Average Error: 27.4 → 1.7
Time: 48.7s
Precision: 64
Internal Precision: 576
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{\sqrt{{cos}^{2}} \cdot \left(\left(\left(sin \cdot x\right) \cdot \left(sin \cdot x\right)\right) \cdot \left|cos\right|\right)} \le 7.052385851237702 \cdot 10^{+163}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{\left|cos \cdot \left(x \cdot sin\right)\right|}}{\left|cos \cdot \left(x \cdot sin\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\left|\left(x \cdot cos\right) \cdot sin\right|} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left|\left(x \cdot cos\right) \cdot sin\right|}\\ \end{array}\]

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)) (* (sqrt (pow cos 2)) (* (* (* sin x) (* sin x)) (fabs cos)))) < 7.052385851237702e+163

    1. Initial program 17.6

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

    3. Applied add-sqr-sqrt17.6

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

    4. Applied simplify17.6

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

    5. Applied simplify2.2

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

    7. Applied add-cube-cbrt2.3

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

    8. Applied times-frac2.0

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

    9. Taylor expanded around inf 23.3

      \[\leadsto \frac{\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}{\left|\left(x \cdot cos\right) \cdot sin\right|} \cdot \frac{\color{blue}{{\left(\cos \left(2 \cdot x\right)\right)}^{\frac{1}{3}}}}{\left|\left(x \cdot cos\right) \cdot sin\right|}\]

    10. Applied simplify0.3

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

    if 7.052385851237702e+163 < (/ (cos (* 2 x)) (* (sqrt (pow cos 2)) (* (* (* sin x) (* sin x)) (fabs cos))))

    1. Initial program 56.4

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

    3. Applied add-sqr-sqrt56.4

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

    4. Applied simplify56.3

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

    5. Applied simplify5.7

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

    7. Applied add-cube-cbrt5.9

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

    8. Applied times-frac5.8

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

Runtime

Time bar (total: 48.7s)Debug logProfile

herbie shell --seed '#(1071821486 549052472 3784827256 1559736200 3548510075 881134285)' +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))))