Average Error: 27.1 → 2.0
Time: 44.9s
Precision: 64
Internal Precision: 384
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\left(\sqrt[3]{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|} \cdot \sqrt[3]{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|}\right) \cdot \sqrt[3]{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|}} \le 4.920500141204781 \cdot 10^{-288}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{\left|\left(cos \cdot sin\right) \cdot x\right|}}{\left|\left(cos \cdot sin\right) \cdot x\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos x \cdot \cos x}{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|} - \frac{\sin x \cdot \sin x}{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|}\\ \end{array}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

  1. Split input into 2 regimes

  2. if (/ (cos (* 2 x)) (* (* (cbrt (* (fabs (* (* x cos) sin)) (fabs (* (* x cos) sin)))) (cbrt (* (fabs (* (* x cos) sin)) (fabs (* (* x cos) sin))))) (cbrt (* (fabs (* (* x cos) sin)) (fabs (* (* x cos) sin)))))) < 4.920500141204781e-288

    1. Initial program 16.7

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

    3. Applied add-sqr-sqrt16.7

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

    4. Applied simplify16.7

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

    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*2.0

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

    9. Applied add-sqr-sqrt2.0

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

    10. Applied associate-/r*2.0

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

    11. Taylor expanded around inf 2.0

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

    12. Applied simplify1.0

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

    if 4.920500141204781e-288 < (/ (cos (* 2 x)) (* (* (cbrt (* (fabs (* (* x cos) sin)) (fabs (* (* x cos) sin)))) (cbrt (* (fabs (* (* x cos) sin)) (fabs (* (* x cos) sin))))) (cbrt (* (fabs (* (* x cos) sin)) (fabs (* (* x cos) sin))))))

    1. Initial program 43.9

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

    3. Applied add-sqr-sqrt44.0

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

    4. Applied 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 simplify3.6

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

    7. Applied cos-23.6

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

    8. Applied div-sub3.6

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

Runtime

Time bar (total: 44.9s)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' 
(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))))