Average Error: 28.2 → 2.4
Time: 8.8s
Precision: binary64
\[[c, s]=\mathsf{sort}([c, s])\]
\[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
↓
\[\begin{array}{l} t_0 := \left|s \cdot \left(x \cdot c\right)\right|\\ \frac{\frac{\cos \left(2 \cdot x\right)}{t_0}}{t_0} \end{array} \]
\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}
↓
\begin{array}{l}
t_0 := \left|s \cdot \left(x \cdot c\right)\right|\\
\frac{\frac{\cos \left(2 \cdot x\right)}{t_0}}{t_0}
\end{array}

(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
 (let* ((t_0 (fabs (* s (* x c))))) (/ (/ (cos (* 2.0 x)) t_0) t_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) {
	double t_0 = fabs(s * (x * c));
	return (cos(2.0 * x) / t_0) / t_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 28.2

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

  3. Applied add-sqr-sqrt_binary6428.2

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

  4. Simplified28.2

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

  5. Simplified2.7

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

  7. Applied associate-/r*_binary642.4

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

  8. Final simplification2.4

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

Reproduce

herbie shell --seed 2021198 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))