Average Error: 36.8 → 13.8
Time: 1.9m
Precision: 64
Internal Precision: 2432
\[\tan \left(x + \varepsilon\right) - \tan x\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \le -1.015348260557885 \cdot 10^{-15}:\\ \;\;\;\;(\left(\tan x + \tan \varepsilon\right) \cdot \left(\frac{1}{1 - \tan x \cdot \tan \varepsilon}\right) + \left(-\tan x\right))_*\\ \mathbf{if}\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x} \le 1.5606657375067698 \cdot 10^{-11}:\\ \;\;\;\;(\left(x \cdot \varepsilon\right) \cdot \left((\left(x \cdot \varepsilon\right) \cdot \varepsilon + \varepsilon)_*\right) + \varepsilon)_*\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\tan x + \tan \varepsilon\right) \cdot \cos x - \left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \sin x}{\left(1 - \tan x \cdot \tan \varepsilon\right) \cdot \cos x}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Target

Original36.8
Target15.2
Herbie13.8
\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (- (* (+ (tan x) (tan eps)) (cos x)) (* (- 1 (* (tan x) (tan eps))) (sin x))) (* (- 1 (* (tan x) (tan eps))) (cos x))) < -1.015348260557885e-15

    1. Initial program 30.2

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

    3. Applied tan-sum1.8

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

    5. Applied div-inv1.8

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

    6. Applied fma-neg1.8

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

    if -1.015348260557885e-15 < (/ (- (* (+ (tan x) (tan eps)) (cos x)) (* (- 1 (* (tan x) (tan eps))) (sin x))) (* (- 1 (* (tan x) (tan eps))) (cos x))) < 1.5606657375067698e-11

    1. Initial program 44.6

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

    2. Taylor expanded around 0 29.1

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

    3. Applied simplify27.9

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

    if 1.5606657375067698e-11 < (/ (- (* (+ (tan x) (tan eps)) (cos x)) (* (- 1 (* (tan x) (tan eps))) (sin x))) (* (- 1 (* (tan x) (tan eps))) (cos x)))

    1. Initial program 29.4

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

    3. Applied tan-quot29.3

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

    4. Applied tan-sum0.7

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

    5. Applied frac-sub0.7

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

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed '#(1070578969 3140398606 632207097 462683394 1189254563 964980650)' +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)))