Jmat.Real.erfi, branch x less than or equal to 0.5

Percentage Accurate: 99.8% → 99.9%

Time: 12.1s

Alternatives: 12

Speedup: 3.2×

• 28.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t_0\right) + \frac{1}{5} \cdot t_1\right) + \frac{1}{21} \cdot \left(\left(t_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
      (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x))))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x))))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t_0\right) + \frac{1}{5} \cdot t_1\right) + \frac{1}{21} \cdot \left(\left(t_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
      (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x))))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x))))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Alternative 1: 99.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ t_0 \cdot \left(x_m \cdot \left(2 + 0.6666666666666666 \cdot {x_m}^{2}\right)\right) + t_0 \cdot \left(0.2 \cdot {x_m}^{5} + 0.047619047619047616 \cdot {x_m}^{7}\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (+
    (* t_0 (* x_m (+ 2.0 (* 0.6666666666666666 (pow x_m 2.0)))))
    (* t_0 (+ (* 0.2 (pow x_m 5.0)) (* 0.047619047619047616 (pow x_m 7.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	return (t_0 * (x_m * (2.0 + (0.6666666666666666 * pow(x_m, 2.0))))) + (t_0 * ((0.2 * pow(x_m, 5.0)) + (0.047619047619047616 * pow(x_m, 7.0))));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	return (t_0 * (x_m * (2.0 + (0.6666666666666666 * Math.pow(x_m, 2.0))))) + (t_0 * ((0.2 * Math.pow(x_m, 5.0)) + (0.047619047619047616 * Math.pow(x_m, 7.0))));
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = math.sqrt((1.0 / math.pi))
	return (t_0 * (x_m * (2.0 + (0.6666666666666666 * math.pow(x_m, 2.0))))) + (t_0 * ((0.2 * math.pow(x_m, 5.0)) + (0.047619047619047616 * math.pow(x_m, 7.0))))

Julia

x_m = abs(x)
function code(x_m)
	t_0 = sqrt(Float64(1.0 / pi))
	return Float64(Float64(t_0 * Float64(x_m * Float64(2.0 + Float64(0.6666666666666666 * (x_m ^ 2.0))))) + Float64(t_0 * Float64(Float64(0.2 * (x_m ^ 5.0)) + Float64(0.047619047619047616 * (x_m ^ 7.0)))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	t_0 = sqrt((1.0 / pi));
	tmp = (t_0 * (x_m * (2.0 + (0.6666666666666666 * (x_m ^ 2.0))))) + (t_0 * ((0.2 * (x_m ^ 5.0)) + (0.047619047619047616 * (x_m ^ 7.0))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, N[(N[(t$95$0 * N[(x$95$m * N[(2.0 + N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(N[(0.2 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 2: 99.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.1:\\ \;\;\;\;\left|\left(x_m \cdot {\pi}^{-0.5}\right) \cdot \left(0.2 \cdot {x_m}^{4} + \mathsf{fma}\left(0.6666666666666666, x_m \cdot x_m, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x_m \cdot \left(0.6666666666666666 \cdot {x_m}^{2} + \mathsf{fma}\left(0.2, {x_m}^{4}, 0.047619047619047616 \cdot {x_m}^{6}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.1)
   (fabs
    (*
     (* x_m (pow PI -0.5))
     (+ (* 0.2 (pow x_m 4.0)) (fma 0.6666666666666666 (* x_m x_m) 2.0))))
   (*
    (pow PI -0.5)
    (*
     x_m
     (+
      (* 0.6666666666666666 (pow x_m 2.0))
      (fma 0.2 (pow x_m 4.0) (* 0.047619047619047616 (pow x_m 6.0))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.1) {
		tmp = fabs(((x_m * pow(((double) M_PI), -0.5)) * ((0.2 * pow(x_m, 4.0)) + fma(0.6666666666666666, (x_m * x_m), 2.0))));
	} else {
		tmp = pow(((double) M_PI), -0.5) * (x_m * ((0.6666666666666666 * pow(x_m, 2.0)) + fma(0.2, pow(x_m, 4.0), (0.047619047619047616 * pow(x_m, 6.0)))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.1)
		tmp = abs(Float64(Float64(x_m * (pi ^ -0.5)) * Float64(Float64(0.2 * (x_m ^ 4.0)) + fma(0.6666666666666666, Float64(x_m * x_m), 2.0))));
	else
		tmp = Float64((pi ^ -0.5) * Float64(x_m * Float64(Float64(0.6666666666666666 * (x_m ^ 2.0)) + fma(0.2, (x_m ^ 4.0), Float64(0.047619047619047616 * (x_m ^ 6.0))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.1], N[Abs[N[(N[(x$95$m * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(0.2 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x$95$m * N[(N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[Power[x$95$m, 4.0], $MachinePrecision] + N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 3: 99.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {\pi}^{-0.5} \cdot \left(x_m \cdot \left(\left(2 + 0.6666666666666666 \cdot {x_m}^{2}\right) + \mathsf{fma}\left(0.2, {x_m}^{4}, 0.047619047619047616 \cdot {x_m}^{6}\right)\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (pow PI -0.5)
  (*
   x_m
   (+
    (+ 2.0 (* 0.6666666666666666 (pow x_m 2.0)))
    (fma 0.2 (pow x_m 4.0) (* 0.047619047619047616 (pow x_m 6.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(((double) M_PI), -0.5) * (x_m * ((2.0 + (0.6666666666666666 * pow(x_m, 2.0))) + fma(0.2, pow(x_m, 4.0), (0.047619047619047616 * pow(x_m, 6.0)))));
}

Julia

x_m = abs(x)
function code(x_m)
	return Float64((pi ^ -0.5) * Float64(x_m * Float64(Float64(2.0 + Float64(0.6666666666666666 * (x_m ^ 2.0))) + fma(0.2, (x_m ^ 4.0), Float64(0.047619047619047616 * (x_m ^ 6.0))))))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x$95$m * N[(N[(2.0 + N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[Power[x$95$m, 4.0], $MachinePrecision] + N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 4: 99.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.1:\\ \;\;\;\;\left|\left(x_m \cdot {\pi}^{-0.5}\right) \cdot \left(0.2 \cdot {x_m}^{4} + \mathsf{fma}\left(0.6666666666666666, x_m \cdot x_m, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x_m\right|}{\sqrt{\pi} \cdot \left(\frac{21}{{x_m}^{6}} + \frac{-88.2}{{x_m}^{8}}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.1)
   (fabs
    (*
     (* x_m (pow PI -0.5))
     (+ (* 0.2 (pow x_m 4.0)) (fma 0.6666666666666666 (* x_m x_m) 2.0))))
   (/
    (fabs x_m)
    (* (sqrt PI) (+ (/ 21.0 (pow x_m 6.0)) (/ -88.2 (pow x_m 8.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.1) {
		tmp = fabs(((x_m * pow(((double) M_PI), -0.5)) * ((0.2 * pow(x_m, 4.0)) + fma(0.6666666666666666, (x_m * x_m), 2.0))));
	} else {
		tmp = fabs(x_m) / (sqrt(((double) M_PI)) * ((21.0 / pow(x_m, 6.0)) + (-88.2 / pow(x_m, 8.0))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.1)
		tmp = abs(Float64(Float64(x_m * (pi ^ -0.5)) * Float64(Float64(0.2 * (x_m ^ 4.0)) + fma(0.6666666666666666, Float64(x_m * x_m), 2.0))));
	else
		tmp = Float64(abs(x_m) / Float64(sqrt(pi) * Float64(Float64(21.0 / (x_m ^ 6.0)) + Float64(-88.2 / (x_m ^ 8.0)))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.1], N[Abs[N[(N[(x$95$m * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(0.2 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[x$95$m], $MachinePrecision] / N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[(21.0 / N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision] + N[(-88.2 / N[Power[x$95$m, 8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 5: 99.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.1:\\ \;\;\;\;{\pi}^{-0.5} \cdot \frac{x_m}{\mathsf{fma}\left({x_m}^{2}, -0.16666666666666666, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x_m\right|}{\sqrt{\pi} \cdot \left(\frac{21}{{x_m}^{6}} + \frac{-88.2}{{x_m}^{8}}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.1)
   (* (pow PI -0.5) (/ x_m (fma (pow x_m 2.0) -0.16666666666666666 0.5)))
   (/
    (fabs x_m)
    (* (sqrt PI) (+ (/ 21.0 (pow x_m 6.0)) (/ -88.2 (pow x_m 8.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.1) {
		tmp = pow(((double) M_PI), -0.5) * (x_m / fma(pow(x_m, 2.0), -0.16666666666666666, 0.5));
	} else {
		tmp = fabs(x_m) / (sqrt(((double) M_PI)) * ((21.0 / pow(x_m, 6.0)) + (-88.2 / pow(x_m, 8.0))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.1)
		tmp = Float64((pi ^ -0.5) * Float64(x_m / fma((x_m ^ 2.0), -0.16666666666666666, 0.5)));
	else
		tmp = Float64(abs(x_m) / Float64(sqrt(pi) * Float64(Float64(21.0 / (x_m ^ 6.0)) + Float64(-88.2 / (x_m ^ 8.0)))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.1], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x$95$m / N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x$95$m], $MachinePrecision] / N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[(21.0 / N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision] + N[(-88.2 / N[Power[x$95$m, 8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 6: 99.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.1:\\ \;\;\;\;{\pi}^{-0.5} \cdot \frac{x_m}{\mathsf{fma}\left({x_m}^{2}, -0.16666666666666666, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x_m}^{5} + 0.047619047619047616 \cdot {x_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.1)
   (* (pow PI -0.5) (/ x_m (fma (pow x_m 2.0) -0.16666666666666666 0.5)))
   (*
    (sqrt (/ 1.0 PI))
    (+ (* 0.2 (pow x_m 5.0)) (* 0.047619047619047616 (pow x_m 7.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.1) {
		tmp = pow(((double) M_PI), -0.5) * (x_m / fma(pow(x_m, 2.0), -0.16666666666666666, 0.5));
	} else {
		tmp = sqrt((1.0 / ((double) M_PI))) * ((0.2 * pow(x_m, 5.0)) + (0.047619047619047616 * pow(x_m, 7.0)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.1)
		tmp = Float64((pi ^ -0.5) * Float64(x_m / fma((x_m ^ 2.0), -0.16666666666666666, 0.5)));
	else
		tmp = Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(0.2 * (x_m ^ 5.0)) + Float64(0.047619047619047616 * (x_m ^ 7.0))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.1], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x$95$m / N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(0.2 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 7: 99.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.1:\\ \;\;\;\;{\pi}^{-0.5} \cdot \frac{x_m}{\mathsf{fma}\left({x_m}^{2}, -0.16666666666666666, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;{x_m}^{6} \cdot \left(0.047619047619047616 \cdot \frac{x_m}{\sqrt{\pi}}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.1)
   (* (pow PI -0.5) (/ x_m (fma (pow x_m 2.0) -0.16666666666666666 0.5)))
   (* (pow x_m 6.0) (* 0.047619047619047616 (/ x_m (sqrt PI))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.1) {
		tmp = pow(((double) M_PI), -0.5) * (x_m / fma(pow(x_m, 2.0), -0.16666666666666666, 0.5));
	} else {
		tmp = pow(x_m, 6.0) * (0.047619047619047616 * (x_m / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.1)
		tmp = Float64((pi ^ -0.5) * Float64(x_m / fma((x_m ^ 2.0), -0.16666666666666666, 0.5)));
	else
		tmp = Float64((x_m ^ 6.0) * Float64(0.047619047619047616 * Float64(x_m / sqrt(pi))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.1], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x$95$m / N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x$95$m, 6.0], $MachinePrecision] * N[(0.047619047619047616 * N[(x$95$m / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 8: 99.3% accurate, 4.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.1:\\ \;\;\;\;\frac{{\pi}^{-0.5} \cdot \left(-x_m\right)}{-0.5 - {x_m}^{2} \cdot -0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;{x_m}^{6} \cdot \left(0.047619047619047616 \cdot \frac{x_m}{\sqrt{\pi}}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.1)
   (/
    (* (pow PI -0.5) (- x_m))
    (- -0.5 (* (pow x_m 2.0) -0.16666666666666666)))
   (* (pow x_m 6.0) (* 0.047619047619047616 (/ x_m (sqrt PI))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.1) {
		tmp = (pow(((double) M_PI), -0.5) * -x_m) / (-0.5 - (pow(x_m, 2.0) * -0.16666666666666666));
	} else {
		tmp = pow(x_m, 6.0) * (0.047619047619047616 * (x_m / sqrt(((double) M_PI))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 0.1) {
		tmp = (Math.pow(Math.PI, -0.5) * -x_m) / (-0.5 - (Math.pow(x_m, 2.0) * -0.16666666666666666));
	} else {
		tmp = Math.pow(x_m, 6.0) * (0.047619047619047616 * (x_m / Math.sqrt(Math.PI)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 0.1:
		tmp = (math.pow(math.pi, -0.5) * -x_m) / (-0.5 - (math.pow(x_m, 2.0) * -0.16666666666666666))
	else:
		tmp = math.pow(x_m, 6.0) * (0.047619047619047616 * (x_m / math.sqrt(math.pi)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.1)
		tmp = Float64(Float64((pi ^ -0.5) * Float64(-x_m)) / Float64(-0.5 - Float64((x_m ^ 2.0) * -0.16666666666666666)));
	else
		tmp = Float64((x_m ^ 6.0) * Float64(0.047619047619047616 * Float64(x_m / sqrt(pi))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (abs(x_m) <= 0.1)
		tmp = ((pi ^ -0.5) * -x_m) / (-0.5 - ((x_m ^ 2.0) * -0.16666666666666666));
	else
		tmp = (x_m ^ 6.0) * (0.047619047619047616 * (x_m / sqrt(pi)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.1], N[(N[(N[Power[Pi, -0.5], $MachinePrecision] * (-x$95$m)), $MachinePrecision] / N[(-0.5 - N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x$95$m, 6.0], $MachinePrecision] * N[(0.047619047619047616 * N[(x$95$m / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 9: 99.0% accurate, 6.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1.86:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(x_m \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;{x_m}^{6} \cdot \left(0.047619047619047616 \cdot \frac{x_m}{\sqrt{\pi}}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.86)
   (* (sqrt (/ 1.0 PI)) (* x_m 2.0))
   (* (pow x_m 6.0) (* 0.047619047619047616 (/ x_m (sqrt PI))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.86) {
		tmp = sqrt((1.0 / ((double) M_PI))) * (x_m * 2.0);
	} else {
		tmp = pow(x_m, 6.0) * (0.047619047619047616 * (x_m / sqrt(((double) M_PI))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.86) {
		tmp = Math.sqrt((1.0 / Math.PI)) * (x_m * 2.0);
	} else {
		tmp = Math.pow(x_m, 6.0) * (0.047619047619047616 * (x_m / Math.sqrt(Math.PI)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.86:
		tmp = math.sqrt((1.0 / math.pi)) * (x_m * 2.0)
	else:
		tmp = math.pow(x_m, 6.0) * (0.047619047619047616 * (x_m / math.sqrt(math.pi)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.86)
		tmp = Float64(sqrt(Float64(1.0 / pi)) * Float64(x_m * 2.0));
	else
		tmp = Float64((x_m ^ 6.0) * Float64(0.047619047619047616 * Float64(x_m / sqrt(pi))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.86)
		tmp = sqrt((1.0 / pi)) * (x_m * 2.0);
	else
		tmp = (x_m ^ 6.0) * (0.047619047619047616 * (x_m / sqrt(pi)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.86], N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[x$95$m, 6.0], $MachinePrecision] * N[(0.047619047619047616 * N[(x$95$m / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 10: 68.4% accurate, 9.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \sqrt{\frac{1}{\pi}} \cdot \left(x_m \cdot 2\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* (sqrt (/ 1.0 PI)) (* x_m 2.0)))

C

x_m = fabs(x);
double code(double x_m) {
	return sqrt((1.0 / ((double) M_PI))) * (x_m * 2.0);
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.sqrt((1.0 / Math.PI)) * (x_m * 2.0);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.sqrt((1.0 / math.pi)) * (x_m * 2.0)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(sqrt(Float64(1.0 / pi)) * Float64(x_m * 2.0))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = sqrt((1.0 / pi)) * (x_m * 2.0);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x$95$m * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

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1. Add Preprocessing

Alternative 11: 68.0% accurate, 9.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{2}{\frac{\sqrt{\pi}}{x_m}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ 2.0 (/ (sqrt PI) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	return 2.0 / (sqrt(((double) M_PI)) / x_m);
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 2.0 / (Math.sqrt(Math.PI) / x_m);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 2.0 / (math.sqrt(math.pi) / x_m)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(2.0 / Float64(sqrt(pi) / x_m))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 2.0 / (sqrt(pi) / x_m);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(2.0 / N[(N[Sqrt[Pi], $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

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1. Add Preprocessing

Alternative 12: 68.0% accurate, 9.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{x_m \cdot 2}{\sqrt{\pi}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ (* x_m 2.0) (sqrt PI)))

C

x_m = fabs(x);
double code(double x_m) {
	return (x_m * 2.0) / sqrt(((double) M_PI));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (x_m * 2.0) / Math.sqrt(Math.PI);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (x_m * 2.0) / math.sqrt(math.pi)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(x_m * 2.0) / sqrt(pi))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (x_m * 2.0) / sqrt(pi);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(x$95$m * 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

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1. Add Preprocessing

Reproduce

herbie shell --seed 2023348 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  :precision binary64
  :pre (<= x 0.5)
  (fabs (* (/ 1.0 (sqrt PI)) (+ (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1.0 5.0) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1.0 21.0) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))