Average Error: 27.3 → 1.3
Time: 1.7m
Precision: 64
Internal Precision: 576
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\cos x \cdot \cos x}{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}} - \left(\sqrt[3]{\frac{\sin x \cdot \sin x}{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}}} \cdot \sqrt[3]{\frac{\sin x \cdot \sin x}{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}}}\right) \cdot \sqrt[3]{\frac{\sin x \cdot \sin x}{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}}} \le 7.902883752881403 \cdot 10^{-308}:\\ \;\;\;\;\left(\frac{\sin x}{\left|x \cdot \left(cos \cdot sin\right)\right|} + \frac{\cos x}{\left|x \cdot \left(cos \cdot sin\right)\right|}\right) \cdot \left(\frac{\cos x}{\left|x \cdot \left(cos \cdot sin\right)\right|} - \frac{\sin x}{\left|\left(\sqrt[3]{x \cdot \left(cos \cdot sin\right)} \cdot \sqrt[3]{x \cdot \left(cos \cdot sin\right)}\right) \cdot \sqrt[3]{x \cdot \left(cos \cdot sin\right)}\right|}\right)\\ \mathbf{if}\;\frac{\cos x \cdot \cos x}{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}} - \left(\sqrt[3]{\frac{\sin x \cdot \sin x}{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}}} \cdot \sqrt[3]{\frac{\sin x \cdot \sin x}{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}}}\right) \cdot \sqrt[3]{\frac{\sin x \cdot \sin x}{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}}} \le 5.711344918322572 \cdot 10^{+242}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{cos \cdot \left(x \cdot sin\right)}}{cos \cdot \left(x \cdot sin\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sin x}{\left|x \cdot \left(cos \cdot sin\right)\right|} + \frac{\cos x}{\left|x \cdot \left(cos \cdot sin\right)\right|}\right) \cdot \left(\frac{\cos x}{\left|x \cdot \left(cos \cdot sin\right)\right|} - \frac{\sin x}{\left|\left(\sqrt[3]{x \cdot \left(cos \cdot sin\right)} \cdot \sqrt[3]{x \cdot \left(cos \cdot sin\right)}\right) \cdot \sqrt[3]{x \cdot \left(cos \cdot sin\right)}\right|}\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (- (/ (* (cos x) (cos x)) (pow (* cos (* x sin)) 2)) (* (* (cbrt (/ (* (sin x) (sin x)) (pow (* cos (* x sin)) 2))) (cbrt (/ (* (sin x) (sin x)) (pow (* cos (* x sin)) 2)))) (cbrt (/ (* (sin x) (sin x)) (pow (* cos (* x sin)) 2))))) < 7.902883752881403e-308 or 5.711344918322572e+242 < (- (/ (* (cos x) (cos x)) (pow (* cos (* x sin)) 2)) (* (* (cbrt (/ (* (sin x) (sin x)) (pow (* cos (* x sin)) 2))) (cbrt (/ (* (sin x) (sin x)) (pow (* cos (* x sin)) 2)))) (cbrt (/ (* (sin x) (sin x)) (pow (* cos (* x sin)) 2)))))

    1. Initial program 18.9

      \[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]

    2. Taylor expanded around 0 3.4

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}}}\]
    3. Using strategy rm

    4. Applied cos-23.4

      \[\leadsto \frac{\color{blue}{\cos x \cdot \cos x - \sin x \cdot \sin x}}{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}}\]

    5. Applied div-sub3.4

      \[\leadsto \color{blue}{\frac{\cos x \cdot \cos x}{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}} - \frac{\sin x \cdot \sin x}{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}}}\]
    6. Using strategy rm

    7. Applied add-sqr-sqrt3.4

      \[\leadsto \frac{\cos x \cdot \cos x}{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}} - \frac{\sin x \cdot \sin x}{\color{blue}{\sqrt{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}} \cdot \sqrt{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}}}}\]

    8. Applied times-frac3.4

      \[\leadsto \frac{\cos x \cdot \cos x}{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}} - \color{blue}{\frac{\sin x}{\sqrt{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}}} \cdot \frac{\sin x}{\sqrt{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}}}}\]

    9. Applied add-sqr-sqrt3.4

      \[\leadsto \frac{\cos x \cdot \cos x}{\color{blue}{\sqrt{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}} \cdot \sqrt{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}}}} - \frac{\sin x}{\sqrt{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}}} \cdot \frac{\sin x}{\sqrt{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}}}\]

    10. Applied times-frac3.4

      \[\leadsto \color{blue}{\frac{\cos x}{\sqrt{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}}} \cdot \frac{\cos x}{\sqrt{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}}}} - \frac{\sin x}{\sqrt{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}}} \cdot \frac{\sin x}{\sqrt{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}}}\]

    11. Applied difference-of-squares3.4

      \[\leadsto \color{blue}{\left(\frac{\cos x}{\sqrt{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}}} + \frac{\sin x}{\sqrt{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}}}\right) \cdot \left(\frac{\cos x}{\sqrt{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}}} - \frac{\sin x}{\sqrt{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}}}\right)}\]

    12. Applied simplify4.3

      \[\leadsto \color{blue}{\left(\frac{\sin x}{\left|x \cdot \left(cos \cdot sin\right)\right|} + \frac{\cos x}{\left|x \cdot \left(cos \cdot sin\right)\right|}\right)} \cdot \left(\frac{\cos x}{\sqrt{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}}} - \frac{\sin x}{\sqrt{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}}}\right)\]

    13. Applied simplify1.3

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

    15. Applied add-cube-cbrt1.4

      \[\leadsto \left(\frac{\sin x}{\left|x \cdot \left(cos \cdot sin\right)\right|} + \frac{\cos x}{\left|x \cdot \left(cos \cdot sin\right)\right|}\right) \cdot \left(\frac{\cos x}{\left|x \cdot \left(cos \cdot sin\right)\right|} - \frac{\sin x}{\left|\color{blue}{\left(\sqrt[3]{x \cdot \left(cos \cdot sin\right)} \cdot \sqrt[3]{x \cdot \left(cos \cdot sin\right)}\right) \cdot \sqrt[3]{x \cdot \left(cos \cdot sin\right)}}\right|}\right)\]

    if 7.902883752881403e-308 < (- (/ (* (cos x) (cos x)) (pow (* cos (* x sin)) 2)) (* (* (cbrt (/ (* (sin x) (sin x)) (pow (* cos (* x sin)) 2))) (cbrt (/ (* (sin x) (sin x)) (pow (* cos (* x sin)) 2)))) (cbrt (/ (* (sin x) (sin x)) (pow (* cos (* x sin)) 2))))) < 5.711344918322572e+242

    1. Initial program 44.1

      \[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]

    2. Taylor expanded around 0 1.0

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{{\left(cos \cdot \left(x \cdot sin\right)\right)}^{2}}}\]
    3. Using strategy rm

    4. Applied unpow21.0

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(cos \cdot \left(x \cdot sin\right)\right) \cdot \left(cos \cdot \left(x \cdot sin\right)\right)}}\]

    5. Applied associate-/r*1.0

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{cos \cdot \left(x \cdot sin\right)}}{cos \cdot \left(x \cdot sin\right)}}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2018195 
(FPCore (x cos sin)
  :name "cos(2*x)/(cos^2(x)*sin^2(x))"
  (/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x))))