Error

Target

Derivation

1. Split input into 3 regimes

2. if (/ (fma (/ (+ (tan eps) (tan x)) (- 1 (* (tan eps) (tan x)))) (/ (+ (tan eps) (tan x)) (- 1 (* (tan eps) (tan x)))) (* (- (tan x)) (tan x))) (+ (/ (+ (tan eps) (tan x)) (- 1 (* (tan eps) (tan x)))) (tan x))) < -2.2600720559677924e-13

   1. Initial program 29.5

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

   3. Applied tan-sum0.8

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

   5. Applied flip--0.9

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

   6. Applied associate-/r/0.9

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

   7. Applied fma-neg0.9

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

   9. Applied add-cube-cbrt1.0

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

   10. Applied associate-*r*1.0

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

   if -2.2600720559677924e-13 < (/ (fma (/ (+ (tan eps) (tan x)) (- 1 (* (tan eps) (tan x)))) (/ (+ (tan eps) (tan x)) (- 1 (* (tan eps) (tan x)))) (* (- (tan x)) (tan x))) (+ (/ (+ (tan eps) (tan x)) (- 1 (* (tan eps) (tan x)))) (tan x))) < 4.571519807336895e-16

   1. Initial program 44.8

      \[\tan \left(x + \varepsilon\right) - \tan x\]

   2. Taylor expanded around 0 29.1

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

   3. Applied simplify28.0

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

   if 4.571519807336895e-16 < (/ (fma (/ (+ (tan eps) (tan x)) (- 1 (* (tan eps) (tan x)))) (/ (+ (tan eps) (tan x)) (- 1 (* (tan eps) (tan x)))) (* (- (tan x)) (tan x))) (+ (/ (+ (tan eps) (tan x)) (- 1 (* (tan eps) (tan x)))) (tan x)))

   1. Initial program 29.6

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

   3. Applied tan-quot29.4

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

   4. Applied tan-sum1.6

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

   5. Applied frac-sub1.6

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

4. Applied simplify13.9

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{(\left(\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\right) \cdot \left(\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\right) + \left(\left(-\tan x\right) \cdot \tan x\right))_*}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \tan x} \le -2.2600720559677924 \cdot 10^{-13}:\\ \;\;\;\;(\left(\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}\right) \cdot \left(\sqrt[3]{\tan \varepsilon} \cdot \left(\tan x \cdot \left(\sqrt[3]{\tan \varepsilon} \cdot \sqrt[3]{\tan \varepsilon}\right)\right) + 1\right) + \left(-\tan x\right))_*\\ \mathbf{if}\;\frac{(\left(\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\right) \cdot \left(\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\right) + \left(\left(-\tan x\right) \cdot \tan x\right))_*}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \tan x} \le 4.571519807336895 \cdot 10^{-16}:\\ \;\;\;\;(\left(\varepsilon \cdot \varepsilon\right) \cdot \left((x \cdot \left(\varepsilon \cdot x\right) + x)_*\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos x \cdot \left(\tan x + \tan \varepsilon\right) - \sin x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}{\cos x \cdot \left(1 - \tan x \cdot \tan \varepsilon\right)}\\ \end{array}}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' +o rules:numerics
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :herbie-expected 28

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

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