Jmat.Real.dawson

Percentage Accurate: 54.1% → 100.0%

Time: 12.7s

Alternatives: 12

Speedup: 23.0×

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: 54.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 10000000:\\ \;\;\;\;\frac{x\_m}{\frac{\mathsf{fma}\left({x\_m}^{12}, 0.0003579942, \mathsf{fma}\left(\mathsf{fma}\left(0.45929036289366354, {x\_m}^{6}, 1\right), {\left(\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot 0.7715471019, \mathsf{fma}\left(x\_m \cdot x\_m, 0.7715471019, -1\right), 1\right)\right)}^{-1}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.0008327945, 0.0140005442\right), x\_m \cdot x\_m, 0.0694555761\right), x\_m \cdot x\_m, 0.2909738639\right) \cdot {x\_m}^{4}\right)\right)}{\mathsf{fma}\left(0.0001789971, {x\_m}^{10}, \mathsf{fma}\left(0.0005064034, {x\_m}^{8}, \mathsf{fma}\left(0.0072644182, {x\_m}^{6}, \mathsf{fma}\left(0.0424060604, {x\_m}^{4}, \mathsf{fma}\left(0.1049934947, x\_m \cdot x\_m, 1\right)\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x\_m}\\ \end{array} \end{array} \]

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 10000000.0)
    (/
     x_m
     (/
      (fma
       (pow x_m 12.0)
       0.0003579942
       (fma
        (fma 0.45929036289366354 (pow x_m 6.0) 1.0)
        (pow
         (fma
          (* (* x_m x_m) 0.7715471019)
          (fma (* x_m x_m) 0.7715471019 -1.0)
          1.0)
         -1.0)
        (*
         (fma
          (fma
           (fma (* x_m x_m) 0.0008327945 0.0140005442)
           (* x_m x_m)
           0.0694555761)
          (* x_m x_m)
          0.2909738639)
         (pow x_m 4.0))))
      (fma
       0.0001789971
       (pow x_m 10.0)
       (fma
        0.0005064034
        (pow x_m 8.0)
        (fma
         0.0072644182
         (pow x_m 6.0)
         (fma
          0.0424060604
          (pow x_m 4.0)
          (fma 0.1049934947 (* x_m x_m) 1.0)))))))
    (/ 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 <= 10000000.0) {
		tmp = x_m / (fma(pow(x_m, 12.0), 0.0003579942, fma(fma(0.45929036289366354, pow(x_m, 6.0), 1.0), pow(fma(((x_m * x_m) * 0.7715471019), fma((x_m * x_m), 0.7715471019, -1.0), 1.0), -1.0), (fma(fma(fma((x_m * x_m), 0.0008327945, 0.0140005442), (x_m * x_m), 0.0694555761), (x_m * x_m), 0.2909738639) * pow(x_m, 4.0)))) / fma(0.0001789971, pow(x_m, 10.0), fma(0.0005064034, pow(x_m, 8.0), fma(0.0072644182, pow(x_m, 6.0), fma(0.0424060604, pow(x_m, 4.0), fma(0.1049934947, (x_m * x_m), 1.0))))));
	} 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 <= 10000000.0)
		tmp = Float64(x_m / Float64(fma((x_m ^ 12.0), 0.0003579942, fma(fma(0.45929036289366354, (x_m ^ 6.0), 1.0), (fma(Float64(Float64(x_m * x_m) * 0.7715471019), fma(Float64(x_m * x_m), 0.7715471019, -1.0), 1.0) ^ -1.0), Float64(fma(fma(fma(Float64(x_m * x_m), 0.0008327945, 0.0140005442), Float64(x_m * x_m), 0.0694555761), Float64(x_m * x_m), 0.2909738639) * (x_m ^ 4.0)))) / fma(0.0001789971, (x_m ^ 10.0), fma(0.0005064034, (x_m ^ 8.0), fma(0.0072644182, (x_m ^ 6.0), fma(0.0424060604, (x_m ^ 4.0), fma(0.1049934947, Float64(x_m * x_m), 1.0)))))));
	else
		tmp = Float64(0.5 / 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, 10000000.0], N[(x$95$m / N[(N[(N[Power[x$95$m, 12.0], $MachinePrecision] * 0.0003579942 + N[(N[(0.45929036289366354 * N[Power[x$95$m, 6.0], $MachinePrecision] + 1.0), $MachinePrecision] * N[Power[N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.7715471019), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.7715471019 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision] + N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0008327945 + 0.0140005442), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.0694555761), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.2909738639), $MachinePrecision] * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.0001789971 * N[Power[x$95$m, 10.0], $MachinePrecision] + N[(0.0005064034 * N[Power[x$95$m, 8.0], $MachinePrecision] + N[(0.0072644182 * N[Power[x$95$m, 6.0], $MachinePrecision] + N[(0.0424060604 * N[Power[x$95$m, 4.0], $MachinePrecision] + N[(0.1049934947 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1e7

   1. Initial program 74.7%

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

   4. Applied rewrites 74.7%

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

   6. Applied rewrites 74.7%

      \[\leadsto \frac{x}{\frac{\mathsf{fma}\left({x}^{12}, 0.0003579942, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.45929036289366354, {x}^{6}, 1\right), \frac{1}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.7715471019, \mathsf{fma}\left(x \cdot x, 0.7715471019, -1\right), 1\right)}, \mathsf{fma}\left({x}^{6}, 0.0694555761, {x}^{4} \cdot 0.2909738639\right) + \mathsf{fma}\left({x}^{10}, 0.0008327945, {x}^{8} \cdot 0.0140005442\right)\right)}\right)}{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left(0.0072644182, {x}^{6}, \mathsf{fma}\left(0.0424060604, {x}^{4}, \mathsf{fma}\left(0.1049934947, x \cdot x, 1\right)\right)\right)\right)\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{x}{\frac{\mathsf{fma}\left({x}^{12}, \frac{1789971}{5000000000}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{459290362893663527321792099859}{1000000000000000000000000000000}, {x}^{6}, 1\right), \frac{1}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{7715471019}{10000000000}, \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000}, -1\right), 1\right)}, \color{blue}{{x}^{4} \cdot \left(\frac{2909738639}{10000000000} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + {x}^{2} \cdot \left(\frac{70002721}{5000000000} + \frac{1665589}{2000000000} \cdot {x}^{2}\right)\right)\right)}\right)\right)}{\mathsf{fma}\left(\frac{1789971}{10000000000}, {x}^{10}, \mathsf{fma}\left(\frac{2532017}{5000000000}, {x}^{8}, \mathsf{fma}\left(\frac{36322091}{5000000000}, {x}^{6}, \mathsf{fma}\left(\frac{106015151}{2500000000}, {x}^{4}, \mathsf{fma}\left(\frac{1049934947}{10000000000}, x \cdot x, 1\right)\right)\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x}{\frac{\mathsf{fma}\left({x}^{12}, \frac{1789971}{5000000000}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{459290362893663527321792099859}{1000000000000000000000000000000}, {x}^{6}, 1\right), \frac{1}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{7715471019}{10000000000}, \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000}, -1\right), 1\right)}, \color{blue}{\left(\frac{2909738639}{10000000000} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + {x}^{2} \cdot \left(\frac{70002721}{5000000000} + \frac{1665589}{2000000000} \cdot {x}^{2}\right)\right)\right) \cdot {x}^{4}}\right)\right)}{\mathsf{fma}\left(\frac{1789971}{10000000000}, {x}^{10}, \mathsf{fma}\left(\frac{2532017}{5000000000}, {x}^{8}, \mathsf{fma}\left(\frac{36322091}{5000000000}, {x}^{6}, \mathsf{fma}\left(\frac{106015151}{2500000000}, {x}^{4}, \mathsf{fma}\left(\frac{1049934947}{10000000000}, x \cdot x, 1\right)\right)\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x}{\frac{\mathsf{fma}\left({x}^{12}, \frac{1789971}{5000000000}, \mathsf{fma}\left(\mathsf{fma}\left(\frac{459290362893663527321792099859}{1000000000000000000000000000000}, {x}^{6}, 1\right), \frac{1}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{7715471019}{10000000000}, \mathsf{fma}\left(x \cdot x, \frac{7715471019}{10000000000}, -1\right), 1\right)}, \color{blue}{\left(\frac{2909738639}{10000000000} + {x}^{2} \cdot \left(\frac{694555761}{10000000000} + {x}^{2} \cdot \left(\frac{70002721}{5000000000} + \frac{1665589}{2000000000} \cdot {x}^{2}\right)\right)\right) \cdot {x}^{4}}\right)\right)}{\mathsf{fma}\left(\frac{1789971}{10000000000}, {x}^{10}, \mathsf{fma}\left(\frac{2532017}{5000000000}, {x}^{8}, \mathsf{fma}\left(\frac{36322091}{5000000000}, {x}^{6}, \mathsf{fma}\left(\frac{106015151}{2500000000}, {x}^{4}, \mathsf{fma}\left(\frac{1049934947}{10000000000}, x \cdot x, 1\right)\right)\right)\right)\right)}} \]

   9. Applied rewrites 74.7%

      \[\leadsto \frac{x}{\frac{\mathsf{fma}\left({x}^{12}, 0.0003579942, \mathsf{fma}\left(\mathsf{fma}\left(0.45929036289366354, {x}^{6}, 1\right), \frac{1}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.7715471019, \mathsf{fma}\left(x \cdot x, 0.7715471019, -1\right), 1\right)}, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.0008327945, 0.0140005442\right), x \cdot x, 0.0694555761\right), x \cdot x, 0.2909738639\right) \cdot {x}^{4}}\right)\right)}{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left(0.0072644182, {x}^{6}, \mathsf{fma}\left(0.0424060604, {x}^{4}, \mathsf{fma}\left(0.1049934947, x \cdot x, 1\right)\right)\right)\right)\right)}} \]

   if 1e7 < x

   1. Initial program 5.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 100.0

         \[\leadsto \color{blue}{\frac{0.5}{x}} \]

   5. Applied rewrites 100.0%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10000000:\\ \;\;\;\;\frac{x}{\frac{\mathsf{fma}\left({x}^{12}, 0.0003579942, \mathsf{fma}\left(\mathsf{fma}\left(0.45929036289366354, {x}^{6}, 1\right), {\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.7715471019, \mathsf{fma}\left(x \cdot x, 0.7715471019, -1\right), 1\right)\right)}^{-1}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.0008327945, 0.0140005442\right), x \cdot x, 0.0694555761\right), x \cdot x, 0.2909738639\right) \cdot {x}^{4}\right)\right)}{\mathsf{fma}\left(0.0001789971, {x}^{10}, \mathsf{fma}\left(0.0005064034, {x}^{8}, \mathsf{fma}\left(0.0072644182, {x}^{6}, \mathsf{fma}\left(0.0424060604, {x}^{4}, \mathsf{fma}\left(0.1049934947, x \cdot x, 1\right)\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \left(x\_m \cdot x\_m\right) \cdot \left(x\_m \cdot x\_m\right)\\ t_1 := t\_0 \cdot \left(x\_m \cdot x\_m\right)\\ t_2 := t\_1 \cdot \left(x\_m \cdot x\_m\right)\\ t_3 := t\_2 \cdot \left(x\_m \cdot x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 10000000:\\ \;\;\;\;\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x\_m \cdot x\_m\right)\right) + 0.0424060604 \cdot t\_0\right) + 0.0072644182 \cdot t\_1\right) + 0.0005064034 \cdot t\_2\right) + 0.0001789971 \cdot t\_3}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x\_m \cdot x\_m\right)\right) + 0.2909738639 \cdot t\_0\right) + 0.0694555761 \cdot t\_1\right) + 0.0140005442 \cdot t\_2\right) + 0.0008327945 \cdot t\_3\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(t\_3 \cdot \left(x\_m \cdot x\_m\right)\right)} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x\_m}\\ \end{array} \end{array} \end{array} \]

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 (* t_0 (* x_m x_m)))
        (t_2 (* t_1 (* x_m x_m)))
        (t_3 (* t_2 (* x_m x_m))))
   (*
    x_s
    (if (<= x_m 10000000.0)
      (*
       (/
        (+
         (+
          (+
           (+ (+ 1.0 (* 0.1049934947 (* x_m x_m))) (* 0.0424060604 t_0))
           (* 0.0072644182 t_1))
          (* 0.0005064034 t_2))
         (* 0.0001789971 t_3))
        (+
         (+
          (+
           (+
            (+ (+ 1.0 (* 0.7715471019 (* x_m x_m))) (* 0.2909738639 t_0))
            (* 0.0694555761 t_1))
           (* 0.0140005442 t_2))
          (* 0.0008327945 t_3))
         (* (* 2.0 0.0001789971) (* t_3 (* x_m x_m)))))
       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 t_0 = (x_m * x_m) * (x_m * x_m);
	double t_1 = t_0 * (x_m * x_m);
	double t_2 = t_1 * (x_m * x_m);
	double t_3 = t_2 * (x_m * x_m);
	double tmp;
	if (x_m <= 10000000.0) {
		tmp = ((((((1.0 + (0.1049934947 * (x_m * x_m))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + (0.7715471019 * (x_m * x_m))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + ((2.0 * 0.0001789971) * (t_3 * (x_m * x_m))))) * 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) :: 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 = t_0 * (x_m * x_m)
    t_2 = t_1 * (x_m * x_m)
    t_3 = t_2 * (x_m * x_m)
    if (x_m <= 10000000.0d0) then
        tmp = ((((((1.0d0 + (0.1049934947d0 * (x_m * x_m))) + (0.0424060604d0 * t_0)) + (0.0072644182d0 * t_1)) + (0.0005064034d0 * t_2)) + (0.0001789971d0 * t_3)) / ((((((1.0d0 + (0.7715471019d0 * (x_m * x_m))) + (0.2909738639d0 * t_0)) + (0.0694555761d0 * t_1)) + (0.0140005442d0 * t_2)) + (0.0008327945d0 * t_3)) + ((2.0d0 * 0.0001789971d0) * (t_3 * (x_m * x_m))))) * 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 t_0 = (x_m * x_m) * (x_m * x_m);
	double t_1 = t_0 * (x_m * x_m);
	double t_2 = t_1 * (x_m * x_m);
	double t_3 = t_2 * (x_m * x_m);
	double tmp;
	if (x_m <= 10000000.0) {
		tmp = ((((((1.0 + (0.1049934947 * (x_m * x_m))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + (0.7715471019 * (x_m * x_m))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + ((2.0 * 0.0001789971) * (t_3 * (x_m * x_m))))) * 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):
	t_0 = (x_m * x_m) * (x_m * x_m)
	t_1 = t_0 * (x_m * x_m)
	t_2 = t_1 * (x_m * x_m)
	t_3 = t_2 * (x_m * x_m)
	tmp = 0
	if x_m <= 10000000.0:
		tmp = ((((((1.0 + (0.1049934947 * (x_m * x_m))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + (0.7715471019 * (x_m * x_m))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + ((2.0 * 0.0001789971) * (t_3 * (x_m * x_m))))) * 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)
	t_0 = Float64(Float64(x_m * x_m) * Float64(x_m * x_m))
	t_1 = Float64(t_0 * Float64(x_m * x_m))
	t_2 = Float64(t_1 * Float64(x_m * x_m))
	t_3 = Float64(t_2 * Float64(x_m * x_m))
	tmp = 0.0
	if (x_m <= 10000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.1049934947 * Float64(x_m * x_m))) + 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_m * x_m))) + 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_m * x_m))))) * 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)
	t_0 = (x_m * x_m) * (x_m * x_m);
	t_1 = t_0 * (x_m * x_m);
	t_2 = t_1 * (x_m * x_m);
	t_3 = t_2 * (x_m * x_m);
	tmp = 0.0;
	if (x_m <= 10000000.0)
		tmp = ((((((1.0 + (0.1049934947 * (x_m * x_m))) + (0.0424060604 * t_0)) + (0.0072644182 * t_1)) + (0.0005064034 * t_2)) + (0.0001789971 * t_3)) / ((((((1.0 + (0.7715471019 * (x_m * x_m))) + (0.2909738639 * t_0)) + (0.0694555761 * t_1)) + (0.0140005442 * t_2)) + (0.0008327945 * t_3)) + ((2.0 * 0.0001789971) * (t_3 * (x_m * x_m))))) * 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_] := Block[{t$95$0 = N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 10000000.0], N[(N[(N[(N[(N[(N[(N[(1.0 + N[(0.1049934947 * N[(x$95$m * x$95$m), $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$95$m * x$95$m), $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$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1e7

   1. Initial program 74.7%

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

   if 1e7 < x

   1. Initial program 5.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 100.0

         \[\leadsto \color{blue}{\frac{0.5}{x}} \]

   5. Applied rewrites 100.0%

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

Alternative 3: 99.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.1:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.015175085973910875, x\_m \cdot x\_m, 0.17858401087518092\right), x\_m \cdot x\_m, 0.6665536072\right), x\_m \cdot x\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{11.259630434457211}{x\_m \cdot x\_m} + 0.15298196345929074}{{x\_m}^{5}} - \frac{\frac{-0.2514179000665374}{x\_m \cdot x\_m}}{x\_m}\right) - \frac{-0.5}{x\_m}\\ \end{array} \end{array} \]

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 2.1)
    (/
     x_m
     (fma
      (fma
       (fma 0.015175085973910875 (* x_m x_m) 0.17858401087518092)
       (* x_m x_m)
       0.6665536072)
      (* x_m x_m)
      1.0))
    (-
     (-
      (/
       (+ (/ 11.259630434457211 (* x_m x_m)) 0.15298196345929074)
       (pow x_m 5.0))
      (/ (/ -0.2514179000665374 (* x_m x_m)) 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 <= 2.1) {
		tmp = x_m / fma(fma(fma(0.015175085973910875, (x_m * x_m), 0.17858401087518092), (x_m * x_m), 0.6665536072), (x_m * x_m), 1.0);
	} else {
		tmp = ((((11.259630434457211 / (x_m * x_m)) + 0.15298196345929074) / pow(x_m, 5.0)) - ((-0.2514179000665374 / (x_m * x_m)) / x_m)) - (-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 <= 2.1)
		tmp = Float64(x_m / fma(fma(fma(0.015175085973910875, Float64(x_m * x_m), 0.17858401087518092), Float64(x_m * x_m), 0.6665536072), Float64(x_m * x_m), 1.0));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(11.259630434457211 / Float64(x_m * x_m)) + 0.15298196345929074) / (x_m ^ 5.0)) - Float64(Float64(-0.2514179000665374 / Float64(x_m * x_m)) / x_m)) - Float64(-0.5 / 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, 2.1], N[(x$95$m / N[(N[(N[(0.015175085973910875 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.17858401087518092), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.6665536072), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(11.259630434457211 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] + 0.15298196345929074), $MachinePrecision] / N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision] - N[(N[(-0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision] - N[(-0.5 / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.10000000000000009

   1. Initial program 74.7%

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

   4. Applied rewrites 74.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{x}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{{x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), {x}^{2}, 1\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) + \frac{833192009}{1250000000}}, {x}^{2}, 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{\left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{833192009}{1250000000}, {x}^{2}, 1\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}, {x}^{2}, \frac{833192009}{1250000000}\right)}, {x}^{2}, 1\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2} + \frac{2232300135939761477}{12500000000000000000}}, {x}^{2}, \frac{833192009}{1250000000}\right), {x}^{2}, 1\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1896885746738859378453363281}{125000000000000000000000000000}, {x}^{2}, \frac{2232300135939761477}{12500000000000000000}\right)}, {x}^{2}, \frac{833192009}{1250000000}\right), {x}^{2}, 1\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1896885746738859378453363281}{125000000000000000000000000000}, \color{blue}{x \cdot x}, \frac{2232300135939761477}{12500000000000000000}\right), {x}^{2}, \frac{833192009}{1250000000}\right), {x}^{2}, 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1896885746738859378453363281}{125000000000000000000000000000}, \color{blue}{x \cdot x}, \frac{2232300135939761477}{12500000000000000000}\right), {x}^{2}, \frac{833192009}{1250000000}\right), {x}^{2}, 1\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1896885746738859378453363281}{125000000000000000000000000000}, x \cdot x, \frac{2232300135939761477}{12500000000000000000}\right), \color{blue}{x \cdot x}, \frac{833192009}{1250000000}\right), {x}^{2}, 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1896885746738859378453363281}{125000000000000000000000000000}, x \cdot x, \frac{2232300135939761477}{12500000000000000000}\right), \color{blue}{x \cdot x}, \frac{833192009}{1250000000}\right), {x}^{2}, 1\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1896885746738859378453363281}{125000000000000000000000000000}, x \cdot x, \frac{2232300135939761477}{12500000000000000000}\right), x \cdot x, \frac{833192009}{1250000000}\right), \color{blue}{x \cdot x}, 1\right)} \]

      14. lower-*.f64 72.6

          \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.015175085973910875, x \cdot x, 0.17858401087518092\right), x \cdot x, 0.6665536072\right), \color{blue}{x \cdot x}, 1\right)} \]

   7. Applied rewrites 72.6%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.015175085973910875, x \cdot x, 0.17858401087518092\right), x \cdot x, 0.6665536072\right), x \cdot x, 1\right)}} \]

   if 2.10000000000000009 < x

   1. Initial program 5.3%

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

   4. Applied rewrites 5.4%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{1307076337763}{8543989815576} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)}{x}} \]

   7. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(\frac{\frac{11.259630434457211}{x \cdot x} + 0.15298196345929074}{{x}^{5}} - \frac{-0.2514179000665374}{{x}^{3}}\right) - \frac{-0.5}{x}} \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

      \[\leadsto \left(\frac{\frac{11.259630434457211}{x \cdot x} + 0.15298196345929074}{{x}^{5}} - \frac{\frac{-0.2514179000665374}{x \cdot x}}{x}\right) - \frac{-0.5}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.1:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.015175085973910875, x\_m \cdot x\_m, 0.17858401087518092\right), x\_m \cdot x\_m, 0.6665536072\right), x\_m \cdot x\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.15298196345929074 - \frac{-11.259630434457211}{x\_m \cdot x\_m}}{{x\_m}^{4}} - \left(-0.5 - \frac{0.2514179000665374}{x\_m \cdot x\_m}\right)}{x\_m}\\ \end{array} \end{array} \]

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 2.1)
    (/
     x_m
     (fma
      (fma
       (fma 0.015175085973910875 (* x_m x_m) 0.17858401087518092)
       (* x_m x_m)
       0.6665536072)
      (* x_m x_m)
      1.0))
    (/
     (-
      (/
       (- 0.15298196345929074 (/ -11.259630434457211 (* x_m x_m)))
       (pow x_m 4.0))
      (- -0.5 (/ 0.2514179000665374 (* x_m x_m))))
     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 <= 2.1) {
		tmp = x_m / fma(fma(fma(0.015175085973910875, (x_m * x_m), 0.17858401087518092), (x_m * x_m), 0.6665536072), (x_m * x_m), 1.0);
	} else {
		tmp = (((0.15298196345929074 - (-11.259630434457211 / (x_m * x_m))) / pow(x_m, 4.0)) - (-0.5 - (0.2514179000665374 / (x_m * x_m)))) / 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 <= 2.1)
		tmp = Float64(x_m / fma(fma(fma(0.015175085973910875, Float64(x_m * x_m), 0.17858401087518092), Float64(x_m * x_m), 0.6665536072), Float64(x_m * x_m), 1.0));
	else
		tmp = Float64(Float64(Float64(Float64(0.15298196345929074 - Float64(-11.259630434457211 / Float64(x_m * x_m))) / (x_m ^ 4.0)) - Float64(-0.5 - Float64(0.2514179000665374 / Float64(x_m * x_m)))) / 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, 2.1], N[(x$95$m / N[(N[(N[(0.015175085973910875 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.17858401087518092), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.6665536072), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.15298196345929074 - N[(-11.259630434457211 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] - N[(-0.5 - N[(0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.10000000000000009

   1. Initial program 74.7%

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

   4. Applied rewrites 74.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{x}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{{x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), {x}^{2}, 1\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) + \frac{833192009}{1250000000}}, {x}^{2}, 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{\left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{833192009}{1250000000}, {x}^{2}, 1\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}, {x}^{2}, \frac{833192009}{1250000000}\right)}, {x}^{2}, 1\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2} + \frac{2232300135939761477}{12500000000000000000}}, {x}^{2}, \frac{833192009}{1250000000}\right), {x}^{2}, 1\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1896885746738859378453363281}{125000000000000000000000000000}, {x}^{2}, \frac{2232300135939761477}{12500000000000000000}\right)}, {x}^{2}, \frac{833192009}{1250000000}\right), {x}^{2}, 1\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1896885746738859378453363281}{125000000000000000000000000000}, \color{blue}{x \cdot x}, \frac{2232300135939761477}{12500000000000000000}\right), {x}^{2}, \frac{833192009}{1250000000}\right), {x}^{2}, 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1896885746738859378453363281}{125000000000000000000000000000}, \color{blue}{x \cdot x}, \frac{2232300135939761477}{12500000000000000000}\right), {x}^{2}, \frac{833192009}{1250000000}\right), {x}^{2}, 1\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1896885746738859378453363281}{125000000000000000000000000000}, x \cdot x, \frac{2232300135939761477}{12500000000000000000}\right), \color{blue}{x \cdot x}, \frac{833192009}{1250000000}\right), {x}^{2}, 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1896885746738859378453363281}{125000000000000000000000000000}, x \cdot x, \frac{2232300135939761477}{12500000000000000000}\right), \color{blue}{x \cdot x}, \frac{833192009}{1250000000}\right), {x}^{2}, 1\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1896885746738859378453363281}{125000000000000000000000000000}, x \cdot x, \frac{2232300135939761477}{12500000000000000000}\right), x \cdot x, \frac{833192009}{1250000000}\right), \color{blue}{x \cdot x}, 1\right)} \]

      14. lower-*.f64 72.6

          \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.015175085973910875, x \cdot x, 0.17858401087518092\right), x \cdot x, 0.6665536072\right), \color{blue}{x \cdot x}, 1\right)} \]

   7. Applied rewrites 72.6%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.015175085973910875, x \cdot x, 0.17858401087518092\right), x \cdot x, 0.6665536072\right), x \cdot x, 1\right)}} \]

   if 2.10000000000000009 < x

   1. Initial program 5.3%

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

   4. Applied rewrites 5.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{x}{\color{blue}{1 + \frac{833192009}{1250000000} \cdot {x}^{2}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{833192009}{1250000000} \cdot {x}^{2} + 1}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(\frac{833192009}{1250000000}, {x}^{2}, 1\right)}} \]

      3. unpow2 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\frac{833192009}{1250000000}, \color{blue}{x \cdot x}, 1\right)} \]

      4. lower-*.f64 12.5

         \[\leadsto \frac{x}{\mathsf{fma}\left(0.6665536072, \color{blue}{x \cdot x}, 1\right)} \]

   7. Applied rewrites 12.5%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(0.6665536072, x \cdot x, 1\right)}} \]

   8. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{1307076337763}{8543989815576} + \frac{344398180852034095277}{30586987988352776592} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}\right)}{x}} \]

   10. Applied rewrites 100.0%

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

Alternative 5: 99.7% accurate, 8.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.85:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.015175085973910875, x\_m \cdot x\_m, 0.17858401087518092\right), x\_m \cdot x\_m, 0.6665536072\right), x\_m \cdot x\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.2514179000665374}{x\_m \cdot x\_m} - -0.5}{x\_m}\\ \end{array} \end{array} \]

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 1.85)
    (/
     x_m
     (fma
      (fma
       (fma 0.015175085973910875 (* x_m x_m) 0.17858401087518092)
       (* x_m x_m)
       0.6665536072)
      (* x_m x_m)
      1.0))
    (/ (- (/ 0.2514179000665374 (* x_m 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 <= 1.85) {
		tmp = x_m / fma(fma(fma(0.015175085973910875, (x_m * x_m), 0.17858401087518092), (x_m * x_m), 0.6665536072), (x_m * x_m), 1.0);
	} else {
		tmp = ((0.2514179000665374 / (x_m * x_m)) - -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 <= 1.85)
		tmp = Float64(x_m / fma(fma(fma(0.015175085973910875, Float64(x_m * x_m), 0.17858401087518092), Float64(x_m * x_m), 0.6665536072), Float64(x_m * x_m), 1.0));
	else
		tmp = Float64(Float64(Float64(0.2514179000665374 / Float64(x_m * x_m)) - -0.5) / 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, 1.85], N[(x$95$m / N[(N[(N[(0.015175085973910875 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.17858401087518092), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.6665536072), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 74.7%

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

   4. Applied rewrites 74.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{x}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{{x}^{2} \cdot \left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(\frac{833192009}{1250000000} + {x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right), {x}^{2}, 1\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) + \frac{833192009}{1250000000}}, {x}^{2}, 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{\left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{833192009}{1250000000}, {x}^{2}, 1\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{2232300135939761477}{12500000000000000000} + \frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2}, {x}^{2}, \frac{833192009}{1250000000}\right)}, {x}^{2}, 1\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1896885746738859378453363281}{125000000000000000000000000000} \cdot {x}^{2} + \frac{2232300135939761477}{12500000000000000000}}, {x}^{2}, \frac{833192009}{1250000000}\right), {x}^{2}, 1\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1896885746738859378453363281}{125000000000000000000000000000}, {x}^{2}, \frac{2232300135939761477}{12500000000000000000}\right)}, {x}^{2}, \frac{833192009}{1250000000}\right), {x}^{2}, 1\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1896885746738859378453363281}{125000000000000000000000000000}, \color{blue}{x \cdot x}, \frac{2232300135939761477}{12500000000000000000}\right), {x}^{2}, \frac{833192009}{1250000000}\right), {x}^{2}, 1\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1896885746738859378453363281}{125000000000000000000000000000}, \color{blue}{x \cdot x}, \frac{2232300135939761477}{12500000000000000000}\right), {x}^{2}, \frac{833192009}{1250000000}\right), {x}^{2}, 1\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1896885746738859378453363281}{125000000000000000000000000000}, x \cdot x, \frac{2232300135939761477}{12500000000000000000}\right), \color{blue}{x \cdot x}, \frac{833192009}{1250000000}\right), {x}^{2}, 1\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1896885746738859378453363281}{125000000000000000000000000000}, x \cdot x, \frac{2232300135939761477}{12500000000000000000}\right), \color{blue}{x \cdot x}, \frac{833192009}{1250000000}\right), {x}^{2}, 1\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1896885746738859378453363281}{125000000000000000000000000000}, x \cdot x, \frac{2232300135939761477}{12500000000000000000}\right), x \cdot x, \frac{833192009}{1250000000}\right), \color{blue}{x \cdot x}, 1\right)} \]

      14. lower-*.f64 72.6

          \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.015175085973910875, x \cdot x, 0.17858401087518092\right), x \cdot x, 0.6665536072\right), \color{blue}{x \cdot x}, 1\right)} \]

   7. Applied rewrites 72.6%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.015175085973910875, x \cdot x, 0.17858401087518092\right), x \cdot x, 0.6665536072\right), x \cdot x, 1\right)}} \]

   if 1.8500000000000001 < x

   1. Initial program 5.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}} + \frac{1}{2}}}{x} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}}{x} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)}{x} \]

      5. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{x} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{x} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{600041}{2386628} \cdot 1}{{x}^{2}}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{600041}{2386628}}}{{x}^{2}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{600041}{2386628}}{{x}^{2}}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      10. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{600041}{2386628}}{\color{blue}{x \cdot x}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{600041}{2386628}}{\color{blue}{x \cdot x}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      12. metadata-eval 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 6: 99.7% accurate, 9.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.15:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0732490286039007, x\_m \cdot x\_m, 0.265709700396151\right), x\_m \cdot x\_m, -0.6665536072\right), x\_m \cdot x\_m, 1\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.2514179000665374}{x\_m \cdot x\_m} - -0.5}{x\_m}\\ \end{array} \end{array} \]

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 1.15)
    (*
     (fma
      (fma
       (fma -0.0732490286039007 (* x_m x_m) 0.265709700396151)
       (* x_m x_m)
       -0.6665536072)
      (* x_m x_m)
      1.0)
     x_m)
    (/ (- (/ 0.2514179000665374 (* x_m 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 <= 1.15) {
		tmp = fma(fma(fma(-0.0732490286039007, (x_m * x_m), 0.265709700396151), (x_m * x_m), -0.6665536072), (x_m * x_m), 1.0) * x_m;
	} else {
		tmp = ((0.2514179000665374 / (x_m * x_m)) - -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 <= 1.15)
		tmp = Float64(fma(fma(fma(-0.0732490286039007, Float64(x_m * x_m), 0.265709700396151), Float64(x_m * x_m), -0.6665536072), Float64(x_m * x_m), 1.0) * x_m);
	else
		tmp = Float64(Float64(Float64(0.2514179000665374 / Float64(x_m * x_m)) - -0.5) / 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, 1.15], N[(N[(N[(N[(-0.0732490286039007 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.265709700396151), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -0.6665536072), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.1499999999999999

   1. Initial program 74.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} + \frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2}\right) - \frac{833192009}{1250000000}\right)\right)} \cdot x \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} + \frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2}\right) - \frac{833192009}{1250000000}\right) + 1\right)} \cdot x \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} + \frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2}\right) - \frac{833192009}{1250000000}\right) \cdot {x}^{2}} + 1\right) \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} + \frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2}\right) - \frac{833192009}{1250000000}, {x}^{2}, 1\right)} \cdot x \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} + \frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right)}, {x}^{2}, 1\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{3321371254951887171}{12500000000000000000} + \frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right), {x}^{2}, 1\right) \cdot x \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{3321371254951887171}{12500000000000000000} + \frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{\frac{-833192009}{1250000000}}, {x}^{2}, 1\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000} + \frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2}, {x}^{2}, \frac{-833192009}{1250000000}\right)}, {x}^{2}, 1\right) \cdot x \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-9156128575487588197208397249}{125000000000000000000000000000} \cdot {x}^{2} + \frac{3321371254951887171}{12500000000000000000}}, {x}^{2}, \frac{-833192009}{1250000000}\right), {x}^{2}, 1\right) \cdot x \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, {x}^{2}, \frac{3321371254951887171}{12500000000000000000}\right)}, {x}^{2}, \frac{-833192009}{1250000000}\right), {x}^{2}, 1\right) \cdot x \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, \color{blue}{x \cdot x}, \frac{3321371254951887171}{12500000000000000000}\right), {x}^{2}, \frac{-833192009}{1250000000}\right), {x}^{2}, 1\right) \cdot x \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, \color{blue}{x \cdot x}, \frac{3321371254951887171}{12500000000000000000}\right), {x}^{2}, \frac{-833192009}{1250000000}\right), {x}^{2}, 1\right) \cdot x \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), \color{blue}{x \cdot x}, \frac{-833192009}{1250000000}\right), {x}^{2}, 1\right) \cdot x \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), \color{blue}{x \cdot x}, \frac{-833192009}{1250000000}\right), {x}^{2}, 1\right) \cdot x \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-9156128575487588197208397249}{125000000000000000000000000000}, x \cdot x, \frac{3321371254951887171}{12500000000000000000}\right), x \cdot x, \frac{-833192009}{1250000000}\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

      15. lower-*.f64 71.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0732490286039007, x \cdot x, 0.265709700396151\right), x \cdot x, -0.6665536072\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

   5. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0732490286039007, x \cdot x, 0.265709700396151\right), x \cdot x, -0.6665536072\right), x \cdot x, 1\right)} \cdot x \]

   if 1.1499999999999999 < x

   1. Initial program 5.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}} + \frac{1}{2}}}{x} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}}{x} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)}{x} \]

      5. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{x} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{x} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{600041}{2386628} \cdot 1}{{x}^{2}}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{600041}{2386628}}}{{x}^{2}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{600041}{2386628}}{{x}^{2}}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      10. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{600041}{2386628}}{\color{blue}{x \cdot x}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{600041}{2386628}}{\color{blue}{x \cdot x}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      12. metadata-eval 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 7: 99.6% accurate, 10.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.4:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(\mathsf{fma}\left(0.17858401087518092, x\_m \cdot x\_m, 0.6665536072\right), x\_m \cdot x\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.2514179000665374}{x\_m \cdot x\_m} - -0.5}{x\_m}\\ \end{array} \end{array} \]

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 2.4)
    (/
     x_m
     (fma (fma 0.17858401087518092 (* x_m x_m) 0.6665536072) (* x_m x_m) 1.0))
    (/ (- (/ 0.2514179000665374 (* x_m 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 <= 2.4) {
		tmp = x_m / fma(fma(0.17858401087518092, (x_m * x_m), 0.6665536072), (x_m * x_m), 1.0);
	} else {
		tmp = ((0.2514179000665374 / (x_m * x_m)) - -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 <= 2.4)
		tmp = Float64(x_m / fma(fma(0.17858401087518092, Float64(x_m * x_m), 0.6665536072), Float64(x_m * x_m), 1.0));
	else
		tmp = Float64(Float64(Float64(0.2514179000665374 / Float64(x_m * x_m)) - -0.5) / 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, 2.4], N[(x$95$m / N[(N[(0.17858401087518092 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.6665536072), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   1. Initial program 74.7%

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

   4. Applied rewrites 74.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{x}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{{x}^{2} \cdot \left(\frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(\frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(\frac{833192009}{1250000000} + \frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2}, {x}^{2}, 1\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{\frac{2232300135939761477}{12500000000000000000} \cdot {x}^{2} + \frac{833192009}{1250000000}}, {x}^{2}, 1\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{2232300135939761477}{12500000000000000000}, {x}^{2}, \frac{833192009}{1250000000}\right)}, {x}^{2}, 1\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2232300135939761477}{12500000000000000000}, \color{blue}{x \cdot x}, \frac{833192009}{1250000000}\right), {x}^{2}, 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2232300135939761477}{12500000000000000000}, \color{blue}{x \cdot x}, \frac{833192009}{1250000000}\right), {x}^{2}, 1\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2232300135939761477}{12500000000000000000}, x \cdot x, \frac{833192009}{1250000000}\right), \color{blue}{x \cdot x}, 1\right)} \]

      9. lower-*.f64 72.6

         \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(0.17858401087518092, x \cdot x, 0.6665536072\right), \color{blue}{x \cdot x}, 1\right)} \]

   7. Applied rewrites 72.6%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.17858401087518092, x \cdot x, 0.6665536072\right), x \cdot x, 1\right)}} \]

   if 2.39999999999999991 < x

   1. Initial program 5.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}} + \frac{1}{2}}}{x} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}}{x} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)}{x} \]

      5. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{x} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{x} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{600041}{2386628} \cdot 1}{{x}^{2}}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{600041}{2386628}}}{{x}^{2}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{600041}{2386628}}{{x}^{2}}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      10. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{600041}{2386628}}{\color{blue}{x \cdot x}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{600041}{2386628}}{\color{blue}{x \cdot x}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      12. metadata-eval 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 8: 99.6% accurate, 11.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.265709700396151, x\_m \cdot x\_m, -0.6665536072\right), x\_m \cdot x\_m, 1\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.2514179000665374}{x\_m \cdot x\_m} - -0.5}{x\_m}\\ \end{array} \end{array} \]

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 1.1)
    (*
     (fma (fma 0.265709700396151 (* x_m x_m) -0.6665536072) (* x_m x_m) 1.0)
     x_m)
    (/ (- (/ 0.2514179000665374 (* x_m 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 <= 1.1) {
		tmp = fma(fma(0.265709700396151, (x_m * x_m), -0.6665536072), (x_m * x_m), 1.0) * x_m;
	} else {
		tmp = ((0.2514179000665374 / (x_m * x_m)) - -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 <= 1.1)
		tmp = Float64(fma(fma(0.265709700396151, Float64(x_m * x_m), -0.6665536072), Float64(x_m * x_m), 1.0) * x_m);
	else
		tmp = Float64(Float64(Float64(0.2514179000665374 / Float64(x_m * x_m)) - -0.5) / 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, 1.1], N[(N[(N[(0.265709700396151 * N[(x$95$m * x$95$m), $MachinePrecision] + -0.6665536072), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(0.2514179000665374 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 74.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)\right)} \cdot x \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right) + 1\right)} \cdot x \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right) \cdot {x}^{2}} + 1\right) \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}, {x}^{2}, 1\right)} \cdot x \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right)}, {x}^{2}, 1\right) \cdot x \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} + \color{blue}{\frac{-833192009}{1250000000}}, {x}^{2}, 1\right) \cdot x \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000}, {x}^{2}, \frac{-833192009}{1250000000}\right)}, {x}^{2}, 1\right) \cdot x \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000}, \color{blue}{x \cdot x}, \frac{-833192009}{1250000000}\right), {x}^{2}, 1\right) \cdot x \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000}, \color{blue}{x \cdot x}, \frac{-833192009}{1250000000}\right), {x}^{2}, 1\right) \cdot x \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000}, x \cdot x, \frac{-833192009}{1250000000}\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

      10. lower-*.f64 71.7

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.265709700396151, x \cdot x, -0.6665536072\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

   5. Applied rewrites 71.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.265709700396151, x \cdot x, -0.6665536072\right), x \cdot x, 1\right)} \cdot x \]

   if 1.1000000000000001 < x

   1. Initial program 5.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} + \frac{600041}{2386628} \cdot \frac{1}{{x}^{2}}}{x}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}} + \frac{1}{2}}}{x} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}}{x} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)}{x} \]

      5. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{x} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{600041}{2386628} \cdot \frac{1}{{x}^{2}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{x} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{600041}{2386628} \cdot 1}{{x}^{2}}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{600041}{2386628}}}{{x}^{2}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{600041}{2386628}}{{x}^{2}}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      10. unpow2 N/A

          \[\leadsto \frac{\frac{\frac{600041}{2386628}}{\color{blue}{x \cdot x}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{600041}{2386628}}{\color{blue}{x \cdot x}} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x} \]

      12. metadata-eval 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 9: 99.4% accurate, 12.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 0.88:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.265709700396151, x\_m \cdot x\_m, -0.6665536072\right), x\_m \cdot x\_m, 1\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x\_m}\\ \end{array} \end{array} \]

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 0.88)
    (*
     (fma (fma 0.265709700396151 (* x_m x_m) -0.6665536072) (* x_m x_m) 1.0)
     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 <= 0.88) {
		tmp = fma(fma(0.265709700396151, (x_m * x_m), -0.6665536072), (x_m * x_m), 1.0) * 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 <= 0.88)
		tmp = Float64(fma(fma(0.265709700396151, Float64(x_m * x_m), -0.6665536072), Float64(x_m * x_m), 1.0) * x_m);
	else
		tmp = Float64(0.5 / 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, 0.88], N[(N[(N[(0.265709700396151 * N[(x$95$m * x$95$m), $MachinePrecision] + -0.6665536072), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision], N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 74.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right)\right)} \cdot x \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right) + 1\right)} \cdot x \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}\right) \cdot {x}^{2}} + 1\right) \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} - \frac{833192009}{1250000000}, {x}^{2}, 1\right)} \cdot x \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{833192009}{1250000000}\right)\right)}, {x}^{2}, 1\right) \cdot x \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000} \cdot {x}^{2} + \color{blue}{\frac{-833192009}{1250000000}}, {x}^{2}, 1\right) \cdot x \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000}, {x}^{2}, \frac{-833192009}{1250000000}\right)}, {x}^{2}, 1\right) \cdot x \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000}, \color{blue}{x \cdot x}, \frac{-833192009}{1250000000}\right), {x}^{2}, 1\right) \cdot x \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000}, \color{blue}{x \cdot x}, \frac{-833192009}{1250000000}\right), {x}^{2}, 1\right) \cdot x \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{3321371254951887171}{12500000000000000000}, x \cdot x, \frac{-833192009}{1250000000}\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

      10. lower-*.f64 71.7

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.265709700396151, x \cdot x, -0.6665536072\right), \color{blue}{x \cdot x}, 1\right) \cdot x \]

   5. Applied rewrites 71.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.265709700396151, x \cdot x, -0.6665536072\right), x \cdot x, 1\right)} \cdot x \]

   if 0.880000000000000004 < x

   1. Initial program 5.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 100.0

         \[\leadsto \color{blue}{\frac{0.5}{x}} \]

   5. Applied rewrites 100.0%

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

Alternative 10: 99.3% accurate, 18.0× speedup

Math

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

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 0.78) (* (fma (* x_m x_m) -0.6665536072 1.0) 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 <= 0.78) {
		tmp = fma((x_m * x_m), -0.6665536072, 1.0) * 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 <= 0.78)
		tmp = Float64(fma(Float64(x_m * x_m), -0.6665536072, 1.0) * x_m);
	else
		tmp = Float64(0.5 / 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, 0.78], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.6665536072 + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision], N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.78000000000000003

   1. Initial program 74.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + \frac{-833192009}{1250000000} \cdot {x}^{2}\right)} \cdot x \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-833192009}{1250000000} \cdot {x}^{2} + 1\right)} \cdot x \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{{x}^{2} \cdot \frac{-833192009}{1250000000}} + 1\right) \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-833192009}{1250000000}, 1\right)} \cdot x \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-833192009}{1250000000}, 1\right) \cdot x \]

      5. lower-*.f64 71.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.6665536072, 1\right) \cdot x \]

   5. Applied rewrites 71.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, -0.6665536072, 1\right)} \cdot x \]

   if 0.78000000000000003 < x

   1. Initial program 5.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 100.0

         \[\leadsto \color{blue}{\frac{0.5}{x}} \]

   5. Applied rewrites 100.0%

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

Alternative 11: 99.1% accurate, 23.0× speedup

Math

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

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) (* 1.0 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 = 1.0 * 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 = 1.0d0 * 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 = 1.0 * 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 = 1.0 * 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 = Float64(1.0 * 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 = 1.0 * 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], N[(1.0 * x$95$m), $MachinePrecision], N[(0.5 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.69999999999999996

   1. Initial program 74.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \cdot x \]
   4. Step-by-step derivation

   5. Applied rewrites 71.3%

      \[\leadsto \color{blue}{1} \cdot x \]

   if 0.69999999999999996 < x

   1. Initial program 5.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 100.0

         \[\leadsto \color{blue}{\frac{0.5}{x}} \]

   5. Applied rewrites 100.0%

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

Alternative 12: 51.4% accurate, 69.2× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (* 1.0 x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (1.0 * 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 * (1.0d0 * 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 * (1.0 * x_m);
}

Python

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

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(1.0 * 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 * (1.0 * 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 * N[(1.0 * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \cdot x \]
4. Step-by-step derivation

5. Applied rewrites 51.8%

   \[\leadsto \color{blue}{1} \cdot x \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024318 
(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))