Jmat.Real.dawson

Percentage Accurate: 53.8% → 100.0%

Time: 15.6s

Alternatives: 8

Speedup: 21.6×

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: 53.8% 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: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 200:\\ \;\;\;\;\frac{x\_m \cdot \left(\mathsf{fma}\left(0.0005064034, {x\_m}^{8}, \mathsf{fma}\left(0.1049934947, x\_m \cdot x\_m, \mathsf{fma}\left(0.0424060604, {x\_m}^{4}, 1\right)\right)\right) + {x\_m}^{6} \cdot \left(0.0072644182 + {x\_m}^{4} \cdot 0.0001789971\right)\right)}{\mathsf{fma}\left(0.0003579942, {x\_m}^{12}, \mathsf{fma}\left({x\_m}^{8}, 0.0140005442, \mathsf{fma}\left(x\_m, x\_m \cdot 0.7715471019, \mathsf{fma}\left(0.2909738639, {x\_m}^{4}, 1\right)\right)\right) + {x\_m}^{6} \cdot \left(0.0694555761 + {x\_m}^{4} \cdot 0.0008327945\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \left(\frac{0.2514179000665374}{{x\_m}^{2}} + \left(\frac{0.15298196345929074}{{x\_m}^{4}} + \frac{11.259630434457211}{{x\_m}^{6}}\right)\right)}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 200.0)
    (/
     (*
      x_m
      (+
       (fma
        0.0005064034
        (pow x_m 8.0)
        (fma 0.1049934947 (* x_m x_m) (fma 0.0424060604 (pow x_m 4.0) 1.0)))
       (* (pow x_m 6.0) (+ 0.0072644182 (* (pow x_m 4.0) 0.0001789971)))))
     (fma
      0.0003579942
      (pow x_m 12.0)
      (+
       (fma
        (pow x_m 8.0)
        0.0140005442
        (fma x_m (* x_m 0.7715471019) (fma 0.2909738639 (pow x_m 4.0) 1.0)))
       (* (pow x_m 6.0) (+ 0.0694555761 (* (pow x_m 4.0) 0.0008327945))))))
    (/
     (+
      0.5
      (+
       (/ 0.2514179000665374 (pow x_m 2.0))
       (+
        (/ 0.15298196345929074 (pow x_m 4.0))
        (/ 11.259630434457211 (pow x_m 6.0)))))
     x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 200.0) {
		tmp = (x_m * (fma(0.0005064034, pow(x_m, 8.0), fma(0.1049934947, (x_m * x_m), fma(0.0424060604, pow(x_m, 4.0), 1.0))) + (pow(x_m, 6.0) * (0.0072644182 + (pow(x_m, 4.0) * 0.0001789971))))) / fma(0.0003579942, pow(x_m, 12.0), (fma(pow(x_m, 8.0), 0.0140005442, fma(x_m, (x_m * 0.7715471019), fma(0.2909738639, pow(x_m, 4.0), 1.0))) + (pow(x_m, 6.0) * (0.0694555761 + (pow(x_m, 4.0) * 0.0008327945)))));
	} else {
		tmp = (0.5 + ((0.2514179000665374 / pow(x_m, 2.0)) + ((0.15298196345929074 / pow(x_m, 4.0)) + (11.259630434457211 / pow(x_m, 6.0))))) / x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 200.0)
		tmp = Float64(Float64(x_m * Float64(fma(0.0005064034, (x_m ^ 8.0), fma(0.1049934947, Float64(x_m * x_m), fma(0.0424060604, (x_m ^ 4.0), 1.0))) + Float64((x_m ^ 6.0) * Float64(0.0072644182 + Float64((x_m ^ 4.0) * 0.0001789971))))) / fma(0.0003579942, (x_m ^ 12.0), Float64(fma((x_m ^ 8.0), 0.0140005442, fma(x_m, Float64(x_m * 0.7715471019), fma(0.2909738639, (x_m ^ 4.0), 1.0))) + Float64((x_m ^ 6.0) * Float64(0.0694555761 + Float64((x_m ^ 4.0) * 0.0008327945))))));
	else
		tmp = Float64(Float64(0.5 + Float64(Float64(0.2514179000665374 / (x_m ^ 2.0)) + Float64(Float64(0.15298196345929074 / (x_m ^ 4.0)) + Float64(11.259630434457211 / (x_m ^ 6.0))))) / x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 200.0], N[(N[(x$95$m * N[(N[(0.0005064034 * N[Power[x$95$m, 8.0], $MachinePrecision] + N[(0.1049934947 * N[(x$95$m * x$95$m), $MachinePrecision] + N[(0.0424060604 * N[Power[x$95$m, 4.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x$95$m, 6.0], $MachinePrecision] * N[(0.0072644182 + N[(N[Power[x$95$m, 4.0], $MachinePrecision] * 0.0001789971), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.0003579942 * N[Power[x$95$m, 12.0], $MachinePrecision] + N[(N[(N[Power[x$95$m, 8.0], $MachinePrecision] * 0.0140005442 + N[(x$95$m * N[(x$95$m * 0.7715471019), $MachinePrecision] + N[(0.2909738639 * N[Power[x$95$m, 4.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x$95$m, 6.0], $MachinePrecision] * N[(0.0694555761 + N[(N[Power[x$95$m, 4.0], $MachinePrecision] * 0.0008327945), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(N[(0.2514179000665374 / N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.15298196345929074 / N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + N[(11.259630434457211 / N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 200

   1. Initial program 62.1%

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

   3. Simplified 62.2%

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

   if 200 < x

   1. Initial program 10.8%

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

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{0.5 + \left(\frac{0.15298196345929074}{{x}^{4}} + \left(0.2514179000665374 \cdot \frac{1}{{x}^{2}} + 11.259630434457211 \cdot \frac{1}{{x}^{6}}\right)\right)}{x}} \]
   4. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \frac{0.5 + \color{blue}{\left(\left(0.2514179000665374 \cdot \frac{1}{{x}^{2}} + 11.259630434457211 \cdot \frac{1}{{x}^{6}}\right) + \frac{0.15298196345929074}{{x}^{4}}\right)}}{x} \]

      2. associate-+l+ 100.0%

         \[\leadsto \frac{0.5 + \color{blue}{\left(0.2514179000665374 \cdot \frac{1}{{x}^{2}} + \left(11.259630434457211 \cdot \frac{1}{{x}^{6}} + \frac{0.15298196345929074}{{x}^{4}}\right)\right)}}{x} \]

      3. associate-*r/ 100.0%

         \[\leadsto \frac{0.5 + \left(\color{blue}{\frac{0.2514179000665374 \cdot 1}{{x}^{2}}} + \left(11.259630434457211 \cdot \frac{1}{{x}^{6}} + \frac{0.15298196345929074}{{x}^{4}}\right)\right)}{x} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{0.5 + \left(\frac{\color{blue}{0.2514179000665374}}{{x}^{2}} + \left(11.259630434457211 \cdot \frac{1}{{x}^{6}} + \frac{0.15298196345929074}{{x}^{4}}\right)\right)}{x} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{0.5 + \left(\frac{0.2514179000665374}{{x}^{2}} + \color{blue}{\left(\frac{0.15298196345929074}{{x}^{4}} + 11.259630434457211 \cdot \frac{1}{{x}^{6}}\right)}\right)}{x} \]

      6. associate-*r/ 100.0%

         \[\leadsto \frac{0.5 + \left(\frac{0.2514179000665374}{{x}^{2}} + \left(\frac{0.15298196345929074}{{x}^{4}} + \color{blue}{\frac{11.259630434457211 \cdot 1}{{x}^{6}}}\right)\right)}{x} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{0.5 + \left(\frac{0.2514179000665374}{{x}^{2}} + \left(\frac{0.15298196345929074}{{x}^{4}} + \frac{\color{blue}{11.259630434457211}}{{x}^{6}}\right)\right)}{x} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{0.5 + \left(\frac{0.2514179000665374}{{x}^{2}} + \left(\frac{0.15298196345929074}{{x}^{4}} + \frac{11.259630434457211}{{x}^{6}}\right)\right)}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 200:\\ \;\;\;\;\frac{x \cdot \left(\mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left(0.1049934947, x \cdot x, \mathsf{fma}\left(0.0424060604, {x}^{4}, 1\right)\right)\right) + {x}^{6} \cdot \left(0.0072644182 + {x}^{4} \cdot 0.0001789971\right)\right)}{\mathsf{fma}\left(0.0003579942, {x}^{12}, \mathsf{fma}\left({x}^{8}, 0.0140005442, \mathsf{fma}\left(x, x \cdot 0.7715471019, \mathsf{fma}\left(0.2909738639, {x}^{4}, 1\right)\right)\right) + {x}^{6} \cdot \left(0.0694555761 + {x}^{4} \cdot 0.0008327945\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \left(\frac{0.2514179000665374}{{x}^{2}} + \left(\frac{0.15298196345929074}{{x}^{4}} + \frac{11.259630434457211}{{x}^{6}}\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\\ t_1 := \left(x\_m \cdot x\_m\right) \cdot t\_0\\ t_2 := \left(x\_m \cdot x\_m\right) \cdot t\_1\\ t_3 := t\_0 \cdot t\_1\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 500:\\ \;\;\;\;\frac{x\_m \cdot \left(0.0001789971 \cdot t\_3 + \left(\left(1 + \left(0.1049934947 \cdot \left(x\_m \cdot x\_m\right) + \left(x\_m \cdot x\_m\right) \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.0424060604\right)\right)\right) + \left(0.0072644182 \cdot t\_1 + 0.0005064034 \cdot t\_2\right)\right)\right)}{0.0003579942 \cdot \left(t\_0 \cdot t\_2\right) + \left(0.0008327945 \cdot t\_3 + \left(0.0140005442 \cdot t\_2 + \left(0.0694555761 \cdot t\_1 + \left(\left(1 + \left(x\_m \cdot x\_m\right) \cdot 0.7715471019\right) + 0.2909738639 \cdot t\_0\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \left(\frac{0.2514179000665374}{{x\_m}^{2}} + \left(\frac{0.15298196345929074}{{x\_m}^{4}} + \frac{11.259630434457211}{{x\_m}^{6}}\right)\right)}{x\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (* x_m (* x_m (* x_m x_m))))
        (t_1 (* (* x_m x_m) t_0))
        (t_2 (* (* x_m x_m) t_1))
        (t_3 (* t_0 t_1)))
   (*
    x_s
    (if (<= x_m 500.0)
      (/
       (*
        x_m
        (+
         (* 0.0001789971 t_3)
         (+
          (+
           1.0
           (+
            (* 0.1049934947 (* x_m x_m))
            (* (* x_m x_m) (* (* x_m x_m) 0.0424060604))))
          (+ (* 0.0072644182 t_1) (* 0.0005064034 t_2)))))
       (+
        (* 0.0003579942 (* t_0 t_2))
        (+
         (* 0.0008327945 t_3)
         (+
          (* 0.0140005442 t_2)
          (+
           (* 0.0694555761 t_1)
           (+ (+ 1.0 (* (* x_m x_m) 0.7715471019)) (* 0.2909738639 t_0)))))))
      (/
       (+
        0.5
        (+
         (/ 0.2514179000665374 (pow x_m 2.0))
         (+
          (/ 0.15298196345929074 (pow x_m 4.0))
          (/ 11.259630434457211 (pow x_m 6.0)))))
       x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = x_m * (x_m * (x_m * x_m));
	double t_1 = (x_m * x_m) * t_0;
	double t_2 = (x_m * x_m) * t_1;
	double t_3 = t_0 * t_1;
	double tmp;
	if (x_m <= 500.0) {
		tmp = (x_m * ((0.0001789971 * t_3) + ((1.0 + ((0.1049934947 * (x_m * x_m)) + ((x_m * x_m) * ((x_m * x_m) * 0.0424060604)))) + ((0.0072644182 * t_1) + (0.0005064034 * t_2))))) / ((0.0003579942 * (t_0 * t_2)) + ((0.0008327945 * t_3) + ((0.0140005442 * t_2) + ((0.0694555761 * t_1) + ((1.0 + ((x_m * x_m) * 0.7715471019)) + (0.2909738639 * t_0))))));
	} else {
		tmp = (0.5 + ((0.2514179000665374 / pow(x_m, 2.0)) + ((0.15298196345929074 / pow(x_m, 4.0)) + (11.259630434457211 / pow(x_m, 6.0))))) / x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = x_m * (x_m * (x_m * x_m))
    t_1 = (x_m * x_m) * t_0
    t_2 = (x_m * x_m) * t_1
    t_3 = t_0 * t_1
    if (x_m <= 500.0d0) then
        tmp = (x_m * ((0.0001789971d0 * t_3) + ((1.0d0 + ((0.1049934947d0 * (x_m * x_m)) + ((x_m * x_m) * ((x_m * x_m) * 0.0424060604d0)))) + ((0.0072644182d0 * t_1) + (0.0005064034d0 * t_2))))) / ((0.0003579942d0 * (t_0 * t_2)) + ((0.0008327945d0 * t_3) + ((0.0140005442d0 * t_2) + ((0.0694555761d0 * t_1) + ((1.0d0 + ((x_m * x_m) * 0.7715471019d0)) + (0.2909738639d0 * t_0))))))
    else
        tmp = (0.5d0 + ((0.2514179000665374d0 / (x_m ** 2.0d0)) + ((0.15298196345929074d0 / (x_m ** 4.0d0)) + (11.259630434457211d0 / (x_m ** 6.0d0))))) / x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = x_m * (x_m * (x_m * x_m));
	double t_1 = (x_m * x_m) * t_0;
	double t_2 = (x_m * x_m) * t_1;
	double t_3 = t_0 * t_1;
	double tmp;
	if (x_m <= 500.0) {
		tmp = (x_m * ((0.0001789971 * t_3) + ((1.0 + ((0.1049934947 * (x_m * x_m)) + ((x_m * x_m) * ((x_m * x_m) * 0.0424060604)))) + ((0.0072644182 * t_1) + (0.0005064034 * t_2))))) / ((0.0003579942 * (t_0 * t_2)) + ((0.0008327945 * t_3) + ((0.0140005442 * t_2) + ((0.0694555761 * t_1) + ((1.0 + ((x_m * x_m) * 0.7715471019)) + (0.2909738639 * t_0))))));
	} else {
		tmp = (0.5 + ((0.2514179000665374 / Math.pow(x_m, 2.0)) + ((0.15298196345929074 / Math.pow(x_m, 4.0)) + (11.259630434457211 / Math.pow(x_m, 6.0))))) / x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = x_m * (x_m * (x_m * x_m))
	t_1 = (x_m * x_m) * t_0
	t_2 = (x_m * x_m) * t_1
	t_3 = t_0 * t_1
	tmp = 0
	if x_m <= 500.0:
		tmp = (x_m * ((0.0001789971 * t_3) + ((1.0 + ((0.1049934947 * (x_m * x_m)) + ((x_m * x_m) * ((x_m * x_m) * 0.0424060604)))) + ((0.0072644182 * t_1) + (0.0005064034 * t_2))))) / ((0.0003579942 * (t_0 * t_2)) + ((0.0008327945 * t_3) + ((0.0140005442 * t_2) + ((0.0694555761 * t_1) + ((1.0 + ((x_m * x_m) * 0.7715471019)) + (0.2909738639 * t_0))))))
	else:
		tmp = (0.5 + ((0.2514179000665374 / math.pow(x_m, 2.0)) + ((0.15298196345929074 / math.pow(x_m, 4.0)) + (11.259630434457211 / math.pow(x_m, 6.0))))) / x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(x_m * Float64(x_m * Float64(x_m * x_m)))
	t_1 = Float64(Float64(x_m * x_m) * t_0)
	t_2 = Float64(Float64(x_m * x_m) * t_1)
	t_3 = Float64(t_0 * t_1)
	tmp = 0.0
	if (x_m <= 500.0)
		tmp = Float64(Float64(x_m * Float64(Float64(0.0001789971 * t_3) + Float64(Float64(1.0 + Float64(Float64(0.1049934947 * Float64(x_m * x_m)) + Float64(Float64(x_m * x_m) * Float64(Float64(x_m * x_m) * 0.0424060604)))) + Float64(Float64(0.0072644182 * t_1) + Float64(0.0005064034 * t_2))))) / Float64(Float64(0.0003579942 * Float64(t_0 * t_2)) + Float64(Float64(0.0008327945 * t_3) + Float64(Float64(0.0140005442 * t_2) + Float64(Float64(0.0694555761 * t_1) + Float64(Float64(1.0 + Float64(Float64(x_m * x_m) * 0.7715471019)) + Float64(0.2909738639 * t_0)))))));
	else
		tmp = Float64(Float64(0.5 + Float64(Float64(0.2514179000665374 / (x_m ^ 2.0)) + Float64(Float64(0.15298196345929074 / (x_m ^ 4.0)) + Float64(11.259630434457211 / (x_m ^ 6.0))))) / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = x_m * (x_m * (x_m * x_m));
	t_1 = (x_m * x_m) * t_0;
	t_2 = (x_m * x_m) * t_1;
	t_3 = t_0 * t_1;
	tmp = 0.0;
	if (x_m <= 500.0)
		tmp = (x_m * ((0.0001789971 * t_3) + ((1.0 + ((0.1049934947 * (x_m * x_m)) + ((x_m * x_m) * ((x_m * x_m) * 0.0424060604)))) + ((0.0072644182 * t_1) + (0.0005064034 * t_2))))) / ((0.0003579942 * (t_0 * t_2)) + ((0.0008327945 * t_3) + ((0.0140005442 * t_2) + ((0.0694555761 * t_1) + ((1.0 + ((x_m * x_m) * 0.7715471019)) + (0.2909738639 * t_0))))));
	else
		tmp = (0.5 + ((0.2514179000665374 / (x_m ^ 2.0)) + ((0.15298196345929074 / (x_m ^ 4.0)) + (11.259630434457211 / (x_m ^ 6.0))))) / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(x$95$m * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * t$95$1), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 500.0], N[(N[(x$95$m * N[(N[(0.0001789971 * t$95$3), $MachinePrecision] + N[(N[(1.0 + N[(N[(0.1049934947 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0424060604), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0072644182 * t$95$1), $MachinePrecision] + N[(0.0005064034 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.0003579942 * N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0008327945 * t$95$3), $MachinePrecision] + N[(N[(0.0140005442 * t$95$2), $MachinePrecision] + N[(N[(0.0694555761 * t$95$1), $MachinePrecision] + N[(N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.7715471019), $MachinePrecision]), $MachinePrecision] + N[(0.2909738639 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(N[(0.2514179000665374 / N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.15298196345929074 / N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + N[(11.259630434457211 / N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 500

   1. Initial program 62.1%

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

   3. Simplified 62.1%

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

   if 500 < x

   1. Initial program 10.8%

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

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{0.5 + \left(\frac{0.15298196345929074}{{x}^{4}} + \left(0.2514179000665374 \cdot \frac{1}{{x}^{2}} + 11.259630434457211 \cdot \frac{1}{{x}^{6}}\right)\right)}{x}} \]
   4. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \frac{0.5 + \color{blue}{\left(\left(0.2514179000665374 \cdot \frac{1}{{x}^{2}} + 11.259630434457211 \cdot \frac{1}{{x}^{6}}\right) + \frac{0.15298196345929074}{{x}^{4}}\right)}}{x} \]

      2. associate-+l+ 100.0%

         \[\leadsto \frac{0.5 + \color{blue}{\left(0.2514179000665374 \cdot \frac{1}{{x}^{2}} + \left(11.259630434457211 \cdot \frac{1}{{x}^{6}} + \frac{0.15298196345929074}{{x}^{4}}\right)\right)}}{x} \]

      3. associate-*r/ 100.0%

         \[\leadsto \frac{0.5 + \left(\color{blue}{\frac{0.2514179000665374 \cdot 1}{{x}^{2}}} + \left(11.259630434457211 \cdot \frac{1}{{x}^{6}} + \frac{0.15298196345929074}{{x}^{4}}\right)\right)}{x} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{0.5 + \left(\frac{\color{blue}{0.2514179000665374}}{{x}^{2}} + \left(11.259630434457211 \cdot \frac{1}{{x}^{6}} + \frac{0.15298196345929074}{{x}^{4}}\right)\right)}{x} \]

      5. +-commutative 100.0%

         \[\leadsto \frac{0.5 + \left(\frac{0.2514179000665374}{{x}^{2}} + \color{blue}{\left(\frac{0.15298196345929074}{{x}^{4}} + 11.259630434457211 \cdot \frac{1}{{x}^{6}}\right)}\right)}{x} \]

      6. associate-*r/ 100.0%

         \[\leadsto \frac{0.5 + \left(\frac{0.2514179000665374}{{x}^{2}} + \left(\frac{0.15298196345929074}{{x}^{4}} + \color{blue}{\frac{11.259630434457211 \cdot 1}{{x}^{6}}}\right)\right)}{x} \]

      7. metadata-eval 100.0%

         \[\leadsto \frac{0.5 + \left(\frac{0.2514179000665374}{{x}^{2}} + \left(\frac{0.15298196345929074}{{x}^{4}} + \frac{\color{blue}{11.259630434457211}}{{x}^{6}}\right)\right)}{x} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{0.5 + \left(\frac{0.2514179000665374}{{x}^{2}} + \left(\frac{0.15298196345929074}{{x}^{4}} + \frac{11.259630434457211}{{x}^{6}}\right)\right)}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 500:\\ \;\;\;\;\frac{x \cdot \left(0.0001789971 \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) + \left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.0424060604\right)\right)\right) + \left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right)}{0.0003579942 \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) + \left(0.0008327945 \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) + \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) + \left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right) + 0.2909738639 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \left(\frac{0.2514179000665374}{{x}^{2}} + \left(\frac{0.15298196345929074}{{x}^{4}} + \frac{11.259630434457211}{{x}^{6}}\right)\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\\ t_1 := \left(x\_m \cdot x\_m\right) \cdot t\_0\\ t_2 := \left(x\_m \cdot x\_m\right) \cdot t\_1\\ t_3 := t\_0 \cdot t\_1\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5000:\\ \;\;\;\;\frac{x\_m \cdot \left(0.0001789971 \cdot t\_3 + \left(\left(1 + \left(0.1049934947 \cdot \left(x\_m \cdot x\_m\right) + \left(x\_m \cdot x\_m\right) \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.0424060604\right)\right)\right) + \left(0.0072644182 \cdot t\_1 + 0.0005064034 \cdot t\_2\right)\right)\right)}{0.0003579942 \cdot \left(t\_0 \cdot t\_2\right) + \left(0.0008327945 \cdot t\_3 + \left(0.0140005442 \cdot t\_2 + \left(0.0694555761 \cdot t\_1 + \left(\left(1 + \left(x\_m \cdot x\_m\right) \cdot 0.7715471019\right) + 0.2909738639 \cdot t\_0\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x\_m} + 0.2514179000665374 \cdot {x\_m}^{-3}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (let* ((t_0 (* x_m (* x_m (* x_m x_m))))
        (t_1 (* (* x_m x_m) t_0))
        (t_2 (* (* x_m x_m) t_1))
        (t_3 (* t_0 t_1)))
   (*
    x_s
    (if (<= x_m 5000.0)
      (/
       (*
        x_m
        (+
         (* 0.0001789971 t_3)
         (+
          (+
           1.0
           (+
            (* 0.1049934947 (* x_m x_m))
            (* (* x_m x_m) (* (* x_m x_m) 0.0424060604))))
          (+ (* 0.0072644182 t_1) (* 0.0005064034 t_2)))))
       (+
        (* 0.0003579942 (* t_0 t_2))
        (+
         (* 0.0008327945 t_3)
         (+
          (* 0.0140005442 t_2)
          (+
           (* 0.0694555761 t_1)
           (+ (+ 1.0 (* (* x_m x_m) 0.7715471019)) (* 0.2909738639 t_0)))))))
      (+ (/ 0.5 x_m) (* 0.2514179000665374 (pow x_m -3.0)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double t_0 = x_m * (x_m * (x_m * x_m));
	double t_1 = (x_m * x_m) * t_0;
	double t_2 = (x_m * x_m) * t_1;
	double t_3 = t_0 * t_1;
	double tmp;
	if (x_m <= 5000.0) {
		tmp = (x_m * ((0.0001789971 * t_3) + ((1.0 + ((0.1049934947 * (x_m * x_m)) + ((x_m * x_m) * ((x_m * x_m) * 0.0424060604)))) + ((0.0072644182 * t_1) + (0.0005064034 * t_2))))) / ((0.0003579942 * (t_0 * t_2)) + ((0.0008327945 * t_3) + ((0.0140005442 * t_2) + ((0.0694555761 * t_1) + ((1.0 + ((x_m * x_m) * 0.7715471019)) + (0.2909738639 * t_0))))));
	} else {
		tmp = (0.5 / x_m) + (0.2514179000665374 * pow(x_m, -3.0));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = x_m * (x_m * (x_m * x_m))
    t_1 = (x_m * x_m) * t_0
    t_2 = (x_m * x_m) * t_1
    t_3 = t_0 * t_1
    if (x_m <= 5000.0d0) then
        tmp = (x_m * ((0.0001789971d0 * t_3) + ((1.0d0 + ((0.1049934947d0 * (x_m * x_m)) + ((x_m * x_m) * ((x_m * x_m) * 0.0424060604d0)))) + ((0.0072644182d0 * t_1) + (0.0005064034d0 * t_2))))) / ((0.0003579942d0 * (t_0 * t_2)) + ((0.0008327945d0 * t_3) + ((0.0140005442d0 * t_2) + ((0.0694555761d0 * t_1) + ((1.0d0 + ((x_m * x_m) * 0.7715471019d0)) + (0.2909738639d0 * t_0))))))
    else
        tmp = (0.5d0 / x_m) + (0.2514179000665374d0 * (x_m ** (-3.0d0)))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double t_0 = x_m * (x_m * (x_m * x_m));
	double t_1 = (x_m * x_m) * t_0;
	double t_2 = (x_m * x_m) * t_1;
	double t_3 = t_0 * t_1;
	double tmp;
	if (x_m <= 5000.0) {
		tmp = (x_m * ((0.0001789971 * t_3) + ((1.0 + ((0.1049934947 * (x_m * x_m)) + ((x_m * x_m) * ((x_m * x_m) * 0.0424060604)))) + ((0.0072644182 * t_1) + (0.0005064034 * t_2))))) / ((0.0003579942 * (t_0 * t_2)) + ((0.0008327945 * t_3) + ((0.0140005442 * t_2) + ((0.0694555761 * t_1) + ((1.0 + ((x_m * x_m) * 0.7715471019)) + (0.2909738639 * t_0))))));
	} else {
		tmp = (0.5 / x_m) + (0.2514179000665374 * Math.pow(x_m, -3.0));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	t_0 = x_m * (x_m * (x_m * x_m))
	t_1 = (x_m * x_m) * t_0
	t_2 = (x_m * x_m) * t_1
	t_3 = t_0 * t_1
	tmp = 0
	if x_m <= 5000.0:
		tmp = (x_m * ((0.0001789971 * t_3) + ((1.0 + ((0.1049934947 * (x_m * x_m)) + ((x_m * x_m) * ((x_m * x_m) * 0.0424060604)))) + ((0.0072644182 * t_1) + (0.0005064034 * t_2))))) / ((0.0003579942 * (t_0 * t_2)) + ((0.0008327945 * t_3) + ((0.0140005442 * t_2) + ((0.0694555761 * t_1) + ((1.0 + ((x_m * x_m) * 0.7715471019)) + (0.2909738639 * t_0))))))
	else:
		tmp = (0.5 / x_m) + (0.2514179000665374 * math.pow(x_m, -3.0))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	t_0 = Float64(x_m * Float64(x_m * Float64(x_m * x_m)))
	t_1 = Float64(Float64(x_m * x_m) * t_0)
	t_2 = Float64(Float64(x_m * x_m) * t_1)
	t_3 = Float64(t_0 * t_1)
	tmp = 0.0
	if (x_m <= 5000.0)
		tmp = Float64(Float64(x_m * Float64(Float64(0.0001789971 * t_3) + Float64(Float64(1.0 + Float64(Float64(0.1049934947 * Float64(x_m * x_m)) + Float64(Float64(x_m * x_m) * Float64(Float64(x_m * x_m) * 0.0424060604)))) + Float64(Float64(0.0072644182 * t_1) + Float64(0.0005064034 * t_2))))) / Float64(Float64(0.0003579942 * Float64(t_0 * t_2)) + Float64(Float64(0.0008327945 * t_3) + Float64(Float64(0.0140005442 * t_2) + Float64(Float64(0.0694555761 * t_1) + Float64(Float64(1.0 + Float64(Float64(x_m * x_m) * 0.7715471019)) + Float64(0.2909738639 * t_0)))))));
	else
		tmp = Float64(Float64(0.5 / x_m) + Float64(0.2514179000665374 * (x_m ^ -3.0)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	t_0 = x_m * (x_m * (x_m * x_m));
	t_1 = (x_m * x_m) * t_0;
	t_2 = (x_m * x_m) * t_1;
	t_3 = t_0 * t_1;
	tmp = 0.0;
	if (x_m <= 5000.0)
		tmp = (x_m * ((0.0001789971 * t_3) + ((1.0 + ((0.1049934947 * (x_m * x_m)) + ((x_m * x_m) * ((x_m * x_m) * 0.0424060604)))) + ((0.0072644182 * t_1) + (0.0005064034 * t_2))))) / ((0.0003579942 * (t_0 * t_2)) + ((0.0008327945 * t_3) + ((0.0140005442 * t_2) + ((0.0694555761 * t_1) + ((1.0 + ((x_m * x_m) * 0.7715471019)) + (0.2909738639 * t_0))))));
	else
		tmp = (0.5 / x_m) + (0.2514179000665374 * (x_m ^ -3.0));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := Block[{t$95$0 = N[(x$95$m * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * t$95$1), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 5000.0], N[(N[(x$95$m * N[(N[(0.0001789971 * t$95$3), $MachinePrecision] + N[(N[(1.0 + N[(N[(0.1049934947 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0424060604), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0072644182 * t$95$1), $MachinePrecision] + N[(0.0005064034 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.0003579942 * N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0008327945 * t$95$3), $MachinePrecision] + N[(N[(0.0140005442 * t$95$2), $MachinePrecision] + N[(N[(0.0694555761 * t$95$1), $MachinePrecision] + N[(N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.7715471019), $MachinePrecision]), $MachinePrecision] + N[(0.2909738639 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / x$95$m), $MachinePrecision] + N[(0.2514179000665374 * N[Power[x$95$m, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 5e3

   1. Initial program 62.3%

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

   3. Simplified 62.3%

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

   if 5e3 < x

   1. Initial program 9.2%

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

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{0.5 + 0.2514179000665374 \cdot \frac{1}{{x}^{2}}}{x}} \]
   4. Step-by-step derivation

      1. associate-*r/ 100.0%

         \[\leadsto \frac{0.5 + \color{blue}{\frac{0.2514179000665374 \cdot 1}{{x}^{2}}}}{x} \]

      2. metadata-eval 100.0%

         \[\leadsto \frac{0.5 + \frac{\color{blue}{0.2514179000665374}}{{x}^{2}}}{x} \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{0.5 + \frac{0.2514179000665374}{{x}^{2}}}{x}} \]
   6. Step-by-step derivation

      1. div-inv 100.0%

         \[\leadsto \color{blue}{\left(0.5 + \frac{0.2514179000665374}{{x}^{2}}\right) \cdot \frac{1}{x}} \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(\frac{0.2514179000665374}{{x}^{2}} + 0.5\right)} \cdot \frac{1}{x} \]

      3. div-inv 100.0%

         \[\leadsto \left(\color{blue}{0.2514179000665374 \cdot \frac{1}{{x}^{2}}} + 0.5\right) \cdot \frac{1}{x} \]

      4. fma-define 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.2514179000665374, \frac{1}{{x}^{2}}, 0.5\right)} \cdot \frac{1}{x} \]

      5. pow-flip 100.0%

         \[\leadsto \mathsf{fma}\left(0.2514179000665374, \color{blue}{{x}^{\left(-2\right)}}, 0.5\right) \cdot \frac{1}{x} \]

      6. metadata-eval 100.0%

         \[\leadsto \mathsf{fma}\left(0.2514179000665374, {x}^{\color{blue}{-2}}, 0.5\right) \cdot \frac{1}{x} \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.2514179000665374, {x}^{-2}, 0.5\right) \cdot \frac{1}{x}} \]
   8. Step-by-step derivation

      1. fma-undefine 100.0%

         \[\leadsto \color{blue}{\left(0.2514179000665374 \cdot {x}^{-2} + 0.5\right)} \cdot \frac{1}{x} \]

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(0.2514179000665374 \cdot {x}^{-2} + 0.5\right)} \cdot \frac{1}{x} \]
   10. Step-by-step derivation

       1. *-commutative 100.0%

          \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(0.2514179000665374 \cdot {x}^{-2} + 0.5\right)} \]

       2. +-commutative 100.0%

          \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(0.5 + 0.2514179000665374 \cdot {x}^{-2}\right)} \]

       3. distribute-lft-in 100.0%

          \[\leadsto \color{blue}{\frac{1}{x} \cdot 0.5 + \frac{1}{x} \cdot \left(0.2514179000665374 \cdot {x}^{-2}\right)} \]

       4. *-commutative 100.0%

          \[\leadsto \color{blue}{0.5 \cdot \frac{1}{x}} + \frac{1}{x} \cdot \left(0.2514179000665374 \cdot {x}^{-2}\right) \]

       5. un-div-inv 100.0%

          \[\leadsto \color{blue}{\frac{0.5}{x}} + \frac{1}{x} \cdot \left(0.2514179000665374 \cdot {x}^{-2}\right) \]

       6. *-commutative 100.0%

          \[\leadsto \frac{0.5}{x} + \color{blue}{\left(0.2514179000665374 \cdot {x}^{-2}\right) \cdot \frac{1}{x}} \]

       7. associate-*l* 100.0%

          \[\leadsto \frac{0.5}{x} + \color{blue}{0.2514179000665374 \cdot \left({x}^{-2} \cdot \frac{1}{x}\right)} \]

       8. inv-pow 100.0%

          \[\leadsto \frac{0.5}{x} + 0.2514179000665374 \cdot \left({x}^{-2} \cdot \color{blue}{{x}^{-1}}\right) \]

       9. pow-prod-up 100.0%

          \[\leadsto \frac{0.5}{x} + 0.2514179000665374 \cdot \color{blue}{{x}^{\left(-2 + -1\right)}} \]

       10. metadata-eval 100.0%

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

   11. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{0.5}{x} + 0.2514179000665374 \cdot {x}^{-3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5000:\\ \;\;\;\;\frac{x \cdot \left(0.0001789971 \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) + \left(\left(1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.0424060604\right)\right)\right) + \left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + 0.0005064034 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right)}{0.0003579942 \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) + \left(0.0008327945 \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) + \left(0.0140005442 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) + \left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(1 + \left(x \cdot x\right) \cdot 0.7715471019\right) + 0.2909738639 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} + 0.2514179000665374 \cdot {x}^{-3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.95:\\ \;\;\;\;x\_m + {x\_m}^{3} \cdot -0.6665536072\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x\_m} + 0.2514179000665374 \cdot {x\_m}^{-3}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.95)
    (+ x_m (* (pow x_m 3.0) -0.6665536072))
    (+ (/ 0.5 x_m) (* 0.2514179000665374 (pow x_m -3.0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.95) {
		tmp = x_m + (pow(x_m, 3.0) * -0.6665536072);
	} else {
		tmp = (0.5 / x_m) + (0.2514179000665374 * pow(x_m, -3.0));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.95d0) then
        tmp = x_m + ((x_m ** 3.0d0) * (-0.6665536072d0))
    else
        tmp = (0.5d0 / x_m) + (0.2514179000665374d0 * (x_m ** (-3.0d0)))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.95) {
		tmp = x_m + (Math.pow(x_m, 3.0) * -0.6665536072);
	} else {
		tmp = (0.5 / x_m) + (0.2514179000665374 * Math.pow(x_m, -3.0));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.95:
		tmp = x_m + (math.pow(x_m, 3.0) * -0.6665536072)
	else:
		tmp = (0.5 / x_m) + (0.2514179000665374 * math.pow(x_m, -3.0))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.95)
		tmp = Float64(x_m + Float64((x_m ^ 3.0) * -0.6665536072));
	else
		tmp = Float64(Float64(0.5 / x_m) + Float64(0.2514179000665374 * (x_m ^ -3.0)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.95)
		tmp = x_m + ((x_m ^ 3.0) * -0.6665536072);
	else
		tmp = (0.5 / x_m) + (0.2514179000665374 * (x_m ^ -3.0));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.95], N[(x$95$m + N[(N[Power[x$95$m, 3.0], $MachinePrecision] * -0.6665536072), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / x$95$m), $MachinePrecision] + N[(0.2514179000665374 * N[Power[x$95$m, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.94999999999999996

   1. Initial program 62.1%

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

   3. Simplified 62.2%

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

   6. Applied egg-rr 62.2%

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

   8. Simplified 62.1%

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

   9. Taylor expanded in x around 0 59.8%

      \[\leadsto \color{blue}{x \cdot \left(1 + -0.6665536072 \cdot {x}^{2}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in 59.8%

          \[\leadsto \color{blue}{x \cdot 1 + x \cdot \left(-0.6665536072 \cdot {x}^{2}\right)} \]

       2. *-rgt-identity 59.8%

          \[\leadsto \color{blue}{x} + x \cdot \left(-0.6665536072 \cdot {x}^{2}\right) \]

       3. *-commutative 59.8%

          \[\leadsto x + x \cdot \color{blue}{\left({x}^{2} \cdot -0.6665536072\right)} \]

       4. associate-*r* 59.8%

          \[\leadsto x + \color{blue}{\left(x \cdot {x}^{2}\right) \cdot -0.6665536072} \]

       5. unpow2 59.8%

          \[\leadsto x + \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot -0.6665536072 \]

       6. cube-mult 59.8%

          \[\leadsto x + \color{blue}{{x}^{3}} \cdot -0.6665536072 \]

   11. Simplified 59.8%

       \[\leadsto \color{blue}{x + {x}^{3} \cdot -0.6665536072} \]

   if 0.94999999999999996 < x

   1. Initial program 10.8%

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

   3. Taylor expanded in x around inf 99.6%

      \[\leadsto \color{blue}{\frac{0.5 + 0.2514179000665374 \cdot \frac{1}{{x}^{2}}}{x}} \]
   4. Step-by-step derivation

      1. associate-*r/ 99.6%

         \[\leadsto \frac{0.5 + \color{blue}{\frac{0.2514179000665374 \cdot 1}{{x}^{2}}}}{x} \]

      2. metadata-eval 99.6%

         \[\leadsto \frac{0.5 + \frac{\color{blue}{0.2514179000665374}}{{x}^{2}}}{x} \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{0.5 + \frac{0.2514179000665374}{{x}^{2}}}{x}} \]
   6. Step-by-step derivation

      1. div-inv 99.6%

         \[\leadsto \color{blue}{\left(0.5 + \frac{0.2514179000665374}{{x}^{2}}\right) \cdot \frac{1}{x}} \]

      2. +-commutative 99.6%

         \[\leadsto \color{blue}{\left(\frac{0.2514179000665374}{{x}^{2}} + 0.5\right)} \cdot \frac{1}{x} \]

      3. div-inv 99.6%

         \[\leadsto \left(\color{blue}{0.2514179000665374 \cdot \frac{1}{{x}^{2}}} + 0.5\right) \cdot \frac{1}{x} \]

      4. fma-define 99.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.2514179000665374, \frac{1}{{x}^{2}}, 0.5\right)} \cdot \frac{1}{x} \]

      5. pow-flip 99.6%

         \[\leadsto \mathsf{fma}\left(0.2514179000665374, \color{blue}{{x}^{\left(-2\right)}}, 0.5\right) \cdot \frac{1}{x} \]

      6. metadata-eval 99.6%

         \[\leadsto \mathsf{fma}\left(0.2514179000665374, {x}^{\color{blue}{-2}}, 0.5\right) \cdot \frac{1}{x} \]

   7. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.2514179000665374, {x}^{-2}, 0.5\right) \cdot \frac{1}{x}} \]
   8. Step-by-step derivation

      1. fma-undefine 99.6%

         \[\leadsto \color{blue}{\left(0.2514179000665374 \cdot {x}^{-2} + 0.5\right)} \cdot \frac{1}{x} \]

   9. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\left(0.2514179000665374 \cdot {x}^{-2} + 0.5\right)} \cdot \frac{1}{x} \]
   10. Step-by-step derivation

       1. *-commutative 99.6%

          \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(0.2514179000665374 \cdot {x}^{-2} + 0.5\right)} \]

       2. +-commutative 99.6%

          \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(0.5 + 0.2514179000665374 \cdot {x}^{-2}\right)} \]

       3. distribute-lft-in 99.6%

          \[\leadsto \color{blue}{\frac{1}{x} \cdot 0.5 + \frac{1}{x} \cdot \left(0.2514179000665374 \cdot {x}^{-2}\right)} \]

       4. *-commutative 99.6%

          \[\leadsto \color{blue}{0.5 \cdot \frac{1}{x}} + \frac{1}{x} \cdot \left(0.2514179000665374 \cdot {x}^{-2}\right) \]

       5. un-div-inv 99.6%

          \[\leadsto \color{blue}{\frac{0.5}{x}} + \frac{1}{x} \cdot \left(0.2514179000665374 \cdot {x}^{-2}\right) \]

       6. *-commutative 99.6%

          \[\leadsto \frac{0.5}{x} + \color{blue}{\left(0.2514179000665374 \cdot {x}^{-2}\right) \cdot \frac{1}{x}} \]

       7. associate-*l* 99.6%

          \[\leadsto \frac{0.5}{x} + \color{blue}{0.2514179000665374 \cdot \left({x}^{-2} \cdot \frac{1}{x}\right)} \]

       8. inv-pow 99.6%

          \[\leadsto \frac{0.5}{x} + 0.2514179000665374 \cdot \left({x}^{-2} \cdot \color{blue}{{x}^{-1}}\right) \]

       9. pow-prod-up 99.6%

          \[\leadsto \frac{0.5}{x} + 0.2514179000665374 \cdot \color{blue}{{x}^{\left(-2 + -1\right)}} \]

       10. metadata-eval 99.6%

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

   11. Applied egg-rr 99.6%

       \[\leadsto \color{blue}{\frac{0.5}{x} + 0.2514179000665374 \cdot {x}^{-3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.95:\\ \;\;\;\;x + {x}^{3} \cdot -0.6665536072\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} + 0.2514179000665374 \cdot {x}^{-3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.95:\\ \;\;\;\;x\_m + {x\_m}^{3} \cdot -0.6665536072\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.2514179000665374}{{x\_m}^{2}}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.95)
    (+ x_m (* (pow x_m 3.0) -0.6665536072))
    (/ (+ 0.5 (/ 0.2514179000665374 (pow x_m 2.0))) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.95) {
		tmp = x_m + (pow(x_m, 3.0) * -0.6665536072);
	} else {
		tmp = (0.5 + (0.2514179000665374 / pow(x_m, 2.0))) / x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.95d0) then
        tmp = x_m + ((x_m ** 3.0d0) * (-0.6665536072d0))
    else
        tmp = (0.5d0 + (0.2514179000665374d0 / (x_m ** 2.0d0))) / x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.95) {
		tmp = x_m + (Math.pow(x_m, 3.0) * -0.6665536072);
	} else {
		tmp = (0.5 + (0.2514179000665374 / Math.pow(x_m, 2.0))) / x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.95:
		tmp = x_m + (math.pow(x_m, 3.0) * -0.6665536072)
	else:
		tmp = (0.5 + (0.2514179000665374 / math.pow(x_m, 2.0))) / x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.95)
		tmp = Float64(x_m + Float64((x_m ^ 3.0) * -0.6665536072));
	else
		tmp = Float64(Float64(0.5 + Float64(0.2514179000665374 / (x_m ^ 2.0))) / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.95)
		tmp = x_m + ((x_m ^ 3.0) * -0.6665536072);
	else
		tmp = (0.5 + (0.2514179000665374 / (x_m ^ 2.0))) / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.95], N[(x$95$m + N[(N[Power[x$95$m, 3.0], $MachinePrecision] * -0.6665536072), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(0.2514179000665374 / N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.94999999999999996

   1. Initial program 62.1%

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

   3. Simplified 62.2%

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

   6. Applied egg-rr 62.2%

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

   8. Simplified 62.1%

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

   9. Taylor expanded in x around 0 59.8%

      \[\leadsto \color{blue}{x \cdot \left(1 + -0.6665536072 \cdot {x}^{2}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in 59.8%

          \[\leadsto \color{blue}{x \cdot 1 + x \cdot \left(-0.6665536072 \cdot {x}^{2}\right)} \]

       2. *-rgt-identity 59.8%

          \[\leadsto \color{blue}{x} + x \cdot \left(-0.6665536072 \cdot {x}^{2}\right) \]

       3. *-commutative 59.8%

          \[\leadsto x + x \cdot \color{blue}{\left({x}^{2} \cdot -0.6665536072\right)} \]

       4. associate-*r* 59.8%

          \[\leadsto x + \color{blue}{\left(x \cdot {x}^{2}\right) \cdot -0.6665536072} \]

       5. unpow2 59.8%

          \[\leadsto x + \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot -0.6665536072 \]

       6. cube-mult 59.8%

          \[\leadsto x + \color{blue}{{x}^{3}} \cdot -0.6665536072 \]

   11. Simplified 59.8%

       \[\leadsto \color{blue}{x + {x}^{3} \cdot -0.6665536072} \]

   if 0.94999999999999996 < x

   1. Initial program 10.8%

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

   3. Taylor expanded in x around inf 99.6%

      \[\leadsto \color{blue}{\frac{0.5 + 0.2514179000665374 \cdot \frac{1}{{x}^{2}}}{x}} \]
   4. Step-by-step derivation

      1. associate-*r/ 99.6%

         \[\leadsto \frac{0.5 + \color{blue}{\frac{0.2514179000665374 \cdot 1}{{x}^{2}}}}{x} \]

      2. metadata-eval 99.6%

         \[\leadsto \frac{0.5 + \frac{\color{blue}{0.2514179000665374}}{{x}^{2}}}{x} \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{0.5 + \frac{0.2514179000665374}{{x}^{2}}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.95:\\ \;\;\;\;x + {x}^{3} \cdot -0.6665536072\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.2514179000665374}{{x}^{2}}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.78:\\ \;\;\;\;x\_m + {x\_m}^{3} \cdot -0.6665536072\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 0.78) (+ x_m (* (pow x_m 3.0) -0.6665536072)) (/ 0.5 x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.78) {
		tmp = x_m + (pow(x_m, 3.0) * -0.6665536072);
	} else {
		tmp = 0.5 / x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.78d0) then
        tmp = x_m + ((x_m ** 3.0d0) * (-0.6665536072d0))
    else
        tmp = 0.5d0 / x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.78) {
		tmp = x_m + (Math.pow(x_m, 3.0) * -0.6665536072);
	} else {
		tmp = 0.5 / x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.78:
		tmp = x_m + (math.pow(x_m, 3.0) * -0.6665536072)
	else:
		tmp = 0.5 / x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.78)
		tmp = Float64(x_m + Float64((x_m ^ 3.0) * -0.6665536072));
	else
		tmp = Float64(0.5 / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.78)
		tmp = x_m + ((x_m ^ 3.0) * -0.6665536072);
	else
		tmp = 0.5 / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.78], N[(x$95$m + N[(N[Power[x$95$m, 3.0], $MachinePrecision] * -0.6665536072), $MachinePrecision]), $MachinePrecision], N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.78000000000000003

   1. Initial program 62.1%

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

   3. Simplified 62.2%

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

   6. Applied egg-rr 62.2%

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

   8. Simplified 62.1%

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

   9. Taylor expanded in x around 0 59.8%

      \[\leadsto \color{blue}{x \cdot \left(1 + -0.6665536072 \cdot {x}^{2}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in 59.8%

          \[\leadsto \color{blue}{x \cdot 1 + x \cdot \left(-0.6665536072 \cdot {x}^{2}\right)} \]

       2. *-rgt-identity 59.8%

          \[\leadsto \color{blue}{x} + x \cdot \left(-0.6665536072 \cdot {x}^{2}\right) \]

       3. *-commutative 59.8%

          \[\leadsto x + x \cdot \color{blue}{\left({x}^{2} \cdot -0.6665536072\right)} \]

       4. associate-*r* 59.8%

          \[\leadsto x + \color{blue}{\left(x \cdot {x}^{2}\right) \cdot -0.6665536072} \]

       5. unpow2 59.8%

          \[\leadsto x + \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot -0.6665536072 \]

       6. cube-mult 59.8%

          \[\leadsto x + \color{blue}{{x}^{3}} \cdot -0.6665536072 \]

   11. Simplified 59.8%

       \[\leadsto \color{blue}{x + {x}^{3} \cdot -0.6665536072} \]

   if 0.78000000000000003 < x

   1. Initial program 10.8%

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

   3. Taylor expanded in x around inf 98.6%

      \[\leadsto \color{blue}{\frac{0.5}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.78:\\ \;\;\;\;x + {x}^{3} \cdot -0.6665536072\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.0% accurate, 21.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.7:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 0.7) x_m (/ 0.5 x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.7) {
		tmp = x_m;
	} else {
		tmp = 0.5 / x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.7d0) then
        tmp = x_m
    else
        tmp = 0.5d0 / x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 0.7) {
		tmp = x_m;
	} else {
		tmp = 0.5 / x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 0.7:
		tmp = x_m
	else:
		tmp = 0.5 / x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 0.7)
		tmp = x_m;
	else
		tmp = Float64(0.5 / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 0.7)
		tmp = x_m;
	else
		tmp = 0.5 / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 0.7], x$95$m, N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.69999999999999996

   1. Initial program 62.1%

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

   3. Taylor expanded in x around 0 60.0%

      \[\leadsto \color{blue}{x} \]

   if 0.69999999999999996 < x

   1. Initial program 10.8%

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

   3. Taylor expanded in x around inf 98.6%

      \[\leadsto \color{blue}{\frac{0.5}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.9% accurate, 173.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot x\_m \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s x_m))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * x_m;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * x_m
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * x_m;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * x_m

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * x_m)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * x_m;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * x$95$m), $MachinePrecision]

Derivation

1. Initial program 51.1%

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

3. Taylor expanded in x around 0 48.0%

   \[\leadsto \color{blue}{x} \]

4. Final simplification 48.0%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024060 
(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))