Average Error: 27.1 → 1.3
Time: 50.0s
Precision: 64
Internal Precision: 384
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\cos x \cdot \cos x}{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|} - \frac{\sin x \cdot \sin x}{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|} \le 4.90694831233465 \cdot 10^{-281}:\\ \;\;\;\;\frac{\frac{\cos x \cdot \cos x}{\left|\left|\left(sin \cdot cos\right) \cdot x\right|\right|} - \frac{\sin x \cdot \sin x}{\left|\left|\left(sin \cdot cos\right) \cdot x\right|\right|}}{\left|\left|\left(sin \cdot cos\right) \cdot x\right|\right|}\\ \mathbf{if}\;\frac{\cos x \cdot \cos x}{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|} - \frac{\sin x \cdot \sin x}{\left|\left(x \cdot cos\right) \cdot sin\right| \cdot \left|\left(x \cdot cos\right) \cdot sin\right|} \le 4.836898521975331 \cdot 10^{+299}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{\left|\left(x \cdot cos\right) \cdot sin\right|}}{\left|\left(x \cdot cos\right) \cdot sin\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos x \cdot \cos x}{\left|\left|\left(sin \cdot cos\right) \cdot x\right|\right|} - \frac{\sin x \cdot \sin x}{\left|\left|\left(sin \cdot cos\right) \cdot x\right|\right|}}{\left|\left|\left(sin \cdot cos\right) \cdot x\right|\right|}\\ \end{array}\]

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Derivation

  1. Split input into 2 regimes

  2. if (- (/ (* (cos x) (cos x)) (* (fabs (* (* x cos) sin)) (fabs (* (* x cos) sin)))) (/ (* (sin x) (sin x)) (* (fabs (* (* x cos) sin)) (fabs (* (* x cos) sin))))) < 4.90694831233465e-281 or 4.836898521975331e+299 < (- (/ (* (cos x) (cos x)) (* (fabs (* (* x cos) sin)) (fabs (* (* x cos) sin)))) (/ (* (sin x) (sin x)) (* (fabs (* (* x cos) sin)) (fabs (* (* x cos) sin)))))

    1. Initial program 18.4

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

    3. Applied add-sqr-sqrt18.4

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

    4. Applied simplify18.4

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

    5. Applied simplify4.1

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

    7. Applied add-sqr-sqrt4.1

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

    8. Applied simplify5.2

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

    9. Applied simplify1.8

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

    11. Applied associate-/r*1.4

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

    13. Applied cos-21.4

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

    14. Applied div-sub1.5

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

    if 4.90694831233465e-281 < (- (/ (* (cos x) (cos x)) (* (fabs (* (* x cos) sin)) (fabs (* (* x cos) sin)))) (/ (* (sin x) (sin x)) (* (fabs (* (* x cos) sin)) (fabs (* (* x cos) sin))))) < 4.836898521975331e+299

    1. Initial program 43.5

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

    3. Applied add-sqr-sqrt43.6

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

    4. Applied simplify43.5

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

    5. Applied simplify1.1

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

    7. Applied associate-/r*1.1

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

Runtime

Time bar (total: 50.0s)Debug logProfile

herbie shell --seed '#(1070386091 2509006183 1430610344 1025408621 36622005 1425925650)' +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))))