Jmat.Real.dawson

Average Error: 29.7 → 0.0

Time: 5.7s

Precision: binary64

Math

\[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -7.312697197229387 \cdot 10^{+22} \lor \neg \left(x \leq 756.1667267563336\right):\\ \;\;\;\;\frac{0.2514179000665375}{{x}^{3}} + \left(\frac{0.15298196345929327}{{x}^{5}} + \frac{0.5}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(0.0424060604 \cdot {x}^{4} + \left(0.0072644182 \cdot {x}^{6} + \left(0.0005064034 \cdot {x}^{8} + 0.0001789971 \cdot {x}^{10}\right)\right)\right)\right)\right)\right) \cdot \frac{1}{1 + \left(\left(x \cdot \left(x \cdot 0.7715471019\right) + {x}^{4} \cdot 0.2909738639\right) + \left({x}^{6} \cdot 0.0694555761 + \left({x}^{8} \cdot 0.0140005442 + \left({x}^{10} \cdot 0.0008327945 + 0.0001789971 \cdot \left(2 \cdot {x}^{12}\right)\right)\right)\right)\right)}\\ \end{array}\]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/
   (+
    (+
     (+
      (+ (+ 1.0 (* 0.1049934947 (* x x))) (* 0.0424060604 (* (* x x) (* x x))))
      (* 0.0072644182 (* (* (* x x) (* x x)) (* x x))))
     (* 0.0005064034 (* (* (* (* x x) (* x x)) (* x x)) (* x x))))
    (* 0.0001789971 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x))))
   (+
    (+
     (+
      (+
       (+
        (+ 1.0 (* 0.7715471019 (* x x)))
        (* 0.2909738639 (* (* x x) (* x x))))
       (* 0.0694555761 (* (* (* x x) (* x x)) (* x x))))
      (* 0.0140005442 (* (* (* (* x x) (* x x)) (* x x)) (* x x))))
     (* 0.0008327945 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x))))
    (*
     (* 2.0 0.0001789971)
     (* (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)) (* x x)))))
  x))

↓

(FPCore (x)
 :precision binary64
 (if (or (<= x -7.312697197229387e+22) (not (<= x 756.1667267563336)))
   (+
    (/ 0.2514179000665375 (pow x 3.0))
    (+ (/ 0.15298196345929327 (pow x 5.0)) (/ 0.5 x)))
   (*
    (*
     x
     (+
      1.0
      (+
       (* 0.1049934947 (* x x))
       (+
        (* 0.0424060604 (pow x 4.0))
        (+
         (* 0.0072644182 (pow x 6.0))
         (+ (* 0.0005064034 (pow x 8.0)) (* 0.0001789971 (pow x 10.0))))))))
    (/
     1.0
     (+
      1.0
      (+
       (+ (* x (* x 0.7715471019)) (* (pow x 4.0) 0.2909738639))
       (+
        (* (pow x 6.0) 0.0694555761)
        (+
         (* (pow x 8.0) 0.0140005442)
         (+
          (* (pow x 10.0) 0.0008327945)
          (* 0.0001789971 (* 2.0 (pow x 12.0))))))))))))

C

double code(double x) {
	return ((double) ((((double) (((double) (((double) (((double) (((double) (1.0 + ((double) (0.1049934947 * ((double) (x * x)))))) + ((double) (0.0424060604 * ((double) (((double) (x * x)) * ((double) (x * x)))))))) + ((double) (0.0072644182 * ((double) (((double) (((double) (x * x)) * ((double) (x * x)))) * ((double) (x * x)))))))) + ((double) (0.0005064034 * ((double) (((double) (((double) (((double) (x * x)) * ((double) (x * x)))) * ((double) (x * x)))) * ((double) (x * x)))))))) + ((double) (0.0001789971 * ((double) (((double) (((double) (((double) (((double) (x * x)) * ((double) (x * x)))) * ((double) (x * x)))) * ((double) (x * x)))) * ((double) (x * x)))))))) / ((double) (((double) (((double) (((double) (((double) (((double) (1.0 + ((double) (0.7715471019 * ((double) (x * x)))))) + ((double) (0.2909738639 * ((double) (((double) (x * x)) * ((double) (x * x)))))))) + ((double) (0.0694555761 * ((double) (((double) (((double) (x * x)) * ((double) (x * x)))) * ((double) (x * x)))))))) + ((double) (0.0140005442 * ((double) (((double) (((double) (((double) (x * x)) * ((double) (x * x)))) * ((double) (x * x)))) * ((double) (x * x)))))))) + ((double) (0.0008327945 * ((double) (((double) (((double) (((double) (((double) (x * x)) * ((double) (x * x)))) * ((double) (x * x)))) * ((double) (x * x)))) * ((double) (x * x)))))))) + ((double) (((double) (2.0 * 0.0001789971)) * ((double) (((double) (((double) (((double) (((double) (((double) (x * x)) * ((double) (x * x)))) * ((double) (x * x)))) * ((double) (x * x)))) * ((double) (x * x)))) * ((double) (x * x))))))))) * x));
}

↓

double code(double x) {
	double tmp;
	if (((x <= -7.312697197229387e+22) || !(x <= 756.1667267563336))) {
		tmp = ((double) ((0.2514179000665375 / ((double) pow(x, 3.0))) + ((double) ((0.15298196345929327 / ((double) pow(x, 5.0))) + (0.5 / x)))));
	} else {
		tmp = ((double) (((double) (x * ((double) (1.0 + ((double) (((double) (0.1049934947 * ((double) (x * x)))) + ((double) (((double) (0.0424060604 * ((double) pow(x, 4.0)))) + ((double) (((double) (0.0072644182 * ((double) pow(x, 6.0)))) + ((double) (((double) (0.0005064034 * ((double) pow(x, 8.0)))) + ((double) (0.0001789971 * ((double) pow(x, 10.0)))))))))))))))) * (1.0 / ((double) (1.0 + ((double) (((double) (((double) (x * ((double) (x * 0.7715471019)))) + ((double) (((double) pow(x, 4.0)) * 0.2909738639)))) + ((double) (((double) (((double) pow(x, 6.0)) * 0.0694555761)) + ((double) (((double) (((double) pow(x, 8.0)) * 0.0140005442)) + ((double) (((double) (((double) pow(x, 10.0)) * 0.0008327945)) + ((double) (0.0001789971 * ((double) (2.0 * ((double) pow(x, 12.0)))))))))))))))))));
	}
	return tmp;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -7.31269719722938702e22 or 756.16672675633356 < x

   1. Initial program Error: 61.3 bits

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]

   2. Simplified Error: 61.3 bits

      \[\leadsto \color{blue}{x \cdot \frac{1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(0.0424060604 \cdot {x}^{4} + \left(0.0072644182 \cdot {x}^{6} + \left(0.0005064034 \cdot {x}^{8} + 0.0001789971 \cdot {x}^{10}\right)\right)\right)\right)}{1 + \left(\left(x \cdot \left(x \cdot 0.7715471019\right) + {x}^{4} \cdot 0.2909738639\right) + \left({x}^{6} \cdot 0.0694555761 + \left({x}^{8} \cdot 0.0140005442 + \left({x}^{10} \cdot 0.0008327945 + 0.0001789971 \cdot \left(2 \cdot {x}^{12}\right)\right)\right)\right)\right)}}\]

   3. Taylor expanded around inf Error: 0.0 bits

      \[\leadsto \color{blue}{0.2514179000665375 \cdot \frac{1}{{x}^{3}} + \left(0.15298196345929327 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)}\]

   4. Simplified Error: 0.0 bits

      \[\leadsto \color{blue}{\frac{0.2514179000665375}{{x}^{3}} + \left(\frac{0.15298196345929327}{{x}^{5}} + \frac{0.5}{x}\right)}\]

   if -7.31269719722938702e22 < x < 756.16672675633356

   1. Initial program Error: 0.0 bits

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]

   2. Simplified Error: 0.0 bits

      \[\leadsto \color{blue}{x \cdot \frac{1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(0.0424060604 \cdot {x}^{4} + \left(0.0072644182 \cdot {x}^{6} + \left(0.0005064034 \cdot {x}^{8} + 0.0001789971 \cdot {x}^{10}\right)\right)\right)\right)}{1 + \left(\left(x \cdot \left(x \cdot 0.7715471019\right) + {x}^{4} \cdot 0.2909738639\right) + \left({x}^{6} \cdot 0.0694555761 + \left({x}^{8} \cdot 0.0140005442 + \left({x}^{10} \cdot 0.0008327945 + 0.0001789971 \cdot \left(2 \cdot {x}^{12}\right)\right)\right)\right)\right)}}\]
   3. Using strategy rm

   4. Applied div-inv Error: 0.0 bits

      \[\leadsto x \cdot \color{blue}{\left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(0.0424060604 \cdot {x}^{4} + \left(0.0072644182 \cdot {x}^{6} + \left(0.0005064034 \cdot {x}^{8} + 0.0001789971 \cdot {x}^{10}\right)\right)\right)\right)\right) \cdot \frac{1}{1 + \left(\left(x \cdot \left(x \cdot 0.7715471019\right) + {x}^{4} \cdot 0.2909738639\right) + \left({x}^{6} \cdot 0.0694555761 + \left({x}^{8} \cdot 0.0140005442 + \left({x}^{10} \cdot 0.0008327945 + 0.0001789971 \cdot \left(2 \cdot {x}^{12}\right)\right)\right)\right)\right)}\right)}\]

   5. Applied associate-*r* Error: 0.0 bits

      \[\leadsto \color{blue}{\left(x \cdot \left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(0.0424060604 \cdot {x}^{4} + \left(0.0072644182 \cdot {x}^{6} + \left(0.0005064034 \cdot {x}^{8} + 0.0001789971 \cdot {x}^{10}\right)\right)\right)\right)\right)\right) \cdot \frac{1}{1 + \left(\left(x \cdot \left(x \cdot 0.7715471019\right) + {x}^{4} \cdot 0.2909738639\right) + \left({x}^{6} \cdot 0.0694555761 + \left({x}^{8} \cdot 0.0140005442 + \left({x}^{10} \cdot 0.0008327945 + 0.0001789971 \cdot \left(2 \cdot {x}^{12}\right)\right)\right)\right)\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification Error: 0.0 bits

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.312697197229387 \cdot 10^{+22} \lor \neg \left(x \leq 756.1667267563336\right):\\ \;\;\;\;\frac{0.2514179000665375}{{x}^{3}} + \left(\frac{0.15298196345929327}{{x}^{5}} + \frac{0.5}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(0.0424060604 \cdot {x}^{4} + \left(0.0072644182 \cdot {x}^{6} + \left(0.0005064034 \cdot {x}^{8} + 0.0001789971 \cdot {x}^{10}\right)\right)\right)\right)\right)\right) \cdot \frac{1}{1 + \left(\left(x \cdot \left(x \cdot 0.7715471019\right) + {x}^{4} \cdot 0.2909738639\right) + \left({x}^{6} \cdot 0.0694555761 + \left({x}^{8} \cdot 0.0140005442 + \left({x}^{10} \cdot 0.0008327945 + 0.0001789971 \cdot \left(2 \cdot {x}^{12}\right)\right)\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(FPCore (x)
  :name "Jmat.Real.dawson"
  :precision binary64
  (* (/ (+ (+ (+ (+ (+ 1.0 (* 0.1049934947 (* x x))) (* 0.0424060604 (* (* x x) (* x x)))) (* 0.0072644182 (* (* (* x x) (* x x)) (* x x)))) (* 0.0005064034 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 0.0001789971 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (+ (+ (+ (+ (+ (+ 1.0 (* 0.7715471019 (* x x))) (* 0.2909738639 (* (* x x) (* x x)))) (* 0.0694555761 (* (* (* x x) (* x x)) (* x x)))) (* 0.0140005442 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 0.0008327945 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (* (* 2.0 0.0001789971) (* (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)) (* x x))))) x))