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

Percentage Accurate: 99.8% → 99.9%

Time: 10.7s

Alternatives: 8

Speedup: 3.7×

• 28.8% 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, 3.7× speedup

Math

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

FPCore

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (x_m * Math.pow(Math.PI, -0.5)) * (2.0 + (((0.6666666666666666 * Math.pow(x_m, 2.0)) + (0.2 * Math.pow(x_m, 4.0))) + (0.047619047619047616 * Math.pow(x_m, 6.0))));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (x_m * math.pow(math.pi, -0.5)) * (2.0 + (((0.6666666666666666 * math.pow(x_m, 2.0)) + (0.2 * math.pow(x_m, 4.0))) + (0.047619047619047616 * math.pow(x_m, 6.0))))

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (x_m * (pi ^ -0.5)) * (2.0 + (((0.6666666666666666 * (x_m ^ 2.0)) + (0.2 * (x_m ^ 4.0))) + (0.047619047619047616 * (x_m ^ 6.0))));
end

Wolfram

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

Derivation

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1. Add Preprocessing

Alternative 2: 99.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x_m}^{7} + \left(0.2 \cdot {x_m}^{5} + x_m \cdot 2\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (pow PI -0.5)
  (+
   (* 0.047619047619047616 (pow x_m 7.0))
   (+ (* 0.2 (pow x_m 5.0)) (* x_m 2.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(((double) M_PI), -0.5) * ((0.047619047619047616 * pow(x_m, 7.0)) + ((0.2 * pow(x_m, 5.0)) + (x_m * 2.0)));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(Math.PI, -0.5) * ((0.047619047619047616 * Math.pow(x_m, 7.0)) + ((0.2 * Math.pow(x_m, 5.0)) + (x_m * 2.0)));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(math.pi, -0.5) * ((0.047619047619047616 * math.pow(x_m, 7.0)) + ((0.2 * math.pow(x_m, 5.0)) + (x_m * 2.0)))

Julia

x_m = abs(x)
function code(x_m)
	return Float64((pi ^ -0.5) * Float64(Float64(0.047619047619047616 * (x_m ^ 7.0)) + Float64(Float64(0.2 * (x_m ^ 5.0)) + Float64(x_m * 2.0))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (pi ^ -0.5) * ((0.047619047619047616 * (x_m ^ 7.0)) + ((0.2 * (x_m ^ 5.0)) + (x_m * 2.0)));
end

Wolfram

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

Derivation

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1. Add Preprocessing

Alternative 3: 99.2% accurate, 6.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 2.2:\\ \;\;\;\;\left(x_m \cdot {\pi}^{-0.5}\right) \cdot \left(2 + 0.6666666666666666 \cdot {x_m}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.047619047619047616 \cdot {x_m}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.2)
   (* (* x_m (pow PI -0.5)) (+ 2.0 (* 0.6666666666666666 (pow x_m 2.0))))
   (* (* 0.047619047619047616 (pow x_m 7.0)) (sqrt (/ 1.0 PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.2) {
		tmp = (x_m * pow(((double) M_PI), -0.5)) * (2.0 + (0.6666666666666666 * pow(x_m, 2.0)));
	} else {
		tmp = (0.047619047619047616 * pow(x_m, 7.0)) * sqrt((1.0 / ((double) M_PI)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.2) {
		tmp = (x_m * Math.pow(Math.PI, -0.5)) * (2.0 + (0.6666666666666666 * Math.pow(x_m, 2.0)));
	} else {
		tmp = (0.047619047619047616 * Math.pow(x_m, 7.0)) * Math.sqrt((1.0 / Math.PI));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.2:
		tmp = (x_m * math.pow(math.pi, -0.5)) * (2.0 + (0.6666666666666666 * math.pow(x_m, 2.0)))
	else:
		tmp = (0.047619047619047616 * math.pow(x_m, 7.0)) * math.sqrt((1.0 / math.pi))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.2)
		tmp = Float64(Float64(x_m * (pi ^ -0.5)) * Float64(2.0 + Float64(0.6666666666666666 * (x_m ^ 2.0))));
	else
		tmp = Float64(Float64(0.047619047619047616 * (x_m ^ 7.0)) * sqrt(Float64(1.0 / pi)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.2)
		tmp = (x_m * (pi ^ -0.5)) * (2.0 + (0.6666666666666666 * (x_m ^ 2.0)));
	else
		tmp = (0.047619047619047616 * (x_m ^ 7.0)) * sqrt((1.0 / pi));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.2], N[(N[(x$95$m * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.047619047619047616 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 4: 98.8% accurate, 6.3× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (* (pow PI -0.5) (* x_m (+ 2.0 (* 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 + (0.047619047619047616 * pow(x_m, 6.0))));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(Math.PI, -0.5) * (x_m * (2.0 + (0.047619047619047616 * Math.pow(x_m, 6.0))));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(math.pi, -0.5) * (x_m * (2.0 + (0.047619047619047616 * math.pow(x_m, 6.0))))

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (pi ^ -0.5) * (x_m * (2.0 + (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[(2.0 + N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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1. Add Preprocessing

Alternative 5: 98.8% accurate, 6.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1.85:\\ \;\;\;\;x_m \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.047619047619047616 \cdot {x_m}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.85)
   (* x_m (/ 2.0 (sqrt PI)))
   (* (* 0.047619047619047616 (pow x_m 7.0)) (sqrt (/ 1.0 PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = (0.047619047619047616 * pow(x_m, 7.0)) * sqrt((1.0 / ((double) M_PI)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = (0.047619047619047616 * Math.pow(x_m, 7.0)) * Math.sqrt((1.0 / Math.PI));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.85:
		tmp = x_m * (2.0 / math.sqrt(math.pi))
	else:
		tmp = (0.047619047619047616 * math.pow(x_m, 7.0)) * math.sqrt((1.0 / math.pi))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.85)
		tmp = Float64(x_m * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(Float64(0.047619047619047616 * (x_m ^ 7.0)) * sqrt(Float64(1.0 / pi)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.85)
		tmp = x_m * (2.0 / sqrt(pi));
	else
		tmp = (0.047619047619047616 * (x_m ^ 7.0)) * sqrt((1.0 / pi));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.85], N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.047619047619047616 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 6: 98.8% accurate, 6.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1.85:\\ \;\;\;\;x_m \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.85)
   (* x_m (/ 2.0 (sqrt PI)))
   (* 0.047619047619047616 (* (pow PI -0.5) (pow x_m 7.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = 0.047619047619047616 * (pow(((double) M_PI), -0.5) * pow(x_m, 7.0));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = 0.047619047619047616 * (Math.pow(Math.PI, -0.5) * Math.pow(x_m, 7.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.85:
		tmp = x_m * (2.0 / math.sqrt(math.pi))
	else:
		tmp = 0.047619047619047616 * (math.pow(math.pi, -0.5) * math.pow(x_m, 7.0))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.85)
		tmp = Float64(x_m * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(0.047619047619047616 * Float64((pi ^ -0.5) * (x_m ^ 7.0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.85)
		tmp = x_m * (2.0 / sqrt(pi));
	else
		tmp = 0.047619047619047616 * ((pi ^ -0.5) * (x_m ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.85], N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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

Alternative 7: 83.6% accurate, 6.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 10^{-22}:\\ \;\;\;\;x_m \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x_m}^{2} \cdot \frac{4}{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1e-22)
   (* x_m (/ 2.0 (sqrt PI)))
   (sqrt (* (pow x_m 2.0) (/ 4.0 PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1e-22) {
		tmp = x_m * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = sqrt((pow(x_m, 2.0) * (4.0 / ((double) M_PI))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1e-22) {
		tmp = x_m * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.sqrt((Math.pow(x_m, 2.0) * (4.0 / Math.PI)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1e-22:
		tmp = x_m * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.sqrt((math.pow(x_m, 2.0) * (4.0 / math.pi)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1e-22)
		tmp = Float64(x_m * Float64(2.0 / sqrt(pi)));
	else
		tmp = sqrt(Float64((x_m ^ 2.0) * Float64(4.0 / pi)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1e-22)
		tmp = x_m * (2.0 / sqrt(pi));
	else
		tmp = sqrt(((x_m ^ 2.0) * (4.0 / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1e-22], N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(4.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

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

Alternative 8: 67.7% accurate, 9.5× speedup

Math

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

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(x_m * Float64(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[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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

Reproduce

herbie shell --seed 2023343 
(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)))))))