Error

Target

Derivation

1. Split input into 3 regimes

2. if (fma (* (cbrt (+ (tan x) (tan eps))) (cbrt (+ (tan x) (tan eps)))) (/ (cbrt (+ (tan x) (tan eps))) (- 1 (* (tan x) (tan eps)))) (- (tan x))) < -2.3868355075214045e-15

   1. Initial program 29.6

      \[\tan \left(x + \varepsilon\right) - \tan x\]
   2. Using strategy rm

   3. Applied tan-sum2.8

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
   4. Using strategy rm

   5. Applied flip3--2.8

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

   6. Applied associate-/r/2.8

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

   7. Simplified2.8

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

   if -2.3868355075214045e-15 < (fma (* (cbrt (+ (tan x) (tan eps))) (cbrt (+ (tan x) (tan eps)))) (/ (cbrt (+ (tan x) (tan eps))) (- 1 (* (tan x) (tan eps)))) (- (tan x))) < 1.2946181514548095e-15

   1. Initial program 44.5

      \[\tan \left(x + \varepsilon\right) - \tan x\]
   2. Using strategy rm

   3. Applied tan-sum44.5

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

   4. Taylor expanded around 0 27.9

      \[\leadsto \color{blue}{x \cdot {\varepsilon}^{2} + \left(\varepsilon + {x}^{2} \cdot {\varepsilon}^{3}\right)}\]

   5. Simplified26.9

      \[\leadsto \color{blue}{(\left(x \cdot \varepsilon\right) \cdot \left((\left(x \cdot \varepsilon\right) \cdot \varepsilon + \varepsilon)_*\right) + \varepsilon)_*}\]

   if 1.2946181514548095e-15 < (fma (* (cbrt (+ (tan x) (tan eps))) (cbrt (+ (tan x) (tan eps)))) (/ (cbrt (+ (tan x) (tan eps))) (- 1 (* (tan x) (tan eps)))) (- (tan x)))

   1. Initial program 32.7

      \[\tan \left(x + \varepsilon\right) - \tan x\]
   2. Using strategy rm

   3. Applied tan-sum4.8

      \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
   4. Using strategy rm

   5. Applied flip--4.8

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

   6. Applied associate-/r/4.8

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

   7. Applied fma-neg4.8

      \[\leadsto \color{blue}{(\left(\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}\right) \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) + \left(-\tan x\right))_*}\]

   8. Simplified4.8

      \[\leadsto (\color{blue}{\left(\frac{\tan x + \tan \varepsilon}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)}\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) + \left(-\tan x\right))_*\]
3. Recombined 3 regimes into one program.

4. Final simplification14.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;(\left(\sqrt[3]{\tan x + \tan \varepsilon} \cdot \sqrt[3]{\tan x + \tan \varepsilon}\right) \cdot \left(\frac{\sqrt[3]{\tan x + \tan \varepsilon}}{1 - \tan \varepsilon \cdot \tan x}\right) + \left(-\tan x\right))_* \le -2.3868355075214045 \cdot 10^{-15}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{{1}^{3} - {\left(\tan \varepsilon \cdot \tan x\right)}^{3}} \cdot (\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left((\left(\tan \varepsilon\right) \cdot \left(\tan x\right) + 1)_*\right))_* - \tan x\\ \mathbf{elif}\;(\left(\sqrt[3]{\tan x + \tan \varepsilon} \cdot \sqrt[3]{\tan x + \tan \varepsilon}\right) \cdot \left(\frac{\sqrt[3]{\tan x + \tan \varepsilon}}{1 - \tan \varepsilon \cdot \tan x}\right) + \left(-\tan x\right))_* \le 1.2946181514548095 \cdot 10^{-15}:\\ \;\;\;\;(\left(x \cdot \varepsilon\right) \cdot \left((\left(x \cdot \varepsilon\right) \cdot \varepsilon + \varepsilon)_*\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\tan x + \tan \varepsilon}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)}\right) \cdot \left(1 + \tan \varepsilon \cdot \tan x\right) + \left(-\tan x\right))_*\\ \end{array}\]

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed 2018225 +o rules:numerics
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))