Average Error: 27.4 → 1.4
Time: 52.1s
Precision: 64
Internal Precision: 320
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{\left|\left(x \cdot cos\right) \cdot sin\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left|\left(x \cdot cos\right) \cdot sin\right|} \le -7.899071278097851 \cdot 10^{-196}:\\ \;\;\;\;\frac{1}{\left|\left(x \cdot cos\right) \cdot sin\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left|\left(x \cdot cos\right) \cdot sin\right|}\\ \mathbf{if}\;\frac{1}{\left|\left(x \cdot cos\right) \cdot sin\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left|\left(x \cdot cos\right) \cdot sin\right|} \le 4.955557478291 \cdot 10^{-318}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{\left|cos \cdot \left(sin \cdot x\right)\right|}}{\left|\left(cos \cdot sin\right) \cdot x\right|}\\ \mathbf{if}\;\frac{1}{\left|\left(x \cdot cos\right) \cdot sin\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left|\left(x \cdot cos\right) \cdot sin\right|} \le 7.955758258939924 \cdot 10^{+40}:\\ \;\;\;\;\frac{1}{\left|\left(x \cdot cos\right) \cdot sin\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left|\left(x \cdot cos\right) \cdot sin\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{\left|\left(\sqrt[3]{\left(cos \cdot sin\right) \cdot x} \cdot \sqrt[3]{\left(cos \cdot sin\right) \cdot x}\right) \cdot \sqrt[3]{\left(cos \cdot sin\right) \cdot x}\right|}}{\left|\left(cos \cdot sin\right) \cdot x\right|}\\ \end{array}\]

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 3 regimes

  2. if (* (/ 1 (fabs (* (* x cos) sin))) (/ (cos (* 2 x)) (fabs (* (* x cos) sin)))) < -7.899071278097851e-196 or 4.955557478291e-318 < (* (/ 1 (fabs (* (* x cos) sin))) (/ (cos (* 2 x)) (fabs (* (* x cos) sin)))) < 7.955758258939924e+40

    1. Initial program 43.8

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

    3. Applied add-sqr-sqrt43.9

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

    4. Applied simplify43.8

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

    5. Applied simplify1.2

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

    7. Applied *-un-lft-identity1.2

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

    8. Applied times-frac0.8

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

    if -7.899071278097851e-196 < (* (/ 1 (fabs (* (* x cos) sin))) (/ (cos (* 2 x)) (fabs (* (* x cos) sin)))) < 4.955557478291e-318

    1. Initial program 11.0

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

    3. Applied add-sqr-sqrt11.0

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

    4. Applied simplify11.0

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

    5. Applied simplify2.8

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

    7. Applied *-un-lft-identity2.8

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

    8. Applied times-frac2.6

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

    9. Taylor expanded around inf 2.6

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

    10. Applied simplify0.1

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

    12. Applied associate-*l*0.7

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

    if 7.955758258939924e+40 < (* (/ 1 (fabs (* (* x cos) sin))) (/ (cos (* 2 x)) (fabs (* (* x cos) sin))))

    1. Initial program 44.3

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

    3. Applied add-sqr-sqrt44.4

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

    4. Applied simplify44.2

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

    5. Applied simplify6.0

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

    7. Applied *-un-lft-identity6.0

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

    8. Applied times-frac6.1

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

    9. Taylor expanded around inf 6.1

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

    10. Applied simplify3.4

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

    12. Applied add-cube-cbrt4.3

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

Runtime

Time bar (total: 52.1s)Debug logProfile

herbie shell --seed 2018167 
(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))))