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

Percentage Accurate: 99.8% → 99.8%

Time: 11.9s

Alternatives: 6

Speedup: 1.5×

• 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: 99.8% 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. Add Preprocessing

3. Final simplification 99.9%

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

Alternative 2: 72.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \left|{\pi}^{-0.5} \cdot \left(\left(\left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (pow PI -0.5)
   (+
    (+
     (+ (* 0.6666666666666666 (pow x 3.0)) (* 2.0 x))
     (* 0.2 (* (* (fabs x) (* x x)) (* x x))))
    (* 0.047619047619047616 (* (* x x) (* (* x x) (* x (* x x)))))))))

C

double code(double x) {
	return fabs((pow(((double) M_PI), -0.5) * ((((0.6666666666666666 * pow(x, 3.0)) + (2.0 * x)) + (0.2 * ((fabs(x) * (x * x)) * (x * x)))) + (0.047619047619047616 * ((x * x) * ((x * x) * (x * (x * x))))))));
}

Java

public static double code(double x) {
	return Math.abs((Math.pow(Math.PI, -0.5) * ((((0.6666666666666666 * Math.pow(x, 3.0)) + (2.0 * x)) + (0.2 * ((Math.abs(x) * (x * x)) * (x * x)))) + (0.047619047619047616 * ((x * x) * ((x * x) * (x * (x * x))))))));
}

Python

def code(x):
	return math.fabs((math.pow(math.pi, -0.5) * ((((0.6666666666666666 * math.pow(x, 3.0)) + (2.0 * x)) + (0.2 * ((math.fabs(x) * (x * x)) * (x * x)))) + (0.047619047619047616 * ((x * x) * ((x * x) * (x * (x * x))))))))

Julia

function code(x)
	return abs(Float64((pi ^ -0.5) * Float64(Float64(Float64(Float64(0.6666666666666666 * (x ^ 3.0)) + Float64(2.0 * x)) + Float64(0.2 * Float64(Float64(abs(x) * Float64(x * x)) * Float64(x * x)))) + Float64(0.047619047619047616 * Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(x * Float64(x * x))))))))
end

MATLAB

function tmp = code(x)
	tmp = abs(((pi ^ -0.5) * ((((0.6666666666666666 * (x ^ 3.0)) + (2.0 * x)) + (0.2 * ((abs(x) * (x * x)) * (x * x)))) + (0.047619047619047616 * ((x * x) * ((x * x) * (x * (x * x))))))));
end

Wolfram

code[x_] := N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(N[(N[(N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[(N[(N[Abs[x], $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $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%

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

   1. fma-undefine 99.9%

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

   2. add-sqr-sqrt 99.5%

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

   3. sqrt-prod 70.2%

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

   4. sqr-abs 70.2%

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

   5. pow2 70.2%

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

   6. sqrt-pow1 99.3%

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

   7. metadata-eval 99.3%

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

   8. pow1 99.3%

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

   9. add-sqr-sqrt 99.3%

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

   10. sqrt-prod 99.3%

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

   11. sqr-abs 99.3%

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

   12. pow2 99.3%

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

   13. sqrt-pow1 74.9%

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

   14. metadata-eval 74.9%

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

   15. pow1 74.9%

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

   16. cube-mult 74.9%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot \color{blue}{{x}^{3}}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

6. Applied egg-rr 74.9%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\color{blue}{\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)} + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]
7. Step-by-step derivation

   1. add-sqr-sqrt 74.9%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\left|x\right|} \cdot \sqrt{\left|x\right|}\right)} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   2. sqrt-prod 74.9%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{\sqrt{\left|x\right| \cdot \left|x\right|}} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   3. sqr-abs 74.9%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\sqrt{\color{blue}{x \cdot x}} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   4. pow2 74.9%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\sqrt{\color{blue}{{x}^{2}}} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   5. sqrt-pow1 71.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{{x}^{\left(\frac{2}{2}\right)}} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   6. metadata-eval 71.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left({x}^{\color{blue}{1}} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   7. pow1 71.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{x} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   8. *-un-lft-identity 71.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{\left(1 \cdot x\right)} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

8. Applied egg-rr 71.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{\left(1 \cdot x\right)} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]
9. Step-by-step derivation

   1. *-lft-identity 71.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{x} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

10. Simplified 71.8%

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{x} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]
11. Step-by-step derivation

    1. *-un-lft-identity 71.8%

       \[\leadsto \left|\color{blue}{\left(1 \cdot \frac{1}{\sqrt{\pi}}\right)} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

    2. pow1/2 71.8%

       \[\leadsto \left|\left(1 \cdot \frac{1}{\color{blue}{{\pi}^{0.5}}}\right) \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

    3. pow-flip 71.8%

       \[\leadsto \left|\left(1 \cdot \color{blue}{{\pi}^{\left(-0.5\right)}}\right) \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

    4. metadata-eval 71.8%

       \[\leadsto \left|\left(1 \cdot {\pi}^{\color{blue}{-0.5}}\right) \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

12. Applied egg-rr 71.8%

    \[\leadsto \left|\color{blue}{\left(1 \cdot {\pi}^{-0.5}\right)} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]
13. Step-by-step derivation

    1. *-lft-identity 71.8%

       \[\leadsto \left|\color{blue}{{\pi}^{-0.5}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

14. Simplified 71.8%

    \[\leadsto \left|\color{blue}{{\pi}^{-0.5}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

15. Final simplification 71.8%

    \[\leadsto \left|{\pi}^{-0.5} \cdot \left(\left(\left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right| \]
16. Add Preprocessing

Alternative 3: 98.8% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + 2 \cdot x\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt PI))
   (+
    (* 0.047619047619047616 (* (* x x) (* (* x x) (* x (* x x)))))
    (* 2.0 x)))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((0.047619047619047616 * ((x * x) * ((x * x) * (x * (x * x))))) + (2.0 * x))));
}

Java

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

Python

def code(x):
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((0.047619047619047616 * ((x * x) * ((x * x) * (x * (x * x))))) + (2.0 * x))))

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(0.047619047619047616 * Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(x * Float64(x * x))))) + Float64(2.0 * x))))
end

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(0.047619047619047616 * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * x), $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%

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

5. Taylor expanded in x around 0 99.2%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{2 \cdot \left|x\right|} + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]
6. Step-by-step derivation

   1. *-commutative 99.2%

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

   2. rem-square-sqrt 35.9%

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

   3. fabs-sqr 35.9%

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

   4. rem-square-sqrt 98.8%

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

7. Simplified 98.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{x \cdot 2} + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]
8. Step-by-step derivation

   1. add-sqr-sqrt 74.9%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\left|x\right|} \cdot \sqrt{\left|x\right|}\right)} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   2. sqrt-prod 74.9%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{\sqrt{\left|x\right| \cdot \left|x\right|}} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   3. sqr-abs 74.9%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\sqrt{\color{blue}{x \cdot x}} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   4. pow2 74.9%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\sqrt{\color{blue}{{x}^{2}}} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   5. sqrt-pow1 71.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{{x}^{\left(\frac{2}{2}\right)}} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   6. metadata-eval 71.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left({x}^{\color{blue}{1}} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   7. pow1 71.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{x} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   8. *-un-lft-identity 71.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{\left(1 \cdot x\right)} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

9. Applied egg-rr 99.2%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(x \cdot 2 + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{\left(1 \cdot x\right)} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]
10. Step-by-step derivation

    1. *-lft-identity 71.8%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{x} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

11. Simplified 99.2%

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

12. Final simplification 99.2%

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) + 2 \cdot x\right)\right| \]
13. Add Preprocessing

Alternative 4: 34.5% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (* x (/ 2.0 (sqrt PI)))
   (* 0.047619047619047616 (/ (pow x 7.0) (sqrt PI)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[x, 7.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   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 99.3%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{3} \cdot \left({\left(\left|x\right|\right)}^{3} \cdot \left|x\right|\right), \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{3} \cdot \left(x \cdot x\right), \mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)\right)\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.1%

      \[\leadsto \left|\frac{\color{blue}{0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7} + \left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}}{\sqrt{\pi}}\right| \]
   6. Step-by-step derivation

      1. fma-define 99.1%

         \[\leadsto \left|\frac{\color{blue}{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}}{\sqrt{\pi}}\right| \]

      2. rem-square-sqrt 36.1%

         \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      3. fabs-sqr 36.1%

         \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      4. rem-square-sqrt 74.1%

         \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {\color{blue}{x}}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      5. *-commutative 74.1%

         \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{{\left(\left|x\right|\right)}^{3} \cdot 0.6666666666666666} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      6. *-commutative 74.1%

         \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      7. unpow3 74.1%

         \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.6666666666666666 \cdot \color{blue}{\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      8. sqr-abs 74.1%

         \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.6666666666666666 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left|x\right|\right) + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      9. unpow2 74.1%

         \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.6666666666666666 \cdot \left(\color{blue}{{x}^{2}} \cdot \left|x\right|\right) + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      10. associate-*r* 74.1%

          \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{\left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \left|x\right|} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      11. rem-square-sqrt 36.1%

          \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      12. fabs-sqr 36.1%

          \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      13. rem-square-sqrt 98.6%

          \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \color{blue}{x} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      14. rem-square-sqrt 36.1%

          \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \left(0.6666666666666666 \cdot {x}^{2}\right) \cdot x + 2 \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}{\sqrt{\pi}}\right| \]

      15. fabs-sqr 36.1%

          \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \left(0.6666666666666666 \cdot {x}^{2}\right) \cdot x + 2 \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right)}{\sqrt{\pi}}\right| \]

      16. rem-square-sqrt 99.1%

          \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \left(0.6666666666666666 \cdot {x}^{2}\right) \cdot x + 2 \cdot \color{blue}{x}\right)}{\sqrt{\pi}}\right| \]

      17. distribute-rgt-out 99.1%

          \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{x \cdot \left(0.6666666666666666 \cdot {x}^{2} + 2\right)}\right)}{\sqrt{\pi}}\right| \]

      18. fma-define 99.1%

          \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, x \cdot \color{blue}{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)}\right)}{\sqrt{\pi}}\right| \]

   7. Simplified 99.1%

      \[\leadsto \left|\frac{\color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, x \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}}{\sqrt{\pi}}\right| \]

   8. Taylor expanded in x around 0 92.6%

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

   9. Taylor expanded in x around 0 69.0%

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

       1. *-commutative 69.0%

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

   11. Simplified 69.0%

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

       1. add-sqr-sqrt 35.7%

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

       2. fabs-sqr 35.7%

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

       3. add-sqr-sqrt 37.1%

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

       4. associate-/l* 37.5%

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

       5. *-commutative 37.5%

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

   13. Applied egg-rr 37.5%

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

   if 1.8500000000000001 < 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 99.3%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{3} \cdot \left({\left(\left|x\right|\right)}^{3} \cdot \left|x\right|\right), \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{3} \cdot \left(x \cdot x\right), \mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)\right)\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.1%

      \[\leadsto \left|\frac{\color{blue}{0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7} + \left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}}{\sqrt{\pi}}\right| \]
   6. Step-by-step derivation

      1. fma-define 99.1%

         \[\leadsto \left|\frac{\color{blue}{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}}{\sqrt{\pi}}\right| \]

      2. rem-square-sqrt 36.1%

         \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      3. fabs-sqr 36.1%

         \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      4. rem-square-sqrt 74.1%

         \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {\color{blue}{x}}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      5. *-commutative 74.1%

         \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{{\left(\left|x\right|\right)}^{3} \cdot 0.6666666666666666} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      6. *-commutative 74.1%

         \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      7. unpow3 74.1%

         \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.6666666666666666 \cdot \color{blue}{\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      8. sqr-abs 74.1%

         \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.6666666666666666 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left|x\right|\right) + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      9. unpow2 74.1%

         \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.6666666666666666 \cdot \left(\color{blue}{{x}^{2}} \cdot \left|x\right|\right) + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      10. associate-*r* 74.1%

          \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{\left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \left|x\right|} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      11. rem-square-sqrt 36.1%

          \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      12. fabs-sqr 36.1%

          \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      13. rem-square-sqrt 98.6%

          \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \color{blue}{x} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      14. rem-square-sqrt 36.1%

          \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \left(0.6666666666666666 \cdot {x}^{2}\right) \cdot x + 2 \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}{\sqrt{\pi}}\right| \]

      15. fabs-sqr 36.1%

          \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \left(0.6666666666666666 \cdot {x}^{2}\right) \cdot x + 2 \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right)}{\sqrt{\pi}}\right| \]

      16. rem-square-sqrt 99.1%

          \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \left(0.6666666666666666 \cdot {x}^{2}\right) \cdot x + 2 \cdot \color{blue}{x}\right)}{\sqrt{\pi}}\right| \]

      17. distribute-rgt-out 99.1%

          \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{x \cdot \left(0.6666666666666666 \cdot {x}^{2} + 2\right)}\right)}{\sqrt{\pi}}\right| \]

      18. fma-define 99.1%

          \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, x \cdot \color{blue}{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)}\right)}{\sqrt{\pi}}\right| \]

   7. Simplified 99.1%

      \[\leadsto \left|\frac{\color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, x \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}}{\sqrt{\pi}}\right| \]

   8. Taylor expanded in x around inf 35.9%

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

      1. add-sqr-sqrt 3.6%

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

      2. fabs-sqr 3.6%

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

      3. add-sqr-sqrt 3.7%

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

      4. associate-/l* 3.7%

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

      5. *-commutative 3.7%

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

   10. Applied egg-rr 3.7%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 34.5% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.78:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{10} \cdot \frac{0.04}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.78) (* x (/ 2.0 (sqrt PI))) (sqrt (* (pow x 10.0) (/ 0.04 PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.78) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = sqrt((pow(x, 10.0) * (0.04 / ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.78) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.sqrt((Math.pow(x, 10.0) * (0.04 / Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.78:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.sqrt((math.pow(x, 10.0) * (0.04 / math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.78)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = sqrt(Float64((x ^ 10.0) * Float64(0.04 / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.78)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = sqrt(((x ^ 10.0) * (0.04 / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.78], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[x, 10.0], $MachinePrecision] * N[(0.04 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.78000000000000003

   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 99.3%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{3} \cdot \left({\left(\left|x\right|\right)}^{3} \cdot \left|x\right|\right), \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{3} \cdot \left(x \cdot x\right), \mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)\right)\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.1%

      \[\leadsto \left|\frac{\color{blue}{0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7} + \left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}}{\sqrt{\pi}}\right| \]
   6. Step-by-step derivation

      1. fma-define 99.1%

         \[\leadsto \left|\frac{\color{blue}{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}}{\sqrt{\pi}}\right| \]

      2. rem-square-sqrt 36.1%

         \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      3. fabs-sqr 36.1%

         \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      4. rem-square-sqrt 74.1%

         \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {\color{blue}{x}}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      5. *-commutative 74.1%

         \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{{\left(\left|x\right|\right)}^{3} \cdot 0.6666666666666666} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      6. *-commutative 74.1%

         \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      7. unpow3 74.1%

         \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.6666666666666666 \cdot \color{blue}{\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      8. sqr-abs 74.1%

         \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.6666666666666666 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left|x\right|\right) + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      9. unpow2 74.1%

         \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.6666666666666666 \cdot \left(\color{blue}{{x}^{2}} \cdot \left|x\right|\right) + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      10. associate-*r* 74.1%

          \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{\left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \left|x\right|} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      11. rem-square-sqrt 36.1%

          \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      12. fabs-sqr 36.1%

          \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      13. rem-square-sqrt 98.6%

          \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \color{blue}{x} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

      14. rem-square-sqrt 36.1%

          \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \left(0.6666666666666666 \cdot {x}^{2}\right) \cdot x + 2 \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}{\sqrt{\pi}}\right| \]

      15. fabs-sqr 36.1%

          \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \left(0.6666666666666666 \cdot {x}^{2}\right) \cdot x + 2 \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right)}{\sqrt{\pi}}\right| \]

      16. rem-square-sqrt 99.1%

          \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \left(0.6666666666666666 \cdot {x}^{2}\right) \cdot x + 2 \cdot \color{blue}{x}\right)}{\sqrt{\pi}}\right| \]

      17. distribute-rgt-out 99.1%

          \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{x \cdot \left(0.6666666666666666 \cdot {x}^{2} + 2\right)}\right)}{\sqrt{\pi}}\right| \]

      18. fma-define 99.1%

          \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, x \cdot \color{blue}{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)}\right)}{\sqrt{\pi}}\right| \]

   7. Simplified 99.1%

      \[\leadsto \left|\frac{\color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, x \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}}{\sqrt{\pi}}\right| \]

   8. Taylor expanded in x around 0 92.6%

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

   9. Taylor expanded in x around 0 69.0%

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

       1. *-commutative 69.0%

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

   11. Simplified 69.0%

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

       1. add-sqr-sqrt 35.7%

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

       2. fabs-sqr 35.7%

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

       3. add-sqr-sqrt 37.1%

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

       4. associate-/l* 37.5%

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

       5. *-commutative 37.5%

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

   13. Applied egg-rr 37.5%

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

   if 1.78000000000000003 < 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 99.3%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{3} \cdot \left({\left(\left|x\right|\right)}^{3} \cdot \left|x\right|\right), \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{3} \cdot \left(x \cdot x\right), \mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)\right)\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 32.4%

      \[\leadsto \left|\color{blue}{0.2 \cdot \left(\left({x}^{2} \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   6. Step-by-step derivation

      1. *-commutative 32.4%

         \[\leadsto \left|\color{blue}{\left(\left({x}^{2} \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}\right) \cdot 0.2}\right| \]

      2. associate-*l* 32.4%

         \[\leadsto \left|\color{blue}{\left({x}^{2} \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.2\right)}\right| \]

      3. rem-square-sqrt 2.1%

         \[\leadsto \left|\left({x}^{2} \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.2\right)\right| \]

      4. fabs-sqr 2.1%

         \[\leadsto \left|\left({x}^{2} \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.2\right)\right| \]

      5. rem-square-sqrt 32.4%

         \[\leadsto \left|\left({x}^{2} \cdot {\color{blue}{x}}^{3}\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.2\right)\right| \]

      6. metadata-eval 32.4%

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

      7. pow-plus 32.4%

         \[\leadsto \left|\left({x}^{2} \cdot \color{blue}{\left({x}^{2} \cdot x\right)}\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.2\right)\right| \]

      8. associate-*r* 32.4%

         \[\leadsto \left|\color{blue}{\left(\left({x}^{2} \cdot {x}^{2}\right) \cdot x\right)} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.2\right)\right| \]

      9. pow-sqr 32.4%

         \[\leadsto \left|\left(\color{blue}{{x}^{\left(2 \cdot 2\right)}} \cdot x\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.2\right)\right| \]

      10. metadata-eval 32.4%

          \[\leadsto \left|\left({x}^{\color{blue}{4}} \cdot x\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.2\right)\right| \]

      11. pow-plus 32.4%

          \[\leadsto \left|\color{blue}{{x}^{\left(4 + 1\right)}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.2\right)\right| \]

      12. metadata-eval 32.4%

          \[\leadsto \left|{x}^{\color{blue}{5}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.2\right)\right| \]

      13. unpow-1 32.4%

          \[\leadsto \left|{x}^{5} \cdot \left(\sqrt{\color{blue}{{\pi}^{-1}}} \cdot 0.2\right)\right| \]

      14. metadata-eval 32.4%

          \[\leadsto \left|{x}^{5} \cdot \left(\sqrt{{\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot 0.2\right)\right| \]

      15. pow-sqr 32.4%

          \[\leadsto \left|{x}^{5} \cdot \left(\sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}} \cdot 0.2\right)\right| \]

      16. rem-sqrt-square 32.4%

          \[\leadsto \left|{x}^{5} \cdot \left(\color{blue}{\left|{\pi}^{-0.5}\right|} \cdot 0.2\right)\right| \]

      17. rem-square-sqrt 32.4%

          \[\leadsto \left|{x}^{5} \cdot \left(\left|\color{blue}{\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}}\right| \cdot 0.2\right)\right| \]

      18. fabs-sqr 32.4%

          \[\leadsto \left|{x}^{5} \cdot \left(\color{blue}{\left(\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}\right)} \cdot 0.2\right)\right| \]

      19. rem-square-sqrt 32.4%

          \[\leadsto \left|{x}^{5} \cdot \left(\color{blue}{{\pi}^{-0.5}} \cdot 0.2\right)\right| \]

   7. Simplified 32.4%

      \[\leadsto \left|\color{blue}{{x}^{5} \cdot \left({\pi}^{-0.5} \cdot 0.2\right)}\right| \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 3.6%

         \[\leadsto \left|\color{blue}{\sqrt{{x}^{5} \cdot \left({\pi}^{-0.5} \cdot 0.2\right)} \cdot \sqrt{{x}^{5} \cdot \left({\pi}^{-0.5} \cdot 0.2\right)}}\right| \]

      2. fabs-sqr 3.6%

         \[\leadsto \color{blue}{\sqrt{{x}^{5} \cdot \left({\pi}^{-0.5} \cdot 0.2\right)} \cdot \sqrt{{x}^{5} \cdot \left({\pi}^{-0.5} \cdot 0.2\right)}} \]

      3. sqrt-unprod 33.8%

         \[\leadsto \color{blue}{\sqrt{\left({x}^{5} \cdot \left({\pi}^{-0.5} \cdot 0.2\right)\right) \cdot \left({x}^{5} \cdot \left({\pi}^{-0.5} \cdot 0.2\right)\right)}} \]

      4. swap-sqr 33.8%

         \[\leadsto \sqrt{\color{blue}{\left({x}^{5} \cdot {x}^{5}\right) \cdot \left(\left({\pi}^{-0.5} \cdot 0.2\right) \cdot \left({\pi}^{-0.5} \cdot 0.2\right)\right)}} \]

      5. pow-prod-up 33.8%

         \[\leadsto \sqrt{\color{blue}{{x}^{\left(5 + 5\right)}} \cdot \left(\left({\pi}^{-0.5} \cdot 0.2\right) \cdot \left({\pi}^{-0.5} \cdot 0.2\right)\right)} \]

      6. metadata-eval 33.8%

         \[\leadsto \sqrt{{x}^{\color{blue}{10}} \cdot \left(\left({\pi}^{-0.5} \cdot 0.2\right) \cdot \left({\pi}^{-0.5} \cdot 0.2\right)\right)} \]

      7. metadata-eval 33.8%

         \[\leadsto \sqrt{{x}^{10} \cdot \left(\left({\pi}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot 0.2\right) \cdot \left({\pi}^{-0.5} \cdot 0.2\right)\right)} \]

      8. sqrt-pow2 33.8%

         \[\leadsto \sqrt{{x}^{10} \cdot \left(\left(\color{blue}{{\left(\sqrt{\pi}\right)}^{-1}} \cdot 0.2\right) \cdot \left({\pi}^{-0.5} \cdot 0.2\right)\right)} \]

      9. inv-pow 33.8%

         \[\leadsto \sqrt{{x}^{10} \cdot \left(\left(\color{blue}{\frac{1}{\sqrt{\pi}}} \cdot 0.2\right) \cdot \left({\pi}^{-0.5} \cdot 0.2\right)\right)} \]

      10. metadata-eval 33.8%

          \[\leadsto \sqrt{{x}^{10} \cdot \left(\left(\frac{1}{\sqrt{\pi}} \cdot 0.2\right) \cdot \left({\pi}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot 0.2\right)\right)} \]

      11. sqrt-pow2 33.8%

          \[\leadsto \sqrt{{x}^{10} \cdot \left(\left(\frac{1}{\sqrt{\pi}} \cdot 0.2\right) \cdot \left(\color{blue}{{\left(\sqrt{\pi}\right)}^{-1}} \cdot 0.2\right)\right)} \]

      12. inv-pow 33.8%

          \[\leadsto \sqrt{{x}^{10} \cdot \left(\left(\frac{1}{\sqrt{\pi}} \cdot 0.2\right) \cdot \left(\color{blue}{\frac{1}{\sqrt{\pi}}} \cdot 0.2\right)\right)} \]

      13. swap-sqr 33.8%

          \[\leadsto \sqrt{{x}^{10} \cdot \color{blue}{\left(\left(\frac{1}{\sqrt{\pi}} \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \left(0.2 \cdot 0.2\right)\right)}} \]

   9. Applied egg-rr 33.8%

      \[\leadsto \color{blue}{\sqrt{{x}^{10} \cdot \left(\frac{1}{\pi} \cdot 0.04\right)}} \]
   10. Step-by-step derivation

       1. associate-*l/ 33.8%

          \[\leadsto \sqrt{{x}^{10} \cdot \color{blue}{\frac{1 \cdot 0.04}{\pi}}} \]

       2. metadata-eval 33.8%

          \[\leadsto \sqrt{{x}^{10} \cdot \frac{\color{blue}{0.04}}{\pi}} \]

   11. Simplified 33.8%

       \[\leadsto \color{blue}{\sqrt{{x}^{10} \cdot \frac{0.04}{\pi}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.78:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{10} \cdot \frac{0.04}{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 34.5% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(x * N[(2.0 / 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 99.3%

   \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{3} \cdot \left({\left(\left|x\right|\right)}^{3} \cdot \left|x\right|\right), \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{3} \cdot \left(x \cdot x\right), \mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)\right)\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 99.1%

   \[\leadsto \left|\frac{\color{blue}{0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7} + \left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}}{\sqrt{\pi}}\right| \]
6. Step-by-step derivation

   1. fma-define 99.1%

      \[\leadsto \left|\frac{\color{blue}{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}}{\sqrt{\pi}}\right| \]

   2. rem-square-sqrt 36.1%

      \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

   3. fabs-sqr 36.1%

      \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

   4. rem-square-sqrt 74.1%

      \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {\color{blue}{x}}^{7}, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

   5. *-commutative 74.1%

      \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{{\left(\left|x\right|\right)}^{3} \cdot 0.6666666666666666} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

   6. *-commutative 74.1%

      \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

   7. unpow3 74.1%

      \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.6666666666666666 \cdot \color{blue}{\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

   8. sqr-abs 74.1%

      \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.6666666666666666 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left|x\right|\right) + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

   9. unpow2 74.1%

      \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.6666666666666666 \cdot \left(\color{blue}{{x}^{2}} \cdot \left|x\right|\right) + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

   10. associate-*r* 74.1%

       \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{\left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \left|x\right|} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

   11. rem-square-sqrt 36.1%

       \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

   12. fabs-sqr 36.1%

       \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

   13. rem-square-sqrt 98.6%

       \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \color{blue}{x} + 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

   14. rem-square-sqrt 36.1%

       \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \left(0.6666666666666666 \cdot {x}^{2}\right) \cdot x + 2 \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}{\sqrt{\pi}}\right| \]

   15. fabs-sqr 36.1%

       \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \left(0.6666666666666666 \cdot {x}^{2}\right) \cdot x + 2 \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right)}{\sqrt{\pi}}\right| \]

   16. rem-square-sqrt 99.1%

       \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \left(0.6666666666666666 \cdot {x}^{2}\right) \cdot x + 2 \cdot \color{blue}{x}\right)}{\sqrt{\pi}}\right| \]

   17. distribute-rgt-out 99.1%

       \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{x \cdot \left(0.6666666666666666 \cdot {x}^{2} + 2\right)}\right)}{\sqrt{\pi}}\right| \]

   18. fma-define 99.1%

       \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, x \cdot \color{blue}{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)}\right)}{\sqrt{\pi}}\right| \]

7. Simplified 99.1%

   \[\leadsto \left|\frac{\color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, x \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}}{\sqrt{\pi}}\right| \]

8. Taylor expanded in x around 0 92.6%

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

9. Taylor expanded in x around 0 69.0%

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

    1. *-commutative 69.0%

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

11. Simplified 69.0%

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

    1. add-sqr-sqrt 35.7%

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

    2. fabs-sqr 35.7%

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

    3. add-sqr-sqrt 37.1%

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

    4. associate-/l* 37.5%

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

    5. *-commutative 37.5%

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

13. Applied egg-rr 37.5%

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

14. Final simplification 37.5%

    \[\leadsto x \cdot \frac{2}{\sqrt{\pi}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(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)))))))