Average Error: 27.6 → 2.6
Time: 6.2s
Precision: binary64
\[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}\]
↓
\[\frac{1}{\left|s \cdot \left(c \cdot x\right)\right|} \cdot \frac{1}{\frac{\left|s \cdot \left(c \cdot x\right)\right|}{\cos \left(x \cdot 2\right)}}\]
\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}
↓
\frac{1}{\left|s \cdot \left(c \cdot x\right)\right|} \cdot \frac{1}{\frac{\left|s \cdot \left(c \cdot x\right)\right|}{\cos \left(x \cdot 2\right)}}
(FPCore (x c s)
 :precision binary64
 (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))
↓
(FPCore (x c s)
 :precision binary64
 (*
  (/ 1.0 (fabs (* s (* c x))))
  (/ 1.0 (/ (fabs (* s (* c x))) (cos (* x 2.0))))))
double code(double x, double c, double s) {
	return cos(2.0 * x) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x));
}
↓
double code(double x, double c, double s) {
	return (1.0 / fabs(s * (c * x))) * (1.0 / (fabs(s * (c * x)) / cos(x * 2.0)));
}

Error

Bits error versus x

Bits error versus c

Bits error versus s

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 27.6

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

  3. Applied add-sqr-sqrt_binary64_43527.6

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

  4. Simplified27.6

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

  5. Simplified2.9

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

  7. Applied *-un-lft-identity_binary64_4142.9

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

  8. Applied times-frac_binary64_4202.6

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

  9. Simplified2.6

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

  11. Applied clear-num_binary64_4132.6

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

  12. Final simplification2.6

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

Reproduce

herbie shell --seed 2020292 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))