Average Error: 27.3 → 1.9
Time: 1.4m
Precision: 64
Internal Precision: 128
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;sin \le -3.0137980008127757 \cdot 10^{-257}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{sin \cdot \left(cos \cdot x\right)}}{sin \cdot \left(cos \cdot x\right)}\\ \mathbf{elif}\;sin \le 1.2693255601299709 \cdot 10^{-197}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(\sqrt[3]{\left(\left(x \cdot sin\right) \cdot cos\right) \cdot \left(\left(x \cdot sin\right) \cdot cos\right)} \cdot \sqrt[3]{\left(\left(x \cdot sin\right) \cdot cos\right) \cdot \left(\left(x \cdot sin\right) \cdot cos\right)}\right) \cdot \sqrt[3]{\left(\left(x \cdot sin\right) \cdot cos\right) \cdot \left(\left(x \cdot sin\right) \cdot cos\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{sin \cdot \left(cos \cdot x\right)}}{sin \cdot \left(cos \cdot x\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

  1. Split input into 2 regimes

  2. if sin < -3.0137980008127757e-257 or 1.2693255601299709e-197 < sin

    1. Initial program 24.7

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

    2. Simplified2.0

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

    4. Applied associate-/r*1.7

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

    if -3.0137980008127757e-257 < sin < 1.2693255601299709e-197

    1. Initial program 61.7

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

    2. Simplified14.4

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

    3. Taylor expanded around -inf 62.1

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

    4. Simplified4.5

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

    6. Applied add-cube-cbrt5.1

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

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;sin \le -3.0137980008127757 \cdot 10^{-257}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{sin \cdot \left(cos \cdot x\right)}}{sin \cdot \left(cos \cdot x\right)}\\ \mathbf{elif}\;sin \le 1.2693255601299709 \cdot 10^{-197}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(\sqrt[3]{\left(\left(x \cdot sin\right) \cdot cos\right) \cdot \left(\left(x \cdot sin\right) \cdot cos\right)} \cdot \sqrt[3]{\left(\left(x \cdot sin\right) \cdot cos\right) \cdot \left(\left(x \cdot sin\right) \cdot cos\right)}\right) \cdot \sqrt[3]{\left(\left(x \cdot sin\right) \cdot cos\right) \cdot \left(\left(x \cdot sin\right) \cdot cos\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{sin \cdot \left(cos \cdot x\right)}}{sin \cdot \left(cos \cdot x\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019090 +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))))