Average Error: 37.1 → 14.5
Time: 1.4m
Precision: 64
Internal Precision: 2432
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\sqrt[3]{\tan x + \tan \varepsilon} \cdot \sqrt[3]{\tan x + \tan \varepsilon}\right) \cdot \sqrt[3]{\tan x + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \le -2.5012544119240587 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{\frac{1 - \tan x \cdot \tan \varepsilon}{\tan x + \tan \varepsilon}} - \tan x\\ \mathbf{if}\;\frac{\left(\sqrt[3]{\tan x + \tan \varepsilon} \cdot \sqrt[3]{\tan x + \tan \varepsilon}\right) \cdot \sqrt[3]{\tan x + \tan \varepsilon}}{1 - \tan x \cdot \tan \varepsilon} - \tan x \le 8.881784197001252 \cdot 10^{-16}:\\ \;\;\;\;{\varepsilon}^{3} \cdot {x}^{2} + \left(\varepsilon + {\varepsilon}^{2} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan x \cdot \tan x - \tan \varepsilon \cdot \tan \varepsilon}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x - \tan \varepsilon\right)} - \tan x\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original37.1
Target15.1
Herbie14.5
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if (- (/ (* (* (cbrt (+ (tan x) (tan eps))) (cbrt (+ (tan x) (tan eps)))) (cbrt (+ (tan x) (tan eps)))) (- 1 (* (tan x) (tan eps)))) (tan x)) < -2.5012544119240587e-16

    1. Initial program 34.3

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

    3. Applied tan-sum9.6

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

    5. Applied clear-num9.7

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

    if -2.5012544119240587e-16 < (- (/ (* (* (cbrt (+ (tan x) (tan eps))) (cbrt (+ (tan x) (tan eps)))) (cbrt (+ (tan x) (tan eps)))) (- 1 (* (tan x) (tan eps)))) (tan x)) < 8.881784197001252e-16

    1. Initial program 42.3

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

    2. Taylor expanded around 0 24.3

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

    if 8.881784197001252e-16 < (- (/ (* (* (cbrt (+ (tan x) (tan eps))) (cbrt (+ (tan x) (tan eps)))) (cbrt (+ (tan x) (tan eps)))) (- 1 (* (tan x) (tan eps)))) (tan x))

    1. Initial program 32.1

      \[\tan \left(x + \varepsilon\right) - \tan x\]
    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 flip-+4.7

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

    6. Applied associate-/l/4.7

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

Runtime

Time bar (total: 1.4m)Debug log

herbie shell --seed '#(212267722 3993171362 1093346726 3605783651 2106536041 3335990851)' 
(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)))