Average Error: 27.3 → 1.4
Time: 45.3s
Precision: 64
Internal Precision: 384
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\cos x \cdot \cos x - \sin x \cdot \sin x}{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|} \le 3.248271077516176 \cdot 10^{-275}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{\left|cos \cdot \left(sin \cdot x\right)\right|}}{{\left(\sqrt{\left|cos \cdot \left(x \cdot sin\right)\right|}\right)}^{2}}\\ \mathbf{if}\;\frac{\cos x \cdot \cos x - \sin x \cdot \sin x}{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|} \le 1.6228453383648464 \cdot 10^{+251}:\\ \;\;\;\;\frac{\cos x \cdot \cos x - \sin x \cdot \sin x}{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left|\left|\left(sin \cdot cos\right) \cdot x\right|\right| \cdot \left|\left|\left(sin \cdot cos\right) \cdot x\right|\right|}\\ \end{array}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

  1. Split input into 3 regimes

  2. if (/ (- (* (cos x) (cos x)) (* (sin x) (sin x))) (* (fabs (* (* x cos) sin)) (fabs (* (* x cos) sin)))) < 3.248271077516176e-275

    1. Initial program 17.0

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

    3. Applied add-sqr-sqrt17.0

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

    4. Applied simplify17.0

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

    5. Applied simplify2.6

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

    6. Taylor expanded around 0 1.6

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

    8. Applied add-sqr-sqrt1.7

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

    9. Applied unpow-prod-down1.7

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

    10. Applied associate-/r*1.4

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

    11. Applied simplify1.3

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

    if 3.248271077516176e-275 < (/ (- (* (cos x) (cos x)) (* (sin x) (sin x))) (* (fabs (* (* x cos) sin)) (fabs (* (* x cos) sin)))) < 1.6228453383648464e+251

    1. Initial program 43.5

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

    3. Applied add-sqr-sqrt43.6

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

    4. Applied simplify43.5

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

    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 double-sum1.0

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

    8. Applied cos-sum1.1

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

    if 1.6228453383648464e+251 < (/ (- (* (cos x) (cos x)) (* (sin x) (sin x))) (* (fabs (* (* x cos) sin)) (fabs (* (* x cos) sin))))

    1. Initial program 49.6

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

    3. Applied add-sqr-sqrt49.7

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

    4. Applied simplify49.3

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

    5. Applied simplify21.5

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

    7. Applied add-sqr-sqrt21.5

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

    8. Applied simplify24.6

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

    9. Applied simplify3.8

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

Runtime

Time bar (total: 45.3s)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' 
(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))))