Average Error: 29.2 → 0.0
Time: 4.2s
Precision: binary64
\[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]
↓
\[\begin{array}{l} t_0 := \frac{0.5}{x} + \frac{0.2514179000665374}{{x}^{3}}\\ \mathbf{if}\;x \leq -3944258682.932016:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9254.197213759346:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left(0.0072644182, {x}^{6}, \mathsf{fma}\left(0.0424060604, {x}^{4}, \mathsf{fma}\left(0.1049934947, x \cdot x, 1\right)\right)\right)\right)\right)}{\mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{4}, 0.2909738639, \mathsf{fma}\left(x \cdot x, 0.7715471019, 1\right)\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
\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}
t_0 := \frac{0.5}{x} + \frac{0.2514179000665374}{{x}^{3}}\\
\mathbf{if}\;x \leq -3944258682.932016:\\
\;\;\;\;t_0\\

\mathbf{elif}\;x \leq 9254.197213759346:\\
\;\;\;\;\frac{x \cdot \mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left(0.0072644182, {x}^{6}, \mathsf{fma}\left(0.0424060604, {x}^{4}, \mathsf{fma}\left(0.1049934947, x \cdot x, 1\right)\right)\right)\right)\right)}{\mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{4}, 0.2909738639, \mathsf{fma}\left(x \cdot x, 0.7715471019, 1\right)\right)\right)\right)\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}

(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
 (let* ((t_0 (+ (/ 0.5 x) (/ 0.2514179000665374 (pow x 3.0)))))
   (if (<= x -3944258682.932016)
     t_0
     (if (<= x 9254.197213759346)
       (/
        (*
         x
         (fma
          0.0001789971
          (pow x 10.0)
          (fma
           0.0005064034
           (pow x 8.0)
           (fma
            0.0072644182
            (pow x 6.0)
            (fma 0.0424060604 (pow x 4.0) (fma 0.1049934947 (* x x) 1.0))))))
        (fma
         0.0003579942
         (pow x 12.0)
         (fma
          (pow x 10.0)
          0.0008327945
          (fma
           (pow x 8.0)
           0.0140005442
           (fma
            (pow x 6.0)
            0.0694555761
            (fma (pow x 4.0) 0.2909738639 (fma (* x x) 0.7715471019 1.0)))))))
       t_0))))
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 t_0 = (0.5 / x) + (0.2514179000665374 / pow(x, 3.0));
	double tmp;
	if (x <= -3944258682.932016) {
		tmp = t_0;
	} else if (x <= 9254.197213759346) {
		tmp = (x * fma(0.0001789971, pow(x, 10.0), fma(0.0005064034, pow(x, 8.0), fma(0.0072644182, pow(x, 6.0), fma(0.0424060604, pow(x, 4.0), fma(0.1049934947, (x * x), 1.0)))))) / fma(0.0003579942, pow(x, 12.0), fma(pow(x, 10.0), 0.0008327945, fma(pow(x, 8.0), 0.0140005442, fma(pow(x, 6.0), 0.0694555761, fma(pow(x, 4.0), 0.2909738639, fma((x * x), 0.7715471019, 1.0))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -3944258682.9320159 or 9254.19721375934569 < x

    1. Initial program 60.0

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

    2. Simplified60.0

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

    3. Taylor expanded in x around inf 0.0

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

    4. Simplified0.0

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

    if -3944258682.9320159 < x < 9254.19721375934569

    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. Simplified0.0

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

    3. Applied associate-*r/_binary640.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3944258682.932016:\\ \;\;\;\;\frac{0.5}{x} + \frac{0.2514179000665374}{{x}^{3}}\\ \mathbf{elif}\;x \leq 9254.197213759346:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left(0.0072644182, {x}^{6}, \mathsf{fma}\left(0.0424060604, {x}^{4}, \mathsf{fma}\left(0.1049934947, x \cdot x, 1\right)\right)\right)\right)\right)}{\mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{10}, 0.0008327945, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left({x}^{6}, 0.0694555761, \mathsf{fma}\left({x}^{4}, 0.2909738639, \mathsf{fma}\left(x \cdot x, 0.7715471019, 1\right)\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} + \frac{0.2514179000665374}{{x}^{3}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(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))