Average Error: 27.1 → 2.1
Time: 44.9s
Precision: 64
Internal Precision: 384
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\cos \left(x \cdot 2\right)}{\left|cos \cdot \left(sin \cdot x\right)\right|}}{\left|cos \cdot \left(sin \cdot x\right)\right|} \le 1.5886540672244532 \cdot 10^{+218}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{\left|cos \cdot \left(sin \cdot x\right)\right|}}{\left|cos \cdot \left(sin \cdot x\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos x \cdot \cos x}{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|} - \frac{\sin x \cdot \sin x}{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|}\\ \end{array}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

  1. Split input into 2 regimes

  2. if (/ (/ (cos (* x 2)) (fabs (* cos (* sin x)))) (fabs (* cos (* sin x)))) < 1.5886540672244532e+218

    1. Initial program 25.4

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

    3. Applied add-sqr-sqrt25.5

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

    4. Applied simplify25.4

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

    5. Applied simplify2.4

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

    7. Applied associate-/r*2.2

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

    9. Applied add-cube-cbrt2.5

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

    10. Taylor expanded around inf 15.3

      \[\leadsto \frac{\left(\sqrt[3]{\frac{\cos \left(2 \cdot x\right)}{\left|\left(x \cdot cos\right) \cdot sin\right|}} \cdot \sqrt[3]{\frac{\cos \left(2 \cdot x\right)}{\left|\left(x \cdot cos\right) \cdot sin\right|}}\right) \cdot \color{blue}{{\left(\frac{\cos \left(2 \cdot x\right)}{\left|sin \cdot \left(x \cdot cos\right)\right|}\right)}^{\frac{1}{3}}}}{\left|\left(x \cdot cos\right) \cdot sin\right|}\]

    11. Applied simplify1.7

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

    if 1.5886540672244532e+218 < (/ (/ (cos (* x 2)) (fabs (* cos (* sin x)))) (fabs (* cos (* sin x))))

    1. Initial program 47.3

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

    3. Applied add-sqr-sqrt47.4

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

    4. Applied simplify47.2

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

    5. Applied 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 cos-27.2

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

    8. Applied div-sub7.2

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

Runtime

Time bar (total: 44.9s)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' +o rules:numerics
(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))))