Jmat.Real.dawson

Percentage Accurate: 55.1% → 55.1%

Time: 22.6s

Alternatives: 4

Speedup: 2.2×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ t_1 := t\_0 \cdot \left(x \cdot x\right)\\ t_2 := t\_1 \cdot \left(x \cdot x\right)\\ t_3 := t\_2 \cdot \left(x \cdot x\right)\\ \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot t\_0\right) + 0.0072644182 \cdot t\_1\right) + 0.0005064034 \cdot t\_2\right) + 0.0001789971 \cdot t\_3}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot t\_0\right) + 0.0694555761 \cdot t\_1\right) + 0.0140005442 \cdot t\_2\right) + 0.0008327945 \cdot t\_3\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(t\_3 \cdot \left(x \cdot x\right)\right)} \cdot x \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) (* x x)))
        (t_1 (* t_0 (* x x)))
        (t_2 (* t_1 (* x x)))
        (t_3 (* t_2 (* x x))))
   (*
    (/
     (+
      (+
       (+
        (+ (+ 1.0 (* 0.1049934947 (* x x))) (* 0.0424060604 t_0))
        (* 0.0072644182 t_1))
       (* 0.0005064034 t_2))
      (* 0.0001789971 t_3))
     (+
      (+
       (+
        (+
         (+ (+ 1.0 (* 0.7715471019 (* x x))) (* 0.2909738639 t_0))
         (* 0.0694555761 t_1))
        (* 0.0140005442 t_2))
       (* 0.0008327945 t_3))
      (* (* 2.0 0.0001789971) (* t_3 (* x x)))))
    x)))

C

double code(double x) {
	double t_0 = (x * x) * (x * x);
	double t_1 = t_0 * (x * x);
	double t_2 = t_1 * (x * x);
	double t_3 = t_2 * (x * x);
	return ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + ((2.0 * 0.0001789971) * (t_3 * (x * x))))) * x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    t_0 = (x * x) * (x * x)
    t_1 = t_0 * (x * x)
    t_2 = t_1 * (x * x)
    t_3 = t_2 * (x * x)
    code = ((((((1.0d0 + (0.1049934947d0 * (x * x))) + (0.0424060604d0 * t_0)) + (0.0072644182d0 * t_1)) + (0.0005064034d0 * t_2)) + (0.0001789971d0 * t_3)) / ((((((1.0d0 + (0.7715471019d0 * (x * x))) + (0.2909738639d0 * t_0)) + (0.0694555761d0 * t_1)) + (0.0140005442d0 * t_2)) + (0.0008327945d0 * t_3)) + ((2.0d0 * 0.0001789971d0) * (t_3 * (x * x))))) * x
end function

Java

public static double code(double x) {
	double t_0 = (x * x) * (x * x);
	double t_1 = t_0 * (x * x);
	double t_2 = t_1 * (x * x);
	double t_3 = t_2 * (x * x);
	return ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + ((2.0 * 0.0001789971) * (t_3 * (x * x))))) * x;
}

Python

def code(x):
	t_0 = (x * x) * (x * x)
	t_1 = t_0 * (x * x)
	t_2 = t_1 * (x * x)
	t_3 = t_2 * (x * x)
	return ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + ((2.0 * 0.0001789971) * (t_3 * (x * x))))) * x

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * Float64(x * x))
	t_1 = Float64(t_0 * Float64(x * x))
	t_2 = Float64(t_1 * Float64(x * x))
	t_3 = Float64(t_2 * Float64(x * x))
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.1049934947 * Float64(x * x))) + Float64(0.0424060604 * t_0)) + Float64(0.0072644182 * t_1)) + Float64(0.0005064034 * t_2)) + Float64(0.0001789971 * t_3)) / Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.7715471019 * Float64(x * x))) + Float64(0.2909738639 * t_0)) + Float64(0.0694555761 * t_1)) + Float64(0.0140005442 * t_2)) + Float64(0.0008327945 * t_3)) + Float64(Float64(2.0 * 0.0001789971) * Float64(t_3 * Float64(x * x))))) * x)
end

MATLAB

function tmp = code(x)
	t_0 = (x * x) * (x * x);
	t_1 = t_0 * (x * x);
	t_2 = t_1 * (x * x);
	t_3 = t_2 * (x * x);
	tmp = ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + ((2.0 * 0.0001789971) * (t_3 * (x * x))))) * x;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(N[(1.0 + N[(0.1049934947 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0424060604 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.0072644182 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.0005064034 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.0001789971 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(1.0 + N[(0.7715471019 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2909738639 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.0694555761 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.0140005442 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.0008327945 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * 0.0001789971), $MachinePrecision] * N[(t$95$3 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]]

Initial Program: 55.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ t_1 := t\_0 \cdot \left(x \cdot x\right)\\ t_2 := t\_1 \cdot \left(x \cdot x\right)\\ t_3 := t\_2 \cdot \left(x \cdot x\right)\\ \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot t\_0\right) + 0.0072644182 \cdot t\_1\right) + 0.0005064034 \cdot t\_2\right) + 0.0001789971 \cdot t\_3}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot t\_0\right) + 0.0694555761 \cdot t\_1\right) + 0.0140005442 \cdot t\_2\right) + 0.0008327945 \cdot t\_3\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(t\_3 \cdot \left(x \cdot x\right)\right)} \cdot x \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) (* x x)))
        (t_1 (* t_0 (* x x)))
        (t_2 (* t_1 (* x x)))
        (t_3 (* t_2 (* x x))))
   (*
    (/
     (+
      (+
       (+
        (+ (+ 1.0 (* 0.1049934947 (* x x))) (* 0.0424060604 t_0))
        (* 0.0072644182 t_1))
       (* 0.0005064034 t_2))
      (* 0.0001789971 t_3))
     (+
      (+
       (+
        (+
         (+ (+ 1.0 (* 0.7715471019 (* x x))) (* 0.2909738639 t_0))
         (* 0.0694555761 t_1))
        (* 0.0140005442 t_2))
       (* 0.0008327945 t_3))
      (* (* 2.0 0.0001789971) (* t_3 (* x x)))))
    x)))

C

double code(double x) {
	double t_0 = (x * x) * (x * x);
	double t_1 = t_0 * (x * x);
	double t_2 = t_1 * (x * x);
	double t_3 = t_2 * (x * x);
	return ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + ((2.0 * 0.0001789971) * (t_3 * (x * x))))) * x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    t_0 = (x * x) * (x * x)
    t_1 = t_0 * (x * x)
    t_2 = t_1 * (x * x)
    t_3 = t_2 * (x * x)
    code = ((((((1.0d0 + (0.1049934947d0 * (x * x))) + (0.0424060604d0 * t_0)) + (0.0072644182d0 * t_1)) + (0.0005064034d0 * t_2)) + (0.0001789971d0 * t_3)) / ((((((1.0d0 + (0.7715471019d0 * (x * x))) + (0.2909738639d0 * t_0)) + (0.0694555761d0 * t_1)) + (0.0140005442d0 * t_2)) + (0.0008327945d0 * t_3)) + ((2.0d0 * 0.0001789971d0) * (t_3 * (x * x))))) * x
end function

Java

public static double code(double x) {
	double t_0 = (x * x) * (x * x);
	double t_1 = t_0 * (x * x);
	double t_2 = t_1 * (x * x);
	double t_3 = t_2 * (x * x);
	return ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + ((2.0 * 0.0001789971) * (t_3 * (x * x))))) * x;
}

Python

def code(x):
	t_0 = (x * x) * (x * x)
	t_1 = t_0 * (x * x)
	t_2 = t_1 * (x * x)
	t_3 = t_2 * (x * x)
	return ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + ((2.0 * 0.0001789971) * (t_3 * (x * x))))) * x

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * Float64(x * x))
	t_1 = Float64(t_0 * Float64(x * x))
	t_2 = Float64(t_1 * Float64(x * x))
	t_3 = Float64(t_2 * Float64(x * x))
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.1049934947 * Float64(x * x))) + Float64(0.0424060604 * t_0)) + Float64(0.0072644182 * t_1)) + Float64(0.0005064034 * t_2)) + Float64(0.0001789971 * t_3)) / Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.7715471019 * Float64(x * x))) + Float64(0.2909738639 * t_0)) + Float64(0.0694555761 * t_1)) + Float64(0.0140005442 * t_2)) + Float64(0.0008327945 * t_3)) + Float64(Float64(2.0 * 0.0001789971) * Float64(t_3 * Float64(x * x))))) * x)
end

MATLAB

function tmp = code(x)
	t_0 = (x * x) * (x * x);
	t_1 = t_0 * (x * x);
	t_2 = t_1 * (x * x);
	t_3 = t_2 * (x * x);
	tmp = ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + (0.7715471019 * (x * x))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + ((2.0 * 0.0001789971) * (t_3 * (x * x))))) * x;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(N[(1.0 + N[(0.1049934947 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0424060604 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.0072644182 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.0005064034 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.0001789971 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(1.0 + N[(0.7715471019 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2909738639 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.0694555761 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.0140005442 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.0008327945 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * 0.0001789971), $MachinePrecision] * N[(t$95$3 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]]

Alternative 1: 55.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := x \cdot t\_0\\ t_2 := x \cdot \left(x \cdot t\_1\right)\\ t_3 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ t_4 := \left(x \cdot x\right) \cdot t\_3\\ t_5 := \left(x \cdot x\right) \cdot t\_4\\ t_6 := \left(x \cdot x\right) \cdot t\_5\\ x \cdot \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot t\_3\right) + 0.0072644182 \cdot t\_4\right) + 0.0005064034 \cdot t\_5\right) + 0.0001789971 \cdot t\_6}{\mathsf{fma}\left(1 - t\_1 \cdot 0.595284930450289, \frac{1}{1 - \left(x \cdot x\right) \cdot 0.7715471019}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.2909738639, t\_0 \cdot \left(x \cdot 0.0694555761\right)\right), \left(x \cdot x\right) \cdot \mathsf{fma}\left(t\_2, 0.0140005442, t\_2 \cdot \left(\left(x \cdot x\right) \cdot 0.0008327945\right)\right)\right)\right) + \left(0.0001789971 \cdot 2\right) \cdot \left(\left(x \cdot x\right) \cdot t\_6\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x)))
        (t_1 (* x t_0))
        (t_2 (* x (* x t_1)))
        (t_3 (* (* x x) (* x x)))
        (t_4 (* (* x x) t_3))
        (t_5 (* (* x x) t_4))
        (t_6 (* (* x x) t_5)))
   (*
    x
    (/
     (+
      (+
       (+
        (+ (+ 1.0 (* 0.1049934947 (* x x))) (* 0.0424060604 t_3))
        (* 0.0072644182 t_4))
       (* 0.0005064034 t_5))
      (* 0.0001789971 t_6))
     (+
      (fma
       (- 1.0 (* t_1 0.595284930450289))
       (/ 1.0 (- 1.0 (* (* x x) 0.7715471019)))
       (fma
        (* x x)
        (fma (* x x) 0.2909738639 (* t_0 (* x 0.0694555761)))
        (* (* x x) (fma t_2 0.0140005442 (* t_2 (* (* x x) 0.0008327945))))))
      (* (* 0.0001789971 2.0) (* (* x x) t_6)))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = x * t_0;
	double t_2 = x * (x * t_1);
	double t_3 = (x * x) * (x * x);
	double t_4 = (x * x) * t_3;
	double t_5 = (x * x) * t_4;
	double t_6 = (x * x) * t_5;
	return x * ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * t_3)) + (0.0072644182 * t_4)) + (0.0005064034 * t_5)) + (0.0001789971 * t_6)) / (fma((1.0 - (t_1 * 0.595284930450289)), (1.0 / (1.0 - ((x * x) * 0.7715471019))), fma((x * x), fma((x * x), 0.2909738639, (t_0 * (x * 0.0694555761))), ((x * x) * fma(t_2, 0.0140005442, (t_2 * ((x * x) * 0.0008327945)))))) + ((0.0001789971 * 2.0) * ((x * x) * t_6))));
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(x * t_0)
	t_2 = Float64(x * Float64(x * t_1))
	t_3 = Float64(Float64(x * x) * Float64(x * x))
	t_4 = Float64(Float64(x * x) * t_3)
	t_5 = Float64(Float64(x * x) * t_4)
	t_6 = Float64(Float64(x * x) * t_5)
	return Float64(x * Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.1049934947 * Float64(x * x))) + Float64(0.0424060604 * t_3)) + Float64(0.0072644182 * t_4)) + Float64(0.0005064034 * t_5)) + Float64(0.0001789971 * t_6)) / Float64(fma(Float64(1.0 - Float64(t_1 * 0.595284930450289)), Float64(1.0 / Float64(1.0 - Float64(Float64(x * x) * 0.7715471019))), fma(Float64(x * x), fma(Float64(x * x), 0.2909738639, Float64(t_0 * Float64(x * 0.0694555761))), Float64(Float64(x * x) * fma(t_2, 0.0140005442, Float64(t_2 * Float64(Float64(x * x) * 0.0008327945)))))) + Float64(Float64(0.0001789971 * 2.0) * Float64(Float64(x * x) * t_6)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x * x), $MachinePrecision] * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[(x * x), $MachinePrecision] * t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(N[(x * x), $MachinePrecision] * t$95$5), $MachinePrecision]}, N[(x * N[(N[(N[(N[(N[(N[(1.0 + N[(0.1049934947 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0424060604 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(0.0072644182 * t$95$4), $MachinePrecision]), $MachinePrecision] + N[(0.0005064034 * t$95$5), $MachinePrecision]), $MachinePrecision] + N[(0.0001789971 * t$95$6), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.0 - N[(t$95$1 * 0.595284930450289), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 - N[(N[(x * x), $MachinePrecision] * 0.7715471019), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.2909738639 + N[(t$95$0 * N[(x * 0.0694555761), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(t$95$2 * 0.0140005442 + N[(t$95$2 * N[(N[(x * x), $MachinePrecision] * 0.0008327945), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0001789971 * 2.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Initial program 54.3%

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

4. Applied egg-rr 54.3%

   \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\color{blue}{\mathsf{fma}\left(1 - \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.595284930450289, \frac{1}{1 - 0.7715471019 \cdot \left(x \cdot x\right)}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.2909738639, \left(0.0694555761 \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right), 0.0140005442, \left(0.0008327945 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right)} + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x \]

5. Final simplification 54.3%

   \[\leadsto x \cdot \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) + 0.0001789971 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)}{\mathsf{fma}\left(1 - \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.595284930450289, \frac{1}{1 - \left(x \cdot x\right) \cdot 0.7715471019}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.2909738639, \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot 0.0694555761\right)\right), \left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right), 0.0140005442, \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 0.0008327945\right)\right)\right)\right) + \left(0.0001789971 \cdot 2\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)} \]
6. Add Preprocessing

Alternative 2: 55.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := x \cdot \left(x \cdot \left(x \cdot t\_0\right)\right)\\ t_2 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ t_3 := \left(x \cdot x\right) \cdot t\_2\\ t_4 := \left(x \cdot x\right) \cdot t\_3\\ t_5 := \left(x \cdot x\right) \cdot t\_4\\ x \cdot \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot t\_2\right) + 0.0072644182 \cdot t\_3\right) + 0.0005064034 \cdot t\_4\right) + 0.0001789971 \cdot t\_5}{\left(0.0001789971 \cdot 2\right) \cdot \left(\left(x \cdot x\right) \cdot t\_5\right) + \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(t\_1, 0.0140005442, t\_1 \cdot \left(\left(x \cdot x\right) \cdot 0.0008327945\right)\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.2909738639, t\_0 \cdot \left(x \cdot 0.0694555761\right)\right), \mathsf{fma}\left(0.7715471019, x \cdot x, 1\right)\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x)))
        (t_1 (* x (* x (* x t_0))))
        (t_2 (* (* x x) (* x x)))
        (t_3 (* (* x x) t_2))
        (t_4 (* (* x x) t_3))
        (t_5 (* (* x x) t_4)))
   (*
    x
    (/
     (+
      (+
       (+
        (+ (+ 1.0 (* 0.1049934947 (* x x))) (* 0.0424060604 t_2))
        (* 0.0072644182 t_3))
       (* 0.0005064034 t_4))
      (* 0.0001789971 t_5))
     (+
      (* (* 0.0001789971 2.0) (* (* x x) t_5))
      (fma
       (* x x)
       (fma t_1 0.0140005442 (* t_1 (* (* x x) 0.0008327945)))
       (fma
        (* x x)
        (fma (* x x) 0.2909738639 (* t_0 (* x 0.0694555761)))
        (fma 0.7715471019 (* x x) 1.0))))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = x * (x * (x * t_0));
	double t_2 = (x * x) * (x * x);
	double t_3 = (x * x) * t_2;
	double t_4 = (x * x) * t_3;
	double t_5 = (x * x) * t_4;
	return x * ((((((1.0 + (0.1049934947 * (x * x))) + (0.0424060604 * t_2)) + (0.0072644182 * t_3)) + (0.0005064034 * t_4)) + (0.0001789971 * t_5)) / (((0.0001789971 * 2.0) * ((x * x) * t_5)) + fma((x * x), fma(t_1, 0.0140005442, (t_1 * ((x * x) * 0.0008327945))), fma((x * x), fma((x * x), 0.2909738639, (t_0 * (x * 0.0694555761))), fma(0.7715471019, (x * x), 1.0)))));
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(x * Float64(x * Float64(x * t_0)))
	t_2 = Float64(Float64(x * x) * Float64(x * x))
	t_3 = Float64(Float64(x * x) * t_2)
	t_4 = Float64(Float64(x * x) * t_3)
	t_5 = Float64(Float64(x * x) * t_4)
	return Float64(x * Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.1049934947 * Float64(x * x))) + Float64(0.0424060604 * t_2)) + Float64(0.0072644182 * t_3)) + Float64(0.0005064034 * t_4)) + Float64(0.0001789971 * t_5)) / Float64(Float64(Float64(0.0001789971 * 2.0) * Float64(Float64(x * x) * t_5)) + fma(Float64(x * x), fma(t_1, 0.0140005442, Float64(t_1 * Float64(Float64(x * x) * 0.0008327945))), fma(Float64(x * x), fma(Float64(x * x), 0.2909738639, Float64(t_0 * Float64(x * 0.0694555761))), fma(0.7715471019, Float64(x * x), 1.0))))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * x), $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x * x), $MachinePrecision] * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[(x * x), $MachinePrecision] * t$95$4), $MachinePrecision]}, N[(x * N[(N[(N[(N[(N[(N[(1.0 + N[(0.1049934947 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0424060604 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.0072644182 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(0.0005064034 * t$95$4), $MachinePrecision]), $MachinePrecision] + N[(0.0001789971 * t$95$5), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(0.0001789971 * 2.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(t$95$1 * 0.0140005442 + N[(t$95$1 * N[(N[(x * x), $MachinePrecision] * 0.0008327945), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.2909738639 + N[(t$95$0 * N[(x * 0.0694555761), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.7715471019 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Initial program 54.3%

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

4. Applied egg-rr 54.3%

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

5. Final simplification 54.3%

   \[\leadsto x \cdot \frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) + 0.0001789971 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)}{\left(0.0001789971 \cdot 2\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) + \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right), 0.0140005442, \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 0.0008327945\right)\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.2909738639, \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot 0.0694555761\right)\right), \mathsf{fma}\left(0.7715471019, x \cdot x, 1\right)\right)\right)} \]
6. Add Preprocessing

Alternative 3: 55.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.0001789971, 0.0005064034\right)\right), \mathsf{fma}\left(x, x \cdot 0.0072644182, 0.0424060604\right)\right), 0.1049934947\right), 1\right) \cdot \frac{x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \left(t\_0 \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.0003579942\right)\right)\right), t\_0 \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.0008327945, 0.0140005442\right)\right)\right), x \cdot \mathsf{fma}\left(x, x \cdot 0.0694555761, 0.2909738639\right)\right), 0.7715471019\right), 1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (*
    (fma
     x
     (*
      x
      (fma
       (* x x)
       (fma
        x
        (* (* x x) (* x (fma (* x x) 0.0001789971 0.0005064034)))
        (fma x (* x 0.0072644182) 0.0424060604))
       0.1049934947))
     1.0)
    (/
     x
     (fma
      x
      (*
       x
       (fma
        x
        (fma
         x
         (fma
          x
          (* x (* t_0 (* x (* (* x x) 0.0003579942))))
          (* t_0 (* x (fma (* x x) 0.0008327945 0.0140005442))))
         (* x (fma x (* x 0.0694555761) 0.2909738639)))
        0.7715471019))
      1.0)))))

C

double code(double x) {
	double t_0 = x * (x * x);
	return fma(x, (x * fma((x * x), fma(x, ((x * x) * (x * fma((x * x), 0.0001789971, 0.0005064034))), fma(x, (x * 0.0072644182), 0.0424060604)), 0.1049934947)), 1.0) * (x / fma(x, (x * fma(x, fma(x, fma(x, (x * (t_0 * (x * ((x * x) * 0.0003579942)))), (t_0 * (x * fma((x * x), 0.0008327945, 0.0140005442)))), (x * fma(x, (x * 0.0694555761), 0.2909738639))), 0.7715471019)), 1.0));
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	return Float64(fma(x, Float64(x * fma(Float64(x * x), fma(x, Float64(Float64(x * x) * Float64(x * fma(Float64(x * x), 0.0001789971, 0.0005064034))), fma(x, Float64(x * 0.0072644182), 0.0424060604)), 0.1049934947)), 1.0) * Float64(x / fma(x, Float64(x * fma(x, fma(x, fma(x, Float64(x * Float64(t_0 * Float64(x * Float64(Float64(x * x) * 0.0003579942)))), Float64(t_0 * Float64(x * fma(Float64(x * x), 0.0008327945, 0.0140005442)))), Float64(x * fma(x, Float64(x * 0.0694555761), 0.2909738639))), 0.7715471019)), 1.0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[(N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * 0.0001789971 + 0.0005064034), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * 0.0072644182), $MachinePrecision] + 0.0424060604), $MachinePrecision]), $MachinePrecision] + 0.1049934947), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x / N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * N[(t$95$0 * N[(x * N[(N[(x * x), $MachinePrecision] * 0.0003579942), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(x * N[(N[(x * x), $MachinePrecision] * 0.0008327945 + 0.0140005442), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * N[(x * 0.0694555761), $MachinePrecision] + 0.2909738639), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.7715471019), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 54.3%

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

4. Applied egg-rr 53.8%

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

6. Applied egg-rr 53.8%

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

8. Applied egg-rr 54.3%

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

10. Applied egg-rr 54.3%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.0001789971, 0.0005064034\right)\right), \mathsf{fma}\left(x, x \cdot 0.0072644182, 0.0424060604\right)\right), 0.1049934947\right), 1\right) \cdot \frac{x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.0003579942\right)\right)\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.0008327945, 0.0140005442\right)\right)\right), x \cdot \mathsf{fma}\left(x, x \cdot 0.0694555761, 0.2909738639\right)\right), 0.7715471019\right), 1\right)}} \]
11. Add Preprocessing

Alternative 4: 55.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ x \cdot \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.0001789971, 0.0005064034\right)\right), \mathsf{fma}\left(x, x \cdot 0.0072644182, 0.0424060604\right)\right), 0.1049934947\right), 1\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \left(t\_0 \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.0003579942\right)\right)\right), t\_0 \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.0008327945, 0.0140005442\right)\right)\right), x \cdot \mathsf{fma}\left(x, x \cdot 0.0694555761, 0.2909738639\right)\right), 0.7715471019\right), 1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (*
    x
    (/
     (fma
      x
      (*
       x
       (fma
        (* x x)
        (fma
         x
         (* (* x x) (* x (fma (* x x) 0.0001789971 0.0005064034)))
         (fma x (* x 0.0072644182) 0.0424060604))
        0.1049934947))
      1.0)
     (fma
      x
      (*
       x
       (fma
        x
        (fma
         x
         (fma
          x
          (* x (* t_0 (* x (* (* x x) 0.0003579942))))
          (* t_0 (* x (fma (* x x) 0.0008327945 0.0140005442))))
         (* x (fma x (* x 0.0694555761) 0.2909738639)))
        0.7715471019))
      1.0)))))

C

double code(double x) {
	double t_0 = x * (x * x);
	return x * (fma(x, (x * fma((x * x), fma(x, ((x * x) * (x * fma((x * x), 0.0001789971, 0.0005064034))), fma(x, (x * 0.0072644182), 0.0424060604)), 0.1049934947)), 1.0) / fma(x, (x * fma(x, fma(x, fma(x, (x * (t_0 * (x * ((x * x) * 0.0003579942)))), (t_0 * (x * fma((x * x), 0.0008327945, 0.0140005442)))), (x * fma(x, (x * 0.0694555761), 0.2909738639))), 0.7715471019)), 1.0));
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	return Float64(x * Float64(fma(x, Float64(x * fma(Float64(x * x), fma(x, Float64(Float64(x * x) * Float64(x * fma(Float64(x * x), 0.0001789971, 0.0005064034))), fma(x, Float64(x * 0.0072644182), 0.0424060604)), 0.1049934947)), 1.0) / fma(x, Float64(x * fma(x, fma(x, fma(x, Float64(x * Float64(t_0 * Float64(x * Float64(Float64(x * x) * 0.0003579942)))), Float64(t_0 * Float64(x * fma(Float64(x * x), 0.0008327945, 0.0140005442)))), Float64(x * fma(x, Float64(x * 0.0694555761), 0.2909738639))), 0.7715471019)), 1.0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[(x * N[(N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * 0.0001789971 + 0.0005064034), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * 0.0072644182), $MachinePrecision] + 0.0424060604), $MachinePrecision]), $MachinePrecision] + 0.1049934947), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * N[(t$95$0 * N[(x * N[(N[(x * x), $MachinePrecision] * 0.0003579942), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(x * N[(N[(x * x), $MachinePrecision] * 0.0008327945 + 0.0140005442), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * N[(x * 0.0694555761), $MachinePrecision] + 0.2909738639), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.7715471019), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 54.3%

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

4. Applied egg-rr 53.8%

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

6. Applied egg-rr 53.8%

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

8. Applied egg-rr 54.3%

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

10. Applied egg-rr 54.3%

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

11. Final simplification 54.3%

    \[\leadsto x \cdot \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.0001789971, 0.0005064034\right)\right), \mathsf{fma}\left(x, x \cdot 0.0072644182, 0.0424060604\right)\right), 0.1049934947\right), 1\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.0003579942\right)\right)\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.0008327945, 0.0140005442\right)\right)\right), x \cdot \mathsf{fma}\left(x, x \cdot 0.0694555761, 0.2909738639\right)\right), 0.7715471019\right), 1\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024208 
(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))