Average Error: 28.1 → 2.9
Time: 16.6s
Precision: binary64
\[[c, s]=\mathsf{sort}([c, s])\]
\[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
↓
\[\frac{\cos x \cdot \cos x - \sin x \cdot \sin x}{{\left(c \cdot \left|x \cdot s\right|\right)}^{2}} \]
\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}
↓
\frac{\cos x \cdot \cos x - \sin x \cdot \sin x}{{\left(c \cdot \left|x \cdot s\right|\right)}^{2}}

(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
 (/
  (- (* (cos x) (cos x)) (* (sin x) (sin x)))
  (pow (* c (fabs (* x s))) 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 ((cos(x) * cos(x)) - (sin(x) * sin(x))) / pow((c * fabs(x * s)), 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 28.1

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

  2. Applied add-sqr-sqrt_binary6428.2

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

  3. Simplified28.2

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

  4. Simplified20.1

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

  5. Applied pow2_binary6420.1

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

  6. Applied pow-prod-down_binary642.9

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

  7. Applied cos-2_binary642.9

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

  8. Final simplification2.9

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

Reproduce

herbie shell --seed 2021275 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))