mixedcos

Average Error: 27.9 → 5.1

Time: 9.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.33159826832465384 \cdot 10^{-194}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot \left(x \cdot {\left(c \cdot s\right)}^{\left(2 \cdot \frac{1}{2}\right)}\right)\right) \cdot {\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}}\\ \mathbf{elif}\;x \le 4.35099145165978745 \cdot 10^{-206}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left({\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {\left(\frac{1}{{\left({\left(\frac{1}{c}\right)}^{1}\right)}^{1} \cdot {\left({\left(\frac{1}{s}\right)}^{1}\right)}^{1}}\right)}^{1}\right)\right)}\\ \mathbf{elif}\;x \le 2.1188823942095531 \cdot 10^{141}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{{\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot \left(x \cdot {\left(\frac{1}{{\left({c}^{\left(-1\right)}\right)}^{1} \cdot {\left({s}^{\left(-1\right)}\right)}^{1}}\right)}^{1}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{x}}{\left(x \cdot {\left(c \cdot s\right)}^{\left(2 \cdot \frac{1}{2}\right)}\right) \cdot {\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus c

Bits error versus s

Derivation

1. Split input into 4 regimes

2. if x < -1.33159826832465384e-194

   1. Initial program 25.7

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

   2. Simplified 25.8

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

   4. Applied pow-prod-down 10.8

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

   6. Applied sqr-pow 10.8

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

   7. Applied associate-*r* 5.3

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

   8. Simplified 5.3

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\color{blue}{\left(x \cdot {\left(c \cdot s\right)}^{\left(\frac{1}{2} \cdot 2\right)}\right)} \cdot {\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}\right)}\]
   9. Using strategy rm

   10. Applied associate-*r* 4.3

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

   if -1.33159826832465384e-194 < x < 4.35099145165978745e-206

   1. Initial program 46.3

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

   2. Simplified 47.6

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

   4. Applied pow-prod-down 40.3

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

   6. Applied sqr-pow 40.3

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

   7. Applied associate-*r* 18.3

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

   8. Simplified 18.3

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

   9. Taylor expanded around inf 50.7

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\color{blue}{\left(x \cdot {\left(\frac{1}{{\left(e^{1 \cdot \left(\log 1 + \log \left(\frac{1}{c}\right)\right)}\right)}^{1} \cdot {\left(e^{1 \cdot \left(\log 1 + \log \left(\frac{1}{s}\right)\right)}\right)}^{1}}\right)}^{1}\right)} \cdot {\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}\right)}\]

   10. Simplified 18.4

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

   if 4.35099145165978745e-206 < x < 2.1188823942095531e141

   1. Initial program 26.2

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

   2. Simplified 26.3

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

   4. Applied pow-prod-down 9.7

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

   6. Applied sqr-pow 9.7

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

   7. Applied associate-*r* 4.5

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

   8. Simplified 4.5

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

   9. Taylor expanded around inf 48.4

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\color{blue}{\left(x \cdot {\left(\frac{1}{{\left(e^{1 \cdot \left(\log 1 + \log \left(\frac{1}{c}\right)\right)}\right)}^{1} \cdot {\left(e^{1 \cdot \left(\log 1 + \log \left(\frac{1}{s}\right)\right)}\right)}^{1}}\right)}^{1}\right)} \cdot {\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}\right)}\]

   10. Simplified 4.5

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\color{blue}{\left(x \cdot {\left(\frac{1}{{\left({\left(\frac{1}{c}\right)}^{1}\right)}^{1} \cdot {\left({\left(\frac{1}{s}\right)}^{1}\right)}^{1}}\right)}^{1}\right)} \cdot {\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}\right)}\]
   11. Using strategy rm

   12. Applied associate-*r* 1.1

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

   13. Simplified 1.1

       \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot \left(x \cdot {\left(\frac{1}{{\left({c}^{\left(-1\right)}\right)}^{1} \cdot {\left({s}^{\left(-1\right)}\right)}^{1}}\right)}^{1}\right)\right)} \cdot {\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}}\]

   if 2.1188823942095531e141 < x

   1. Initial program 26.6

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

   2. Simplified 26.6

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

   4. Applied pow-prod-down 15.0

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

   6. Applied sqr-pow 15.0

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

   7. Applied associate-*r* 6.4

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

   8. Simplified 6.4

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\color{blue}{\left(x \cdot {\left(c \cdot s\right)}^{\left(\frac{1}{2} \cdot 2\right)}\right)} \cdot {\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}\right)}\]
   9. Using strategy rm

   10. Applied associate-/r* 6.3

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

   11. Simplified 6.3

       \[\leadsto \frac{\color{blue}{\frac{\cos \left(x \cdot 2\right)}{x}}}{\left(x \cdot {\left(c \cdot s\right)}^{\left(\frac{1}{2} \cdot 2\right)}\right) \cdot {\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 5.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.33159826832465384 \cdot 10^{-194}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot \left(x \cdot {\left(c \cdot s\right)}^{\left(2 \cdot \frac{1}{2}\right)}\right)\right) \cdot {\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}}\\ \mathbf{elif}\;x \le 4.35099145165978745 \cdot 10^{-206}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{x \cdot \left({\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {\left(\frac{1}{{\left({\left(\frac{1}{c}\right)}^{1}\right)}^{1} \cdot {\left({\left(\frac{1}{s}\right)}^{1}\right)}^{1}}\right)}^{1}\right)\right)}\\ \mathbf{elif}\;x \le 2.1188823942095531 \cdot 10^{141}:\\ \;\;\;\;\frac{\cos \left(x \cdot 2\right)}{{\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot \left(x \cdot {\left(\frac{1}{{\left({c}^{\left(-1\right)}\right)}^{1} \cdot {\left({s}^{\left(-1\right)}\right)}^{1}}\right)}^{1}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot 2\right)}{x}}{\left(x \cdot {\left(c \cdot s\right)}^{\left(2 \cdot \frac{1}{2}\right)}\right) \cdot {\left(c \cdot s\right)}^{\left(\frac{2}{2}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))