Average Error: 28.4 → 2.6
Time: 8.5s
Precision: binary64
\[[c, s]=\mathsf{sort}([c, s])\]
\[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
↓
\[\begin{array}{l} t_0 := \left|x \cdot s\right|\\ \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{c}}{t_0}}{c \cdot t_0} \end{array} \]
\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|x \cdot s\right|\\
\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{c}}{t_0}}{c \cdot 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 (* x s)))) (/ (/ (/ (cos (* 2.0 x)) c) t_0) (* c 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(x * s);
	return ((cos(2.0 * x) / c) / t_0) / (c * 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 28.4

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

  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. Simplified28.4

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

  4. Simplified20.3

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

  5. Applied pow2_binary6420.3

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

  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 unpow2_binary642.9

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

  8. Applied associate-/r*_binary642.7

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

  9. Applied associate-/r*_binary642.6

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

  10. Final simplification2.6

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

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))))