Average Error: 27.2 → 7.7
Time: 39.8s
Precision: 64
Internal Precision: 384
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\cos \left(x + x\right)}{cos}}{\left(\left(sin \cdot x\right) \cdot \left(sin \cdot x\right)\right) \cdot cos} \le -0.0:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{cos \cdot \left(\left(\left(cos \cdot \left(x \cdot sin\right)\right) \cdot sin\right) \cdot x\right)}\\ \mathbf{if}\;\frac{\frac{\cos \left(x + x\right)}{cos}}{\left(\left(sin \cdot x\right) \cdot \left(sin \cdot x\right)\right) \cdot cos} \le 1.4545281233436587 \cdot 10^{-194}:\\ \;\;\;\;\frac{1}{cos} \cdot \frac{\cos \left(2 \cdot x\right)}{cos \cdot \left(\left(\left(x \cdot sin\right) \cdot sin\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{cos \cdot \left(\left(\left(cos \cdot \left(x \cdot sin\right)\right) \cdot sin\right) \cdot x\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

  1. Split input into 2 regimes

  2. if (/ (/ (cos (+ x x)) cos) (* (* (* sin x) (* sin x)) cos)) < -0.0 or 1.4545281233436587e-194 < (/ (/ (cos (+ x x)) cos) (* (* (* sin x) (* sin x)) cos))

    1. Initial program 26.4

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

    3. Applied unpow226.4

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

    4. Applied associate-*l*22.7

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

    6. Applied unpow222.7

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

    7. Applied associate-*r*16.3

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

    9. Applied associate-*r*13.0

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

    11. Applied associate-*r*7.6

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

    if -0.0 < (/ (/ (cos (+ x x)) cos) (* (* (* sin x) (* sin x)) cos)) < 1.4545281233436587e-194

    1. Initial program 41.9

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

    3. Applied unpow241.9

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

    4. Applied associate-*l*27.2

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

    6. Applied unpow227.2

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

    7. Applied associate-*r*12.5

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

    9. Applied *-un-lft-identity12.5

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

    10. Applied times-frac10.0

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

Runtime

Time bar (total: 39.8s)Debug logProfile

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' 
(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))))