Average Error: 27.2 → 1.8
Time: 1.0m
Precision: 64
Internal Precision: 576
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\cos x \cdot \cos x}{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|} - \frac{\sin x \cdot \sin x}{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|} \le 2.6423667567049956 \cdot 10^{-293}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(\left|\left|\left(sin \cdot cos\right) \cdot x\right|\right| \cdot \sqrt{\left|\left|\left(sin \cdot cos\right) \cdot x\right|\right|}\right) \cdot \sqrt{\left|\left|\left(sin \cdot cos\right) \cdot x\right|\right|}}\\ \mathbf{if}\;\frac{\cos x \cdot \cos x}{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|} - \frac{\sin x \cdot \sin x}{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|} \le 6.996832546986506 \cdot 10^{+283}:\\ \;\;\;\;\frac{\cos x \cdot \cos x}{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|} - \frac{\sin x \cdot \sin x}{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left|\left|\left(x \cdot sin\right) \cdot cos\right|\right| \cdot \left|\left|\left(sin \cdot cos\right) \cdot x\right|\right|}\\ \end{array}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (- (/ (* (cos x) (cos x)) (* (fabs (* (* x cos) sin)) (fabs (* (* x cos) sin)))) (/ (* (sin x) (sin x)) (* (fabs (* (* x cos) sin)) (fabs (* (* x cos) sin))))) < 2.6423667567049956e-293

    1. Initial program 16.4

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

    3. Applied add-sqr-sqrt16.4

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

    4. Applied simplify16.4

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

    5. Applied simplify2.2

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

    7. Applied add-sqr-sqrt2.2

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

    8. Applied simplify3.2

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

    9. Applied simplify1.8

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

    11. Applied add-sqr-sqrt1.8

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

    12. Applied associate-*r*1.8

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

    if 2.6423667567049956e-293 < (- (/ (* (cos x) (cos x)) (* (fabs (* (* x cos) sin)) (fabs (* (* x cos) sin)))) (/ (* (sin x) (sin x)) (* (fabs (* (* x cos) sin)) (fabs (* (* x cos) sin))))) < 6.996832546986506e+283

    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 simplify1.0

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

    7. Applied cos-21.1

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

    8. Applied div-sub1.1

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

    if 6.996832546986506e+283 < (- (/ (* (cos x) (cos x)) (* (fabs (* (* x cos) sin)) (fabs (* (* x cos) sin)))) (/ (* (sin x) (sin x)) (* (fabs (* (* x cos) sin)) (fabs (* (* x cos) sin)))))

    1. Initial program 53.0

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

    3. Applied add-sqr-sqrt53.0

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

    4. Applied simplify52.3

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

    5. Applied simplify30.5

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

    7. Applied add-sqr-sqrt30.5

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

    8. Applied simplify32.4

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

    9. Applied simplify2.8

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

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

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

    12. Applied associate-*r*2.8

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

    13. Applied simplify7.4

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

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed '#(1072967564 1937075727 894099792 790700740 1036514779 1027793188)' +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))))