Average Error: 14.7 → 1.8
Time: 6.1s
Precision: 64
Internal Precision: 1344
\[\tan^{-1} \left(x + y\right) - \tan^{-1} \left(x - y\right)\]
\[\tan^{-1}_* \frac{y + y}{\left(x + y\right) \cdot \left(x - y\right) + 1}\]

Error

Bits error versus x

Bits error versus y

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Results

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Derivation

  1. Initial program 14.7

    \[\tan^{-1} \left(x + y\right) - \tan^{-1} \left(x - y\right)\]
  2. Initial simplification14.7

    \[\leadsto \tan^{-1} \left(y + x\right) - \tan^{-1} \left(x - y\right)\]
  3. Using strategy rm
  4. Applied diff-atan14.4

    \[\leadsto \color{blue}{\tan^{-1}_* \frac{\left(y + x\right) - \left(x - y\right)}{1 + \left(y + x\right) \cdot \left(x - y\right)}}\]
  5. Simplified1.8

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{y + y}}{1 + \left(y + x\right) \cdot \left(x - y\right)}\]
  6. Final simplification1.8

    \[\leadsto \tan^{-1}_* \frac{y + y}{\left(x + y\right) \cdot \left(x - y\right) + 1}\]

Runtime

Time bar (total: 6.1s)Debug log

herbie shell --seed '#(2775764126 3555076145 3898259844 1891440260 2599947619 1948460636)' 
(FPCore (x y)
  :name "atan(x+y)-atan(x-y)"
  (- (atan (+ x y)) (atan (- x y))))