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))) < -1.8725134023530507e-16

   1. Initial program 31.6

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

   3. Applied tan-sum2.5

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

   5. Applied flip--2.5

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

   6. Applied associate-/r/2.5

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

   7. Applied fma-neg2.5

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

   if -1.8725134023530507e-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))) < 1.2082914156281068e-16

   1. Initial program 43.8

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

   2. Taylor expanded around 0 27.4

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

   3. Applied simplify26.4

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

   if 1.2082914156281068e-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 31.7

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

   3. Applied tan-sum3.6

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

   5. Applied add-cbrt-cube3.7

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

   6. Applied add-cbrt-cube3.7

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

   7. Applied cbrt-unprod3.7

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

   8. Applied simplify3.7

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

4. Applied simplify13.6

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{(\left(\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\right) \cdot \left(\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\right) + \left(\tan x \cdot \left(-\tan x\right)\right))_*}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \tan x} \le -1.8725134023530507 \cdot 10^{-16}:\\ \;\;\;\;(\left(\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}\right) \cdot \left(\tan x \cdot \tan \varepsilon + 1\right) + \left(-\tan x\right))_*\\ \mathbf{if}\;\frac{(\left(\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\right) \cdot \left(\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}\right) + \left(\tan x \cdot \left(-\tan x\right)\right))_*}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} + \tan x} \le 1.2082914156281068 \cdot 10^{-16}:\\ \;\;\;\;(\varepsilon \cdot \left((\left(\varepsilon \cdot x\right) \cdot \left(\varepsilon \cdot x\right) + \left(\varepsilon \cdot x\right))_*\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \sqrt[3]{{\left(\tan x \cdot \tan \varepsilon\right)}^{3}}} - \tan x\\ \end{array}}\]

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +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)))