mixedcos

Average Error: 21.5 → 0.9

Time: 12.2s

Precision: binary64

\[[c, s] = \mathsf{sort}([c, s]) \\]

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(2 \cdot x\right)\\ \mathbf{if}\;\frac{t_0}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq 6.733258681127257 \cdot 10^{+264}:\\ \;\;\;\;\begin{array}{l} t_1 := \left|x \cdot s\right|\\ \frac{\frac{1}{c}}{t_1} \cdot \frac{t_0}{c \cdot t_1} \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{{\left(\left(c \cdot \left|s\right|\right) \cdot \left|x\right|\right)}^{2}}\\ \end{array} \]

FPCore

(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 (cos (* 2.0 x))))
   (if (<=
        (/ t_0 (* (pow c 2.0) (* x (* x (pow s 2.0)))))
        6.733258681127257e+264)
     (let* ((t_1 (fabs (* x s)))) (* (/ (/ 1.0 c) t_1) (/ t_0 (* c t_1))))
     (/ t_0 (pow (* (* c (fabs s)) (fabs x)) 2.0)))))

C

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 = cos(2.0 * x);
	double tmp;
	if ((t_0 / (pow(c, 2.0) * (x * (x * pow(s, 2.0))))) <= 6.733258681127257e+264) {
		double t_1_1 = fabs(x * s);
		tmp = ((1.0 / c) / t_1_1) * (t_0 / (c * t_1_1));
	} else {
		tmp = t_0 / pow(((c * fabs(s)) * fabs(x)), 2.0);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus c

Bits error versus s

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 (pow.f64 c 2) (*.f64 (*.f64 x (pow.f64 s 2)) x))) < 6.73325868112725697e264

   1. Initial program 13.5

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

   2. Applied add-sqr-sqrt_binary64 13.6

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

   3. Simplified 13.5

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

   4. Simplified 6.7

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

   5. Applied pow2_binary64 6.7

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

   6. Applied pow-prod-down_binary64 0.6

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

   7. Applied unpow2_binary64 0.6

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

   8. Applied *-un-lft-identity_binary64 0.6

      \[\leadsto \frac{\color{blue}{1 \cdot \cos \left(2 \cdot x\right)}}{\left(c \cdot \left|s \cdot x\right|\right) \cdot \left(c \cdot \left|s \cdot x\right|\right)} \]

   9. Applied times-frac_binary64 0.3

      \[\leadsto \color{blue}{\frac{1}{c \cdot \left|s \cdot x\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{c \cdot \left|s \cdot x\right|}} \]

   10. Applied associate-/r*_binary64 0.3

       \[\leadsto \color{blue}{\frac{\frac{1}{c}}{\left|s \cdot x\right|}} \cdot \frac{\cos \left(2 \cdot x\right)}{c \cdot \left|s \cdot x\right|} \]

   if 6.73325868112725697e264 < (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 (pow.f64 c 2) (*.f64 (*.f64 x (pow.f64 s 2)) x)))

   1. Initial program 29.9

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

   2. Applied add-sqr-sqrt_binary64 29.9

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

   3. Simplified 29.9

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

   4. Simplified 22.5

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

   5. Applied pow2_binary64 22.5

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

   6. Applied pow-prod-down_binary64 3.3

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

   7. Applied fabs-mul_binary64 3.3

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

   8. Applied associate-*r*_binary64 1.6

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{\color{blue}{\left(\left(c \cdot \left|s\right|\right) \cdot \left|x\right|\right)}}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq 6.733258681127257 \cdot 10^{+264}:\\ \;\;\;\;\frac{\frac{1}{c}}{\left|x \cdot s\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{c \cdot \left|x \cdot s\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(\left(c \cdot \left|s\right|\right) \cdot \left|x\right|\right)}^{2}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))