Jmat.Real.dawson

Average Error: 28.8 → 0.0

Time: 9.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1924412471488402 \lor \neg \left(x \leq 11811.963660157699\right):\\ \;\;\;\;\frac{0.5}{x} + \frac{\sqrt{0.2514179000665374}}{x \cdot x} \cdot \frac{\sqrt{0.2514179000665374}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\left(-1 - \left(\left(x \cdot x\right) \cdot 0.1049934947 + {x}^{4} \cdot 0.0424060604\right)\right) - \left({x}^{6} \cdot 0.0072644182 + \left(0.0005064034 \cdot {x}^{8} + 0.0001789971 \cdot {x}^{10}\right)\right)}{\left(-1 - \left(\left(\left(x \cdot x\right) \cdot 0.7715471019 + {x}^{4} \cdot 0.2909738639\right) + \left({x}^{6} \cdot 0.0694555761 + {x}^{8} \cdot 0.0140005442\right)\right)\right) + \left({x}^{12} \cdot -0.0003579942 - {x}^{10} \cdot 0.0008327945\right)}\\ \end{array}\]

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 -1924412471488402.0) (not (<= x 11811.963660157699)))
   (+
    (/ 0.5 x)
    (* (/ (sqrt 0.2514179000665374) (* x x)) (/ (sqrt 0.2514179000665374) x)))
   (*
    x
    (/
     (-
      (- -1.0 (+ (* (* x x) 0.1049934947) (* (pow x 4.0) 0.0424060604)))
      (+
       (* (pow x 6.0) 0.0072644182)
       (+ (* 0.0005064034 (pow x 8.0)) (* 0.0001789971 (pow x 10.0)))))
     (+
      (-
       -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 12.0) -0.0003579942) (* (pow x 10.0) 0.0008327945)))))))

C

double code(double x) {
	return ((((((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;
}

↓

double code(double x) {
	double tmp;
	if ((x <= -1924412471488402.0) || !(x <= 11811.963660157699)) {
		tmp = (0.5 / x) + ((sqrt(0.2514179000665374) / (x * x)) * (sqrt(0.2514179000665374) / x));
	} else {
		tmp = x * (((-1.0 - (((x * x) * 0.1049934947) + (pow(x, 4.0) * 0.0424060604))) - ((pow(x, 6.0) * 0.0072644182) + ((0.0005064034 * pow(x, 8.0)) + (0.0001789971 * pow(x, 10.0))))) / ((-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, 12.0) * -0.0003579942) - (pow(x, 10.0) * 0.0008327945))));
	}
	return tmp;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -1924412471488402 or 11811.9636601576985 < x

   1. Initial program 60.5

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

   2. Simplified 60.5

      \[\leadsto \color{blue}{x \cdot \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot {x}^{4}\right) + 0.0072644182 \cdot {x}^{6}\right) + 0.0005064034 \cdot {x}^{8}\right) + 0.0001789971 \cdot {x}^{10}}{\left(\left(\left(\left(\left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right) + {x}^{4} \cdot 0.2909738639\right) + {x}^{6} \cdot 0.0694555761\right) + {x}^{8} \cdot 0.0140005442\right) + {x}^{10} \cdot 0.0008327945\right) + 0.0003579942 \cdot {x}^{12}}}\]
   3. Using strategy rm

   4. Applied frac-2neg_binary64_3840 60.5

      \[\leadsto x \cdot \color{blue}{\frac{-\left(\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot {x}^{4}\right) + 0.0072644182 \cdot {x}^{6}\right) + 0.0005064034 \cdot {x}^{8}\right) + 0.0001789971 \cdot {x}^{10}\right)}{-\left(\left(\left(\left(\left(\left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right) + {x}^{4} \cdot 0.2909738639\right) + {x}^{6} \cdot 0.0694555761\right) + {x}^{8} \cdot 0.0140005442\right) + {x}^{10} \cdot 0.0008327945\right) + 0.0003579942 \cdot {x}^{12}\right)}}\]

   5. Simplified 60.5

      \[\leadsto x \cdot \frac{\color{blue}{\left(-1 - \left(\left(x \cdot x\right) \cdot 0.1049934947 + {x}^{4} \cdot 0.0424060604\right)\right) - \left({x}^{6} \cdot 0.0072644182 + \left(0.0005064034 \cdot {x}^{8} + 0.0001789971 \cdot {x}^{10}\right)\right)}}{-\left(\left(\left(\left(\left(\left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right) + {x}^{4} \cdot 0.2909738639\right) + {x}^{6} \cdot 0.0694555761\right) + {x}^{8} \cdot 0.0140005442\right) + {x}^{10} \cdot 0.0008327945\right) + 0.0003579942 \cdot {x}^{12}\right)}\]

   6. Simplified 60.5

      \[\leadsto x \cdot \frac{\left(-1 - \left(\left(x \cdot x\right) \cdot 0.1049934947 + {x}^{4} \cdot 0.0424060604\right)\right) - \left({x}^{6} \cdot 0.0072644182 + \left(0.0005064034 \cdot {x}^{8} + 0.0001789971 \cdot {x}^{10}\right)\right)}{\color{blue}{\left(-1 - \left(\left(\left(x \cdot x\right) \cdot 0.7715471019 + {x}^{4} \cdot 0.2909738639\right) + \left({x}^{6} \cdot 0.0694555761 + {x}^{8} \cdot 0.0140005442\right)\right)\right) + \left({x}^{12} \cdot -0.0003579942 - {x}^{10} \cdot 0.0008327945\right)}}\]

   7. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{0.5 \cdot \frac{1}{x} + 0.2514179000665374 \cdot \frac{1}{{x}^{3}}}\]

   8. Simplified 0.0

      \[\leadsto \color{blue}{\frac{0.5}{x} + \frac{0.2514179000665374}{{x}^{3}}}\]
   9. Using strategy rm

   10. Applied unpow3_binary64_3895 0.0

       \[\leadsto \frac{0.5}{x} + \frac{0.2514179000665374}{\color{blue}{\left(x \cdot x\right) \cdot x}}\]

   11. Applied add-sqr-sqrt_binary64_3851 0.0

       \[\leadsto \frac{0.5}{x} + \frac{\color{blue}{\sqrt{0.2514179000665374} \cdot \sqrt{0.2514179000665374}}}{\left(x \cdot x\right) \cdot x}\]

   12. Applied times-frac_binary64_3835 0.0

       \[\leadsto \frac{0.5}{x} + \color{blue}{\frac{\sqrt{0.2514179000665374}}{x \cdot x} \cdot \frac{\sqrt{0.2514179000665374}}{x}}\]

   if -1924412471488402 < x < 11811.9636601576985

   1. Initial program 0.0

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

   2. Simplified 0.0

      \[\leadsto \color{blue}{x \cdot \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot {x}^{4}\right) + 0.0072644182 \cdot {x}^{6}\right) + 0.0005064034 \cdot {x}^{8}\right) + 0.0001789971 \cdot {x}^{10}}{\left(\left(\left(\left(\left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right) + {x}^{4} \cdot 0.2909738639\right) + {x}^{6} \cdot 0.0694555761\right) + {x}^{8} \cdot 0.0140005442\right) + {x}^{10} \cdot 0.0008327945\right) + 0.0003579942 \cdot {x}^{12}}}\]
   3. Using strategy rm

   4. Applied frac-2neg_binary64_3840 0.0

      \[\leadsto x \cdot \color{blue}{\frac{-\left(\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot {x}^{4}\right) + 0.0072644182 \cdot {x}^{6}\right) + 0.0005064034 \cdot {x}^{8}\right) + 0.0001789971 \cdot {x}^{10}\right)}{-\left(\left(\left(\left(\left(\left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right) + {x}^{4} \cdot 0.2909738639\right) + {x}^{6} \cdot 0.0694555761\right) + {x}^{8} \cdot 0.0140005442\right) + {x}^{10} \cdot 0.0008327945\right) + 0.0003579942 \cdot {x}^{12}\right)}}\]

   5. Simplified 0.0

      \[\leadsto x \cdot \frac{\color{blue}{\left(-1 - \left(\left(x \cdot x\right) \cdot 0.1049934947 + {x}^{4} \cdot 0.0424060604\right)\right) - \left({x}^{6} \cdot 0.0072644182 + \left(0.0005064034 \cdot {x}^{8} + 0.0001789971 \cdot {x}^{10}\right)\right)}}{-\left(\left(\left(\left(\left(\left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right) + {x}^{4} \cdot 0.2909738639\right) + {x}^{6} \cdot 0.0694555761\right) + {x}^{8} \cdot 0.0140005442\right) + {x}^{10} \cdot 0.0008327945\right) + 0.0003579942 \cdot {x}^{12}\right)}\]

   6. Simplified 0.0

      \[\leadsto x \cdot \frac{\left(-1 - \left(\left(x \cdot x\right) \cdot 0.1049934947 + {x}^{4} \cdot 0.0424060604\right)\right) - \left({x}^{6} \cdot 0.0072644182 + \left(0.0005064034 \cdot {x}^{8} + 0.0001789971 \cdot {x}^{10}\right)\right)}{\color{blue}{\left(-1 - \left(\left(\left(x \cdot x\right) \cdot 0.7715471019 + {x}^{4} \cdot 0.2909738639\right) + \left({x}^{6} \cdot 0.0694555761 + {x}^{8} \cdot 0.0140005442\right)\right)\right) + \left({x}^{12} \cdot -0.0003579942 - {x}^{10} \cdot 0.0008327945\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1924412471488402 \lor \neg \left(x \leq 11811.963660157699\right):\\ \;\;\;\;\frac{0.5}{x} + \frac{\sqrt{0.2514179000665374}}{x \cdot x} \cdot \frac{\sqrt{0.2514179000665374}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\left(-1 - \left(\left(x \cdot x\right) \cdot 0.1049934947 + {x}^{4} \cdot 0.0424060604\right)\right) - \left({x}^{6} \cdot 0.0072644182 + \left(0.0005064034 \cdot {x}^{8} + 0.0001789971 \cdot {x}^{10}\right)\right)}{\left(-1 - \left(\left(\left(x \cdot x\right) \cdot 0.7715471019 + {x}^{4} \cdot 0.2909738639\right) + \left({x}^{6} \cdot 0.0694555761 + {x}^{8} \cdot 0.0140005442\right)\right)\right) + \left({x}^{12} \cdot -0.0003579942 - {x}^{10} \cdot 0.0008327945\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021075 
(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))