Average Error: 28.4 → 2.7
Time: 16.7s
Precision: binary64
\[[c, s]=\mathsf{sort}([c, s])\]
\[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
↓
\[{\left(c \cdot \left|s \cdot x\right|\right)}^{-2} \cdot \cos \left(x \cdot 2\right) \]
\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}
↓
{\left(c \cdot \left|s \cdot x\right|\right)}^{-2} \cdot \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
 (* (pow (* c (fabs (* s x))) -2.0) (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 pow((c * fabs(s * x)), -2.0) * 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 28.4

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

  2. Applied add-sqr-sqrt_binary6428.4

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

  3. Applied unpow2_binary6428.4

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

  4. Applied unswap-sqr_binary6423.6

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

  5. Simplified23.6

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

  6. Simplified2.9

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

  7. Taylor expanded in x around inf 20.3

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

  8. Simplified2.7

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

  9. Final simplification2.7

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

Reproduce

herbie shell --seed 2021215 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))