Average Error: 26.9 → 1.2
Time: 46.6s
Precision: 64
Internal Precision: 576
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{\frac{\cos \left(2 \cdot x\right)}{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|}} \cdot \sqrt{\frac{\cos \left(2 \cdot x\right)}{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|}} \le 6.843294812283126 \cdot 10^{-305}:\\ \;\;\;\;\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\left|\left|\left(sin \cdot cos\right) \cdot x\right|\right|} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left|\left|\left(sin \cdot cos\right) \cdot x\right|\right|}\\ \mathbf{if}\;\sqrt{\frac{\cos \left(2 \cdot x\right)}{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|}} \cdot \sqrt{\frac{\cos \left(2 \cdot x\right)}{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|}} \le 1.3836647264228708 \cdot 10^{+290}:\\ \;\;\;\;\sqrt{\frac{\cos \left(2 \cdot x\right)}{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|}} \cdot \sqrt{\frac{\cos \left(2 \cdot x\right)}{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos x \cdot \cos x}{\left|\left|\left(sin \cdot cos\right) \cdot x\right|\right| \cdot \left|\left|\left(sin \cdot cos\right) \cdot x\right|\right|} - \frac{\sin x \cdot \sin x}{\left|\left|\left(sin \cdot cos\right) \cdot x\right|\right| \cdot \left|\left|\left(sin \cdot cos\right) \cdot x\right|\right|}\\ \end{array}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

  1. Split input into 3 regimes

  2. if (* (sqrt (/ (cos (* 2 x)) (* (fabs (* (* x cos) sin)) (fabs (* (* x cos) sin))))) (sqrt (/ (cos (* 2 x)) (* (fabs (* (* x cos) sin)) (fabs (* (* x cos) sin)))))) < 6.843294812283126e-305

    1. Initial program 9.2

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

    3. Applied add-sqr-sqrt9.2

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

    4. Applied simplify9.2

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

    5. Applied simplify2.6

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

    7. Applied add-sqr-sqrt2.6

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

    8. Applied simplify2.7

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

    9. Applied simplify0.7

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

    11. Applied add-cube-cbrt0.7

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}}{\left|\left|\left(sin \cdot cos\right) \cdot x\right|\right| \cdot \left|\left|\left(sin \cdot cos\right) \cdot x\right|\right|}\]

    12. Applied times-frac0.2

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\left|\left|\left(sin \cdot cos\right) \cdot x\right|\right|} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left|\left|\left(sin \cdot cos\right) \cdot x\right|\right|}}\]

    if 6.843294812283126e-305 < (* (sqrt (/ (cos (* 2 x)) (* (fabs (* (* x cos) sin)) (fabs (* (* x cos) sin))))) (sqrt (/ (cos (* 2 x)) (* (fabs (* (* x cos) sin)) (fabs (* (* x cos) sin)))))) < 1.3836647264228708e+290

    1. Initial program 43.2

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

    3. Applied add-sqr-sqrt43.3

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

    4. Applied simplify43.2

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

    5. Applied simplify0.9

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

    7. Applied add-sqr-sqrt1.1

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

    if 1.3836647264228708e+290 < (* (sqrt (/ (cos (* 2 x)) (* (fabs (* (* x cos) sin)) (fabs (* (* x cos) sin))))) (sqrt (/ (cos (* 2 x)) (* (fabs (* (* x cos) sin)) (fabs (* (* x cos) sin))))))

    1. Initial program 45.5

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

    3. Applied add-sqr-sqrt45.5

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

    4. Applied simplify45.3

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

    5. Applied simplify7.7

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

    7. Applied add-sqr-sqrt7.7

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

    8. Applied simplify11.0

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

    9. Applied simplify4.3

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

    11. Applied cos-24.5

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

    12. Applied div-sub4.7

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

Runtime

Time bar (total: 46.6s)Debug logProfile

herbie shell --seed '#(1072361757 3390613284 2339397988 1175251238 145061547 3101881848)' 
(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))))