Average Error: 27.1 → 2.5
Time: 46.7s
Precision: 64
Internal Precision: 384
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{cos \cdot \left(cos \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)\right)} \le -1.7987478660952937 \cdot 10^{-258}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left|\left(cos \cdot sin\right) \cdot x\right| \cdot \left|\left(cos \cdot sin\right) \cdot x\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\left|\left(x \cdot cos\right) \cdot sin\right|} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left|\left(x \cdot cos\right) \cdot sin\right|}\\ \end{array}\]

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)) (* cos (* cos (* (* x (pow sin 2)) x)))) < -1.7987478660952937e-258

    1. Initial program 40.2

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

    3. Applied add-sqr-sqrt40.2

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

    4. Applied simplify40.2

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

    5. Applied simplify6.3

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

    7. Applied add-cube-cbrt6.8

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

    8. Applied times-frac6.8

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

    9. Taylor expanded around inf 6.8

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

    10. Applied simplify3.8

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

    if -1.7987478660952937e-258 < (/ (cos (* 2 x)) (* cos (* cos (* (* x (pow sin 2)) x))))

    1. Initial program 25.7

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

    3. Applied add-sqr-sqrt25.7

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

    4. Applied simplify25.6

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

    5. Applied simplify2.5

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

    7. Applied add-cube-cbrt2.6

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

    8. Applied times-frac2.3

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

Runtime

Time bar (total: 46.7s)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))))