Average Error: 36.7 → 13.2
Time: 1.2m
Precision: 64
Internal Precision: 2432
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}} - \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}}\right) \cdot \left(\left(\frac{\frac{\cos x \cdot \sin x}{\cos x}}{1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}} - \sin x\right) + \frac{\frac{\cos x \cdot \sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}}\right)}{\left(\frac{\sin x}{\left(1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}\right) \cdot \cos x} - \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}\right)}\right) \cdot \cos x} \le -4.0953336377332593 \cdot 10^{-215}:\\ \;\;\;\;\frac{\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}} - \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}}\right) \cdot \left(\left(\frac{\frac{\cos x \cdot \sin x}{\cos x}}{1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}} - \sin x\right) + \frac{\frac{\cos x \cdot \sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}}\right)}{\left(\frac{\sin x}{\left(1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}\right) \cdot \cos x} - \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}\right)}\right) \cdot \cos x}\\ \mathbf{if}\;\frac{\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}} - \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}}\right) \cdot \left(\left(\frac{\frac{\cos x \cdot \sin x}{\cos x}}{1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}} - \sin x\right) + \frac{\frac{\cos x \cdot \sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}}\right)}{\left(\frac{\sin x}{\left(1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}\right) \cdot \cos x} - \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}\right)}\right) \cdot \cos x} \le 3.8418546076681496 \cdot 10^{-305}:\\ \;\;\;\;\varepsilon + \left({\varepsilon}^{3} \cdot {x}^{2} + {\varepsilon}^{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{\sin x}{\cos x}}{1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}} - \frac{\frac{\sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}}\right) \cdot \left(\left(\frac{\frac{\cos x \cdot \sin x}{\cos x}}{1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}} - \sin x\right) + \frac{\frac{\cos x \cdot \sin \varepsilon}{\cos \varepsilon}}{1 - \frac{\sin \varepsilon}{\cos x} \cdot \frac{\sin x}{\cos \varepsilon}}\right)}{\left(\frac{\sin x}{\left(1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}\right) \cdot \cos x} - \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}\right)}\right) \cdot \cos x}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original36.7
Target15.2
Herbie13.2
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* (- (/ (/ (sin x) (cos x)) (- 1 (* (/ (sin eps) (cos x)) (/ (sin x) (cos eps))))) (/ (/ (sin eps) (cos eps)) (- 1 (* (/ (sin eps) (cos x)) (/ (sin x) (cos eps)))))) (+ (- (/ (/ (* (cos x) (sin x)) (cos x)) (- 1 (* (/ (sin eps) (cos x)) (/ (sin x) (cos eps))))) (sin x)) (/ (/ (* (cos x) (sin eps)) (cos eps)) (- 1 (* (/ (sin eps) (cos x)) (/ (sin x) (cos eps))))))) (* (- (/ (sin x) (* (- 1 (/ (* (sin eps) (sin x)) (* (cos eps) (cos x)))) (cos x))) (/ (sin eps) (* (cos eps) (- 1 (/ (* (sin eps) (sin x)) (* (cos eps) (cos x))))))) (cos x))) < -4.0953336377332593e-215 or 3.8418546076681496e-305 < (/ (* (- (/ (/ (sin x) (cos x)) (- 1 (* (/ (sin eps) (cos x)) (/ (sin x) (cos eps))))) (/ (/ (sin eps) (cos eps)) (- 1 (* (/ (sin eps) (cos x)) (/ (sin x) (cos eps)))))) (+ (- (/ (/ (* (cos x) (sin x)) (cos x)) (- 1 (* (/ (sin eps) (cos x)) (/ (sin x) (cos eps))))) (sin x)) (/ (/ (* (cos x) (sin eps)) (cos eps)) (- 1 (* (/ (sin eps) (cos x)) (/ (sin x) (cos eps))))))) (* (- (/ (sin x) (* (- 1 (/ (* (sin eps) (sin x)) (* (cos eps) (cos x)))) (cos x))) (/ (sin eps) (* (cos eps) (- 1 (/ (* (sin eps) (sin x)) (* (cos eps) (cos x))))))) (cos x)))

    1. Initial program 35.6

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

    3. Applied tan-sum18.2

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

    4. Taylor expanded around inf 18.4

      \[\leadsto \color{blue}{\left(\frac{\sin x}{\left(1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}\right) \cdot \cos x} + \frac{\sin \varepsilon}{\cos \varepsilon \cdot \left(1 - \frac{\sin \varepsilon \cdot \sin x}{\cos \varepsilon \cdot \cos x}\right)}\right) - \frac{\sin x}{\cos x}}\]
    5. Using strategy rm

    6. Applied flip-+18.5

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

    7. Applied frac-sub18.6

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

    8. Applied simplify13.0

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

    if -4.0953336377332593e-215 < (/ (* (- (/ (/ (sin x) (cos x)) (- 1 (* (/ (sin eps) (cos x)) (/ (sin x) (cos eps))))) (/ (/ (sin eps) (cos eps)) (- 1 (* (/ (sin eps) (cos x)) (/ (sin x) (cos eps)))))) (+ (- (/ (/ (* (cos x) (sin x)) (cos x)) (- 1 (* (/ (sin eps) (cos x)) (/ (sin x) (cos eps))))) (sin x)) (/ (/ (* (cos x) (sin eps)) (cos eps)) (- 1 (* (/ (sin eps) (cos x)) (/ (sin x) (cos eps))))))) (* (- (/ (sin x) (* (- 1 (/ (* (sin eps) (sin x)) (* (cos eps) (cos x)))) (cos x))) (/ (sin eps) (* (cos eps) (- 1 (/ (* (sin eps) (sin x)) (* (cos eps) (cos x))))))) (cos x))) < 3.8418546076681496e-305

    1. Initial program 44.4

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

    2. Taylor expanded around 0 14.6

      \[\leadsto \color{blue}{\varepsilon + \left({\varepsilon}^{3} \cdot {x}^{2} + {\varepsilon}^{2} \cdot x\right)}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed '#(1064397287 3527694221 3797617954 1138343853 2854031332 1153838279)' 
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"

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

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