Average Error: 27.4 → 2.1
Time: 51.4s
Precision: 64
Internal Precision: 576
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\cos \left(x \cdot 2\right)}{\left|cos \cdot \left(sin \cdot x\right)\right| \cdot \sqrt[3]{\left|cos \cdot \left(sin \cdot x\right)\right|}}}{{\left(\sqrt[3]{\left|cos \cdot \left(x \cdot sin\right)\right|}\right)}^{2}} \le -1.3491970326677394 \cdot 10^{-296}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(\left|cos \cdot \left(x \cdot sin\right)\right|\right)}^{2}}\\ \mathbf{if}\;\frac{\frac{\cos \left(x \cdot 2\right)}{\left|cos \cdot \left(sin \cdot x\right)\right| \cdot \sqrt[3]{\left|cos \cdot \left(sin \cdot x\right)\right|}}}{{\left(\sqrt[3]{\left|cos \cdot \left(x \cdot sin\right)\right|}\right)}^{2}} \le 1.203217726821972 \cdot 10^{-225}:\\ \;\;\;\;\cos \left(2 \cdot x\right) \cdot \frac{1}{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(\left|cos \cdot \left(x \cdot sin\right)\right|\right)}^{2}}\\ \end{array}\]

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 2 regimes

  2. if (/ (/ (cos (* x 2)) (* (fabs (* cos (* sin x))) (cbrt (fabs (* cos (* sin x)))))) (pow (cbrt (fabs (* cos (* x sin)))) 2)) < -1.3491970326677394e-296 or 1.203217726821972e-225 < (/ (/ (cos (* x 2)) (* (fabs (* cos (* sin x))) (cbrt (fabs (* cos (* sin x)))))) (pow (cbrt (fabs (* cos (* x sin)))) 2))

    1. Initial program 44.2

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

    3. Applied add-sqr-sqrt44.3

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

    4. Applied simplify44.2

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

    5. Applied simplify4.5

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

    6. Taylor expanded around 0 3.0

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

    if -1.3491970326677394e-296 < (/ (/ (cos (* x 2)) (* (fabs (* cos (* sin x))) (cbrt (fabs (* cos (* sin x)))))) (pow (cbrt (fabs (* cos (* x sin)))) 2)) < 1.203217726821972e-225

    1. Initial program 12.2

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

    3. Applied add-sqr-sqrt12.2

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

    4. Applied simplify12.2

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

    5. Applied simplify1.3

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

    7. Applied div-inv1.3

      \[\leadsto \color{blue}{\cos \left(2 \cdot x\right) \cdot \frac{1}{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|}}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 51.4s)Debug logProfile

herbie shell --seed 2018166 +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))))