Average Error: 37.5 → 14.4
Time: 1.4m
Precision: 64
Internal Precision: 2432
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(\tan \varepsilon + \tan x\right) \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)\right) \cdot \cos x - \left(1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \sin x}{\left(1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \cos x} \le -4.885050503140088 \cdot 10^{-16}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \frac{\tan x \cdot \sin \varepsilon}{\cos \varepsilon}} - \tan x\\ \mathbf{if}\;\frac{\left(\left(\tan \varepsilon + \tan x\right) \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)\right) \cdot \cos x - \left(1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \sin x}{\left(1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \cos x} \le 2.271267445739451 \cdot 10^{-16}:\\ \;\;\;\;\varepsilon + \left({\varepsilon}^{3} \cdot {x}^{2} + {\varepsilon}^{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\tan x \cdot \tan x - \tan \varepsilon \cdot \tan \varepsilon}{\tan x - \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original37.5
Target15.4
Herbie14.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (- (* (* (+ (tan eps) (tan x)) (+ 1 (* (tan x) (tan eps)))) (cos x)) (* (- 1 (* (* (tan eps) (tan x)) (* (tan eps) (tan x)))) (sin x))) (* (- 1 (* (* (tan eps) (tan x)) (* (tan eps) (tan x)))) (cos x))) < -4.885050503140088e-16

    1. Initial program 30.8

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

    3. Applied tan-sum2.7

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

    5. Applied tan-quot2.7

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

    6. Applied associate-*r/2.7

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

    if -4.885050503140088e-16 < (/ (- (* (* (+ (tan eps) (tan x)) (+ 1 (* (tan x) (tan eps)))) (cos x)) (* (- 1 (* (* (tan eps) (tan x)) (* (tan eps) (tan x)))) (sin x))) (* (- 1 (* (* (tan eps) (tan x)) (* (tan eps) (tan x)))) (cos x))) < 2.271267445739451e-16

    1. Initial program 44.6

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

    2. Taylor expanded around 0 27.6

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

    if 2.271267445739451e-16 < (/ (- (* (* (+ (tan eps) (tan x)) (+ 1 (* (tan x) (tan eps)))) (cos x)) (* (- 1 (* (* (tan eps) (tan x)) (* (tan eps) (tan x)))) (sin x))) (* (- 1 (* (* (tan eps) (tan x)) (* (tan eps) (tan x)))) (cos x)))

    1. Initial program 32.7

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

    3. Applied tan-sum4.9

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

    5. Applied flip-+5.0

      \[\leadsto \frac{\color{blue}{\frac{\tan x \cdot \tan x - \tan \varepsilon \cdot \tan \varepsilon}{\tan x - \tan \varepsilon}}}{1 - \tan x \cdot \tan \varepsilon} - \tan x\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' 
(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)))