCore (x)
  :name "Trigonometry B"
  (/ (- 1 (* (tan x) (tan x))) (+ 1 (* (tan x) (tan x)))))