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

Percentage Accurate: 99.8% → 99.8%

Time: 8.2s

Alternatives: 8

Speedup: 1.5×

• 29.3% 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: 99.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot \left(x \cdot x\right)\\ t_1 := \left|x\right| \cdot \left(\left|x\right| \cdot t_0\right)\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + 0.6666666666666666 \cdot t_0\right) + 0.2 \cdot t_1\right) + 0.047619047619047616 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot t_1\right)\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (* x x))) (t_1 (* (fabs x) (* (fabs x) t_0))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* 0.6666666666666666 t_0)) (* 0.2 t_1))
      (* 0.047619047619047616 (* (fabs x) (* (fabs x) t_1))))))))

C

double code(double x) {
	double t_0 = fabs(x) * (x * x);
	double t_1 = fabs(x) * (fabs(x) * t_0);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (fabs(x) * (fabs(x) * t_1))))));
}

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * (x * x);
	double t_1 = Math.abs(x) * (Math.abs(x) * t_0);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (Math.abs(x) * (Math.abs(x) * t_1))))));
}

Python

def code(x):
	t_0 = math.fabs(x) * (x * x)
	t_1 = math.fabs(x) * (math.fabs(x) * t_0)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (math.fabs(x) * (math.fabs(x) * t_1))))))

Julia

function code(x)
	t_0 = Float64(abs(x) * Float64(x * x))
	t_1 = Float64(abs(x) * Float64(abs(x) * t_0))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(0.6666666666666666 * t_0)) + Float64(0.2 * t_1)) + Float64(0.047619047619047616 * Float64(abs(x) * Float64(abs(x) * t_1))))))
end

MATLAB

function tmp = code(x)
	t_0 = abs(x) * (x * x);
	t_1 = abs(x) * (abs(x) * t_0);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (abs(x) * (abs(x) * t_1))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$0), $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[(0.6666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.2 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

2. Final simplification 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right)\right)\right) + 0.047619047619047616 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right| \]

Alternative 2: 99.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot \left(x \cdot x\right)\\ t_1 := \left(x \cdot x\right) \cdot t_0\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot t_0\right) + \left(0.2 \cdot t_1 + 0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot t_1\right)\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (* x x))) (t_1 (* (* x x) t_0)))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (fma 2.0 (fabs x) (* 0.6666666666666666 t_0))
      (+ (* 0.2 t_1) (* 0.047619047619047616 (* (* x x) t_1))))))))

C

double code(double x) {
	double t_0 = fabs(x) * (x * x);
	double t_1 = (x * x) * t_0;
	return fabs(((1.0 / sqrt(((double) M_PI))) * (fma(2.0, fabs(x), (0.6666666666666666 * t_0)) + ((0.2 * t_1) + (0.047619047619047616 * ((x * x) * t_1))))));
}

Julia

function code(x)
	t_0 = Float64(abs(x) * Float64(x * x))
	t_1 = Float64(Float64(x * x) * t_0)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(fma(2.0, abs(x), Float64(0.6666666666666666 * t_0)) + Float64(Float64(0.2 * t_1) + Float64(0.047619047619047616 * Float64(Float64(x * x) * t_1))))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[Abs[x], $MachinePrecision] + N[(0.6666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(0.2 * t$95$1), $MachinePrecision] + N[(0.047619047619047616 * N[(N[(x * x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + \left(0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right|} \]

4. Final simplification 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + \left(0.2 \cdot \left(\left(x \cdot x\right) \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right| \]

Alternative 3: 34.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (/
   (+
    (+ (* 0.2 (pow x 4.0)) (* 0.047619047619047616 (pow x 6.0)))
    (+ 2.0 (* 0.6666666666666666 (pow x 2.0))))
   (sqrt PI))))

C

double code(double x) {
	return x * ((((0.2 * pow(x, 4.0)) + (0.047619047619047616 * pow(x, 6.0))) + (2.0 + (0.6666666666666666 * pow(x, 2.0)))) / sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return x * ((((0.2 * Math.pow(x, 4.0)) + (0.047619047619047616 * Math.pow(x, 6.0))) + (2.0 + (0.6666666666666666 * Math.pow(x, 2.0)))) / Math.sqrt(Math.PI));
}

Python

def code(x):
	return x * ((((0.2 * math.pow(x, 4.0)) + (0.047619047619047616 * math.pow(x, 6.0))) + (2.0 + (0.6666666666666666 * math.pow(x, 2.0)))) / math.sqrt(math.pi))

Julia

function code(x)
	return Float64(x * Float64(Float64(Float64(Float64(0.2 * (x ^ 4.0)) + Float64(0.047619047619047616 * (x ^ 6.0))) + Float64(2.0 + Float64(0.6666666666666666 * (x ^ 2.0)))) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = x * ((((0.2 * (x ^ 4.0)) + (0.047619047619047616 * (x ^ 6.0))) + (2.0 + (0.6666666666666666 * (x ^ 2.0)))) / sqrt(pi));
end

Wolfram

code[x_] := N[(x * N[(N[(N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|}} \]
4. Step-by-step derivation

   1. div-inv 99.9%

      \[\leadsto \color{blue}{\left|x\right| \cdot \frac{1}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|}} \]

   2. add-sqr-sqrt 33.3%

      \[\leadsto \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \frac{1}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|} \]

   3. fabs-sqr 33.3%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \frac{1}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|} \]

   4. add-sqr-sqrt 34.6%

      \[\leadsto \color{blue}{x} \cdot \frac{1}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|} \]

   5. *-commutative 34.6%

      \[\leadsto \color{blue}{\frac{1}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|} \cdot x} \]

5. Applied egg-rr 34.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)}{\sqrt{\pi}} \cdot x} \]
6. Step-by-step derivation

   1. fma-udef 34.6%

      \[\leadsto \frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \color{blue}{\left(0.6666666666666666 \cdot {x}^{2} + 2\right)}}{\sqrt{\pi}} \cdot x \]

7. Applied egg-rr 34.6%

   \[\leadsto \frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \color{blue}{\left(0.6666666666666666 \cdot {x}^{2} + 2\right)}}{\sqrt{\pi}} \cdot x \]
8. Step-by-step derivation

   1. fma-udef 34.6%

      \[\leadsto \frac{\color{blue}{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)} + \left(0.6666666666666666 \cdot {x}^{2} + 2\right)}{\sqrt{\pi}} \cdot x \]

9. Applied egg-rr 34.6%

   \[\leadsto \frac{\color{blue}{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)} + \left(0.6666666666666666 \cdot {x}^{2} + 2\right)}{\sqrt{\pi}} \cdot x \]

10. Final simplification 34.6%

    \[\leadsto x \cdot \frac{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)}{\sqrt{\pi}} \]

Alternative 4: 34.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{0.047619047619047616 \cdot {x}^{6} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (/
   (+
    (* 0.047619047619047616 (pow x 6.0))
    (fma 0.6666666666666666 (pow x 2.0) 2.0))
   (sqrt PI))))

C

double code(double x) {
	return x * (((0.047619047619047616 * pow(x, 6.0)) + fma(0.6666666666666666, pow(x, 2.0), 2.0)) / sqrt(((double) M_PI)));
}

Julia

function code(x)
	return Float64(x * Float64(Float64(Float64(0.047619047619047616 * (x ^ 6.0)) + fma(0.6666666666666666, (x ^ 2.0), 2.0)) / sqrt(pi)))
end

Wolfram

code[x_] := N[(x * N[(N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|}} \]
4. Step-by-step derivation

   1. div-inv 99.9%

      \[\leadsto \color{blue}{\left|x\right| \cdot \frac{1}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|}} \]

   2. add-sqr-sqrt 33.3%

      \[\leadsto \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \frac{1}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|} \]

   3. fabs-sqr 33.3%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \frac{1}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|} \]

   4. add-sqr-sqrt 34.6%

      \[\leadsto \color{blue}{x} \cdot \frac{1}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|} \]

   5. *-commutative 34.6%

      \[\leadsto \color{blue}{\frac{1}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|} \cdot x} \]

5. Applied egg-rr 34.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)}{\sqrt{\pi}} \cdot x} \]

6. Taylor expanded in x around inf 34.4%

   \[\leadsto \frac{\color{blue}{0.047619047619047616 \cdot {x}^{6}} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)}{\sqrt{\pi}} \cdot x \]

7. Final simplification 34.4%

   \[\leadsto x \cdot \frac{0.047619047619047616 \cdot {x}^{6} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)}{\sqrt{\pi}} \]

Alternative 5: 34.3% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{2 + 0.047619047619047616 \cdot {x}^{6}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (/ (+ 2.0 (* 0.047619047619047616 (pow x 6.0))) (sqrt PI))))

C

double code(double x) {
	return x * ((2.0 + (0.047619047619047616 * pow(x, 6.0))) / sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return x * ((2.0 + (0.047619047619047616 * Math.pow(x, 6.0))) / Math.sqrt(Math.PI));
}

Python

def code(x):
	return x * ((2.0 + (0.047619047619047616 * math.pow(x, 6.0))) / math.sqrt(math.pi))

Julia

function code(x)
	return Float64(x * Float64(Float64(2.0 + Float64(0.047619047619047616 * (x ^ 6.0))) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = x * ((2.0 + (0.047619047619047616 * (x ^ 6.0))) / sqrt(pi));
end

Wolfram

code[x_] := N[(x * N[(N[(2.0 + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|}} \]
4. Step-by-step derivation

   1. div-inv 99.9%

      \[\leadsto \color{blue}{\left|x\right| \cdot \frac{1}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|}} \]

   2. add-sqr-sqrt 33.3%

      \[\leadsto \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \frac{1}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|} \]

   3. fabs-sqr 33.3%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \frac{1}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|} \]

   4. add-sqr-sqrt 34.6%

      \[\leadsto \color{blue}{x} \cdot \frac{1}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|} \]

   5. *-commutative 34.6%

      \[\leadsto \color{blue}{\frac{1}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|} \cdot x} \]

5. Applied egg-rr 34.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)}{\sqrt{\pi}} \cdot x} \]

6. Taylor expanded in x around inf 34.4%

   \[\leadsto \frac{\color{blue}{0.047619047619047616 \cdot {x}^{6}} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)}{\sqrt{\pi}} \cdot x \]

7. Taylor expanded in x around 0 34.0%

   \[\leadsto \frac{0.047619047619047616 \cdot {x}^{6} + \color{blue}{2}}{\sqrt{\pi}} \cdot x \]

8. Final simplification 34.0%

   \[\leadsto x \cdot \frac{2 + 0.047619047619047616 \cdot {x}^{6}}{\sqrt{\pi}} \]

Alternative 6: 34.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\left(2 \cdot x\right) \cdot {\pi}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616}{\frac{\sqrt{\pi}}{{x}^{7}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.9)
   (* (* 2.0 x) (pow PI -0.5))
   (/ 0.047619047619047616 (/ (sqrt PI) (pow x 7.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = (2.0 * x) * pow(((double) M_PI), -0.5);
	} else {
		tmp = 0.047619047619047616 / (sqrt(((double) M_PI)) / pow(x, 7.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = (2.0 * x) * Math.pow(Math.PI, -0.5);
	} else {
		tmp = 0.047619047619047616 / (Math.sqrt(Math.PI) / Math.pow(x, 7.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.9:
		tmp = (2.0 * x) * math.pow(math.pi, -0.5)
	else:
		tmp = 0.047619047619047616 / (math.sqrt(math.pi) / math.pow(x, 7.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.9)
		tmp = Float64(Float64(2.0 * x) * (pi ^ -0.5));
	else
		tmp = Float64(0.047619047619047616 / Float64(sqrt(pi) / (x ^ 7.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.9)
		tmp = (2.0 * x) * (pi ^ -0.5);
	else
		tmp = 0.047619047619047616 / (sqrt(pi) / (x ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.9], N[(N[(2.0 * x), $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 / N[(N[Sqrt[Pi], $MachinePrecision] / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 33.3%

         \[\leadsto \frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|} \]

      2. fabs-sqr 33.3%

         \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|} \]

      3. add-sqr-sqrt 33.3%

         \[\leadsto \frac{\sqrt{x} \cdot \sqrt{x}}{\left|\color{blue}{\sqrt{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}} \cdot \sqrt{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}}}\right|} \]

      4. fabs-sqr 33.3%

         \[\leadsto \frac{\sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}} \cdot \sqrt{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}}}} \]

      5. add-sqr-sqrt 34.5%

         \[\leadsto \frac{\color{blue}{x}}{\sqrt{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}} \cdot \sqrt{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}}} \]

      6. add-sqr-sqrt 34.4%

         \[\leadsto \frac{x}{\color{blue}{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}}} \]

      7. *-un-lft-identity 34.4%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}} \]

   5. Applied egg-rr 34.6%

      \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \frac{x}{\frac{1}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)}}} \]
   6. Step-by-step derivation

      1. associate-/r/ 34.6%

         \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(\frac{x}{1} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\right)} \]

      2. /-rgt-identity 34.6%

         \[\leadsto {\pi}^{-0.5} \cdot \left(\color{blue}{x} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\right) \]

      3. *-commutative 34.6%

         \[\leadsto \color{blue}{\left(x \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\right) \cdot {\pi}^{-0.5}} \]

      4. associate-*l* 34.6%

         \[\leadsto \color{blue}{x \cdot \left(\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot {\pi}^{-0.5}\right)} \]

      5. fma-def 34.6%

         \[\leadsto x \cdot \left(\left(\color{blue}{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot {\pi}^{-0.5}\right) \]

      6. +-commutative 34.6%

         \[\leadsto x \cdot \left(\left(\color{blue}{\left(0.047619047619047616 \cdot {x}^{6} + 0.2 \cdot {x}^{4}\right)} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot {\pi}^{-0.5}\right) \]

      7. fma-def 34.6%

         \[\leadsto x \cdot \left(\left(\color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 0.2 \cdot {x}^{4}\right)} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot {\pi}^{-0.5}\right) \]

      8. *-commutative 34.6%

         \[\leadsto x \cdot \left(\left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \color{blue}{{x}^{4} \cdot 0.2}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot {\pi}^{-0.5}\right) \]

   7. Simplified 34.6%

      \[\leadsto \color{blue}{x \cdot \left(\left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, {x}^{4} \cdot 0.2\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot {\pi}^{-0.5}\right)} \]

   8. Taylor expanded in x around 0 34.1%

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 34.1%

         \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} \]

   10. Simplified 34.1%

       \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} \]
   11. Step-by-step derivation

       1. sqrt-div 34.1%

          \[\leadsto \left(2 \cdot x\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}} \]

       2. metadata-eval 34.1%

          \[\leadsto \left(2 \cdot x\right) \cdot \frac{\color{blue}{1}}{\sqrt{\pi}} \]

       3. un-div-inv 33.9%

          \[\leadsto \color{blue}{\frac{2 \cdot x}{\sqrt{\pi}}} \]

       4. *-commutative 33.9%

          \[\leadsto \frac{\color{blue}{x \cdot 2}}{\sqrt{\pi}} \]

   12. Applied egg-rr 33.9%

       \[\leadsto \color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}} \]
   13. Step-by-step derivation

       1. div-inv 34.1%

          \[\leadsto \color{blue}{\left(x \cdot 2\right) \cdot \frac{1}{\sqrt{\pi}}} \]

       2. pow1/2 34.1%

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

       3. pow-flip 34.1%

          \[\leadsto \left(x \cdot 2\right) \cdot \color{blue}{{\pi}^{\left(-0.5\right)}} \]

       4. metadata-eval 34.1%

          \[\leadsto \left(x \cdot 2\right) \cdot {\pi}^{\color{blue}{-0.5}} \]

   14. Applied egg-rr 34.1%

       \[\leadsto \color{blue}{\left(x \cdot 2\right) \cdot {\pi}^{-0.5}} \]

   if 1.8999999999999999 < x

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 33.3%

         \[\leadsto \frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|} \]

      2. fabs-sqr 33.3%

         \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|} \]

      3. add-sqr-sqrt 33.3%

         \[\leadsto \frac{\sqrt{x} \cdot \sqrt{x}}{\left|\color{blue}{\sqrt{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}} \cdot \sqrt{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}}}\right|} \]

      4. fabs-sqr 33.3%

         \[\leadsto \frac{\sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}} \cdot \sqrt{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}}}} \]

      5. add-sqr-sqrt 34.5%

         \[\leadsto \frac{\color{blue}{x}}{\sqrt{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}} \cdot \sqrt{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}}} \]

      6. add-sqr-sqrt 34.4%

         \[\leadsto \frac{x}{\color{blue}{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}}} \]

      7. *-un-lft-identity 34.4%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}} \]

   5. Applied egg-rr 34.6%

      \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \frac{x}{\frac{1}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)}}} \]
   6. Step-by-step derivation

      1. associate-/r/ 34.6%

         \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(\frac{x}{1} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\right)} \]

      2. /-rgt-identity 34.6%

         \[\leadsto {\pi}^{-0.5} \cdot \left(\color{blue}{x} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\right) \]

      3. *-commutative 34.6%

         \[\leadsto \color{blue}{\left(x \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\right) \cdot {\pi}^{-0.5}} \]

      4. associate-*l* 34.6%

         \[\leadsto \color{blue}{x \cdot \left(\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot {\pi}^{-0.5}\right)} \]

      5. fma-def 34.6%

         \[\leadsto x \cdot \left(\left(\color{blue}{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot {\pi}^{-0.5}\right) \]

      6. +-commutative 34.6%

         \[\leadsto x \cdot \left(\left(\color{blue}{\left(0.047619047619047616 \cdot {x}^{6} + 0.2 \cdot {x}^{4}\right)} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot {\pi}^{-0.5}\right) \]

      7. fma-def 34.6%

         \[\leadsto x \cdot \left(\left(\color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 0.2 \cdot {x}^{4}\right)} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot {\pi}^{-0.5}\right) \]

      8. *-commutative 34.6%

         \[\leadsto x \cdot \left(\left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \color{blue}{{x}^{4} \cdot 0.2}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot {\pi}^{-0.5}\right) \]

   7. Simplified 34.6%

      \[\leadsto \color{blue}{x \cdot \left(\left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, {x}^{4} \cdot 0.2\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot {\pi}^{-0.5}\right)} \]

   8. Taylor expanded in x around inf 3.4%

      \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 3.4%

         \[\leadsto \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}} \]

   10. Simplified 3.4%

       \[\leadsto \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 3.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]

       2. expm1-udef 3.3%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}\right)} - 1} \]

       3. sqrt-div 3.3%

          \[\leadsto e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)} - 1 \]

       4. metadata-eval 3.3%

          \[\leadsto e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)} - 1 \]

       5. un-div-inv 3.3%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}}\right)} - 1 \]

   12. Applied egg-rr 3.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 3.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right)\right)} \]

       2. expm1-log1p 3.4%

          \[\leadsto \color{blue}{\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}} \]

       3. associate-/l* 3.4%

          \[\leadsto \color{blue}{\frac{0.047619047619047616}{\frac{\sqrt{\pi}}{{x}^{7}}}} \]

   14. Simplified 3.4%

       \[\leadsto \color{blue}{\frac{0.047619047619047616}{\frac{\sqrt{\pi}}{{x}^{7}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\left(2 \cdot x\right) \cdot {\pi}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616}{\frac{\sqrt{\pi}}{{x}^{7}}}\\ \end{array} \]

Alternative 7: 34.4% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \left(2 \cdot x\right) \cdot {\pi}^{-0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* 2.0 x) (pow PI -0.5)))

C

double code(double x) {
	return (2.0 * x) * pow(((double) M_PI), -0.5);
}

Java

public static double code(double x) {
	return (2.0 * x) * Math.pow(Math.PI, -0.5);
}

Python

def code(x):
	return (2.0 * x) * math.pow(math.pi, -0.5)

Julia

function code(x)
	return Float64(Float64(2.0 * x) * (pi ^ -0.5))
end

MATLAB

function tmp = code(x)
	tmp = (2.0 * x) * (pi ^ -0.5);
end

Wolfram

code[x_] := N[(N[(2.0 * x), $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|}} \]
4. Step-by-step derivation

   1. add-sqr-sqrt 33.3%

      \[\leadsto \frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|} \]

   2. fabs-sqr 33.3%

      \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|} \]

   3. add-sqr-sqrt 33.3%

      \[\leadsto \frac{\sqrt{x} \cdot \sqrt{x}}{\left|\color{blue}{\sqrt{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}} \cdot \sqrt{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}}}\right|} \]

   4. fabs-sqr 33.3%

      \[\leadsto \frac{\sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}} \cdot \sqrt{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}}}} \]

   5. add-sqr-sqrt 34.5%

      \[\leadsto \frac{\color{blue}{x}}{\sqrt{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}} \cdot \sqrt{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}}} \]

   6. add-sqr-sqrt 34.4%

      \[\leadsto \frac{x}{\color{blue}{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}}} \]

   7. *-un-lft-identity 34.4%

      \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}} \]

5. Applied egg-rr 34.6%

   \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \frac{x}{\frac{1}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)}}} \]
6. Step-by-step derivation

   1. associate-/r/ 34.6%

      \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(\frac{x}{1} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\right)} \]

   2. /-rgt-identity 34.6%

      \[\leadsto {\pi}^{-0.5} \cdot \left(\color{blue}{x} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\right) \]

   3. *-commutative 34.6%

      \[\leadsto \color{blue}{\left(x \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\right) \cdot {\pi}^{-0.5}} \]

   4. associate-*l* 34.6%

      \[\leadsto \color{blue}{x \cdot \left(\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot {\pi}^{-0.5}\right)} \]

   5. fma-def 34.6%

      \[\leadsto x \cdot \left(\left(\color{blue}{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot {\pi}^{-0.5}\right) \]

   6. +-commutative 34.6%

      \[\leadsto x \cdot \left(\left(\color{blue}{\left(0.047619047619047616 \cdot {x}^{6} + 0.2 \cdot {x}^{4}\right)} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot {\pi}^{-0.5}\right) \]

   7. fma-def 34.6%

      \[\leadsto x \cdot \left(\left(\color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 0.2 \cdot {x}^{4}\right)} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot {\pi}^{-0.5}\right) \]

   8. *-commutative 34.6%

      \[\leadsto x \cdot \left(\left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \color{blue}{{x}^{4} \cdot 0.2}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot {\pi}^{-0.5}\right) \]

7. Simplified 34.6%

   \[\leadsto \color{blue}{x \cdot \left(\left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, {x}^{4} \cdot 0.2\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot {\pi}^{-0.5}\right)} \]

8. Taylor expanded in x around 0 34.1%

   \[\leadsto \color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)} \]
9. Step-by-step derivation

   1. associate-*r* 34.1%

      \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} \]

10. Simplified 34.1%

    \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} \]
11. Step-by-step derivation

    1. sqrt-div 34.1%

       \[\leadsto \left(2 \cdot x\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}} \]

    2. metadata-eval 34.1%

       \[\leadsto \left(2 \cdot x\right) \cdot \frac{\color{blue}{1}}{\sqrt{\pi}} \]

    3. un-div-inv 33.9%

       \[\leadsto \color{blue}{\frac{2 \cdot x}{\sqrt{\pi}}} \]

    4. *-commutative 33.9%

       \[\leadsto \frac{\color{blue}{x \cdot 2}}{\sqrt{\pi}} \]

12. Applied egg-rr 33.9%

    \[\leadsto \color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}} \]
13. Step-by-step derivation

    1. div-inv 34.1%

       \[\leadsto \color{blue}{\left(x \cdot 2\right) \cdot \frac{1}{\sqrt{\pi}}} \]

    2. pow1/2 34.1%

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

    3. pow-flip 34.1%

       \[\leadsto \left(x \cdot 2\right) \cdot \color{blue}{{\pi}^{\left(-0.5\right)}} \]

    4. metadata-eval 34.1%

       \[\leadsto \left(x \cdot 2\right) \cdot {\pi}^{\color{blue}{-0.5}} \]

14. Applied egg-rr 34.1%

    \[\leadsto \color{blue}{\left(x \cdot 2\right) \cdot {\pi}^{-0.5}} \]

15. Final simplification 34.1%

    \[\leadsto \left(2 \cdot x\right) \cdot {\pi}^{-0.5} \]

Alternative 8: 34.2% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \frac{2 \cdot x}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* 2.0 x) (sqrt PI)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(2.0 * x), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|}} \]
4. Step-by-step derivation

   1. add-sqr-sqrt 33.3%

      \[\leadsto \frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|} \]

   2. fabs-sqr 33.3%

      \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|} \]

   3. add-sqr-sqrt 33.3%

      \[\leadsto \frac{\sqrt{x} \cdot \sqrt{x}}{\left|\color{blue}{\sqrt{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}} \cdot \sqrt{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}}}\right|} \]

   4. fabs-sqr 33.3%

      \[\leadsto \frac{\sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}} \cdot \sqrt{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}}}} \]

   5. add-sqr-sqrt 34.5%

      \[\leadsto \frac{\color{blue}{x}}{\sqrt{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}} \cdot \sqrt{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}}} \]

   6. add-sqr-sqrt 34.4%

      \[\leadsto \frac{x}{\color{blue}{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}}} \]

   7. *-un-lft-identity 34.4%

      \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}} \]

5. Applied egg-rr 34.6%

   \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \frac{x}{\frac{1}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)}}} \]
6. Step-by-step derivation

   1. associate-/r/ 34.6%

      \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(\frac{x}{1} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\right)} \]

   2. /-rgt-identity 34.6%

      \[\leadsto {\pi}^{-0.5} \cdot \left(\color{blue}{x} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\right) \]

   3. *-commutative 34.6%

      \[\leadsto \color{blue}{\left(x \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\right) \cdot {\pi}^{-0.5}} \]

   4. associate-*l* 34.6%

      \[\leadsto \color{blue}{x \cdot \left(\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot {\pi}^{-0.5}\right)} \]

   5. fma-def 34.6%

      \[\leadsto x \cdot \left(\left(\color{blue}{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot {\pi}^{-0.5}\right) \]

   6. +-commutative 34.6%

      \[\leadsto x \cdot \left(\left(\color{blue}{\left(0.047619047619047616 \cdot {x}^{6} + 0.2 \cdot {x}^{4}\right)} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot {\pi}^{-0.5}\right) \]

   7. fma-def 34.6%

      \[\leadsto x \cdot \left(\left(\color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 0.2 \cdot {x}^{4}\right)} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot {\pi}^{-0.5}\right) \]

   8. *-commutative 34.6%

      \[\leadsto x \cdot \left(\left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \color{blue}{{x}^{4} \cdot 0.2}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot {\pi}^{-0.5}\right) \]

7. Simplified 34.6%

   \[\leadsto \color{blue}{x \cdot \left(\left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, {x}^{4} \cdot 0.2\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot {\pi}^{-0.5}\right)} \]

8. Taylor expanded in x around 0 34.1%

   \[\leadsto \color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)} \]
9. Step-by-step derivation

   1. associate-*r* 34.1%

      \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} \]

10. Simplified 34.1%

    \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} \]
11. Step-by-step derivation

    1. sqrt-div 34.1%

       \[\leadsto \left(2 \cdot x\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}} \]

    2. metadata-eval 34.1%

       \[\leadsto \left(2 \cdot x\right) \cdot \frac{\color{blue}{1}}{\sqrt{\pi}} \]

    3. un-div-inv 33.9%

       \[\leadsto \color{blue}{\frac{2 \cdot x}{\sqrt{\pi}}} \]

    4. *-commutative 33.9%

       \[\leadsto \frac{\color{blue}{x \cdot 2}}{\sqrt{\pi}} \]

12. Applied egg-rr 33.9%

    \[\leadsto \color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}} \]

13. Final simplification 33.9%

    \[\leadsto \frac{2 \cdot x}{\sqrt{\pi}} \]

Reproduce

herbie shell --seed 2023336 
(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)))))))