Average Error: 36.7 → 14.4
Time: 1.5m
Precision: 64
Internal Precision: 2432
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \left(\cos x \cdot {\left(\tan \varepsilon \cdot \tan x\right)}^{3} + \cos x\right) - \left(1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \left(\sin x \cdot \left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left(1 - \tan \varepsilon \cdot \tan x\right)\right)\right)}{\left(\left(1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \left(\left(1 - \tan \varepsilon \cdot \tan x\right) + \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right) \cdot \cos x} \le -3.562198966313061 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(\frac{\tan \varepsilon + \tan x}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)\right) \cdot \left(\frac{\tan \varepsilon + \tan x}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)\right) - \tan x \cdot \tan x}{\frac{\tan \varepsilon + \tan x}{1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right) + \tan x}\\ \mathbf{if}\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \left(\cos x \cdot {\left(\tan \varepsilon \cdot \tan x\right)}^{3} + \cos x\right) - \left(1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \left(\sin x \cdot \left(\left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right) + \left(1 - \tan \varepsilon \cdot \tan x\right)\right)\right)}{\left(\left(1 - \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right) \cdot \left(\left(1 - \tan \varepsilon \cdot \tan x\right) + \left(\tan \varepsilon \cdot \tan x\right) \cdot \left(\tan \varepsilon \cdot \tan x\right)\right)\right) \cdot \cos x} \le 2.1322419203862984 \cdot 10^{-16}:\\ \;\;\;\;\varepsilon + \left({\varepsilon}^{3} \cdot {x}^{2} + {\varepsilon}^{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x + \tan \varepsilon}{1 - \left(\sqrt[3]{\tan x \cdot \tan \varepsilon} \cdot \sqrt[3]{\tan x \cdot \tan \varepsilon}\right) \cdot \sqrt[3]{\tan x \cdot \tan \varepsilon}} - \tan x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original36.7
Target14.9
Herbie14.4
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (- (* (+ (tan x) (tan eps)) (+ (* (cos x) (pow (* (tan eps) (tan x)) 3)) (cos x))) (* (- 1 (* (* (tan eps) (tan x)) (* (tan eps) (tan x)))) (* (sin x) (+ (* (* (tan eps) (tan x)) (* (tan eps) (tan x))) (- 1 (* (tan eps) (tan x))))))) (* (* (- 1 (* (* (tan eps) (tan x)) (* (tan eps) (tan x)))) (+ (- 1 (* (tan eps) (tan x))) (* (* (tan eps) (tan x)) (* (tan eps) (tan x))))) (cos x))) < -3.562198966313061e-16

    1. Initial program 30.9

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

    3. Applied tan-sum3.7

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

    5. Applied flip--3.7

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

    6. Applied associate-/r/3.7

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

    7. Applied simplify3.7

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

    9. Applied flip--3.8

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

    if -3.562198966313061e-16 < (/ (- (* (+ (tan x) (tan eps)) (+ (* (cos x) (pow (* (tan eps) (tan x)) 3)) (cos x))) (* (- 1 (* (* (tan eps) (tan x)) (* (tan eps) (tan x)))) (* (sin x) (+ (* (* (tan eps) (tan x)) (* (tan eps) (tan x))) (- 1 (* (tan eps) (tan x))))))) (* (* (- 1 (* (* (tan eps) (tan x)) (* (tan eps) (tan x)))) (+ (- 1 (* (tan eps) (tan x))) (* (* (tan eps) (tan x)) (* (tan eps) (tan x))))) (cos x))) < 2.1322419203862984e-16

    1. Initial program 43.8

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

    2. Taylor expanded around 0 27.3

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

    if 2.1322419203862984e-16 < (/ (- (* (+ (tan x) (tan eps)) (+ (* (cos x) (pow (* (tan eps) (tan x)) 3)) (cos x))) (* (- 1 (* (* (tan eps) (tan x)) (* (tan eps) (tan x)))) (* (sin x) (+ (* (* (tan eps) (tan x)) (* (tan eps) (tan x))) (- 1 (* (tan eps) (tan x))))))) (* (* (- 1 (* (* (tan eps) (tan x)) (* (tan eps) (tan x)))) (+ (- 1 (* (tan eps) (tan x))) (* (* (tan eps) (tan x)) (* (tan eps) (tan x))))) (cos x)))

    1. Initial program 31.2

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

    3. Applied tan-sum4.6

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

    5. Applied add-cube-cbrt4.8

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

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' 
(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)))