Average Error: 27.9 → 2.5
Time: 15.1s
Precision: binary64
\[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
↓
\[\begin{array}{l} t_0 := \left|c \cdot \left(x \cdot s\right)\right|\\ \frac{\frac{\cos \left(2 \cdot x\right)}{t_0}}{t_0} \end{array} \]
\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|c \cdot \left(x \cdot s\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 (* c (* x s))))) (/ (/ (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(c * (x * s));
	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 27.9

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

  3. Applied add-sqr-sqrt_binary6427.9

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

  4. Simplified27.9

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

  5. Simplified19.8

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

  7. Applied add-sqr-sqrt_binary6419.8

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

  8. Simplified19.8

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

  9. Simplified2.7

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

  11. Applied associate-/r*_binary642.5

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

  12. Final simplification2.5

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

Reproduce

herbie shell --seed 2021210 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))