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

Percentage Accurate: 99.8% → 99.8%

Time: 12.8s

Alternatives: 8

Speedup: 1.5×

• 28.6% 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: 99.8% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot {x}^{5}\right) + 0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt PI))
   (+
    (+ (+ (* 2.0 x) (* 0.6666666666666666 (pow x 3.0))) (* 0.2 (pow x 5.0)))
    (* 0.047619047619047616 (* (* x x) (* (* x x) (* x (* x x)))))))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * x) + (0.6666666666666666 * pow(x, 3.0))) + (0.2 * pow(x, 5.0))) + (0.047619047619047616 * ((x * x) * ((x * x) * (x * (x * x))))))));
}

Java

public static double code(double x) {
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * x) + (0.6666666666666666 * Math.pow(x, 3.0))) + (0.2 * Math.pow(x, 5.0))) + (0.047619047619047616 * ((x * x) * ((x * x) * (x * (x * x))))))));
}

Python

def code(x):
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * x) + (0.6666666666666666 * math.pow(x, 3.0))) + (0.2 * math.pow(x, 5.0))) + (0.047619047619047616 * ((x * x) * ((x * x) * (x * (x * x))))))))

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * x) + Float64(0.6666666666666666 * (x ^ 3.0))) + Float64(0.2 * (x ^ 5.0))) + Float64(0.047619047619047616 * Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(x * Float64(x * x))))))))
end

MATLAB

function tmp = code(x)
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * x) + (0.6666666666666666 * (x ^ 3.0))) + (0.2 * (x ^ 5.0))) + (0.047619047619047616 * ((x * x) * ((x * x) * (x * (x * x))))))));
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * x), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[Power[x, 5.0], $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 34.5%

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

   3. fabs-sqr 34.5%

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

   4. add-sqr-sqrt 99.6%

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

   5. add-sqr-sqrt 34.7%

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

   6. fabs-sqr 34.7%

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

   7. add-sqr-sqrt 75.9%

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

   8. cube-mult 75.9%

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

6. Applied egg-rr 75.9%

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

7. Taylor expanded in x around 0 75.9%

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

   1. rem-square-sqrt 34.7%

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

   2. fabs-sqr 34.7%

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

   3. rem-square-sqrt 69.2%

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

   4. pow-plus 69.2%

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

   5. metadata-eval 69.2%

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

9. Simplified 69.2%

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

    1. add-sqr-sqrt 34.7%

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

    2. fabs-sqr 34.7%

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

    3. add-sqr-sqrt 99.9%

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

    4. *-un-lft-identity 99.9%

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

11. Applied egg-rr 99.9%

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

    1. *-lft-identity 99.9%

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

13. Simplified 99.9%

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

14. Taylor expanded in x around 0 99.9%

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + \color{blue}{0.2 \cdot {x}^{5}}\right) + 0.047619047619047616 \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

15. Final simplification 99.9%

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot {x}^{5}\right) + 0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right| \]
16. Add Preprocessing

Alternative 3: 34.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;x \cdot \left|\frac{0.2 \cdot {x}^{4} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{7} \cdot \left({\pi}^{-0.5} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (*
    x
    (fabs
     (/
      (+ (* 0.2 (pow x 4.0)) (fma 0.6666666666666666 (* x x) 2.0))
      (sqrt PI))))
   (fabs
    (*
     (pow x 7.0)
     (* (pow PI -0.5) (+ 0.047619047619047616 (/ 0.2 (pow x 2.0))))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = x * fabs((((0.2 * pow(x, 4.0)) + fma(0.6666666666666666, (x * x), 2.0)) / sqrt(((double) M_PI))));
	} else {
		tmp = fabs((pow(x, 7.0) * (pow(((double) M_PI), -0.5) * (0.047619047619047616 + (0.2 / pow(x, 2.0))))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = Float64(x * abs(Float64(Float64(Float64(0.2 * (x ^ 4.0)) + fma(0.6666666666666666, Float64(x * x), 2.0)) / sqrt(pi))));
	else
		tmp = abs(Float64((x ^ 7.0) * Float64((pi ^ -0.5) * Float64(0.047619047619047616 + Float64(0.2 / (x ^ 2.0))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[(x * N[Abs[N[(N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[Power[x, 7.0], $MachinePrecision] * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(0.047619047619047616 + N[(0.2 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   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%

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

   5. Taylor expanded in x around 0 95.1%

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

      1. add-sqr-sqrt 34.7%

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

      2. fabs-sqr 34.7%

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

      3. add-sqr-sqrt 99.9%

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

      4. *-un-lft-identity 99.9%

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

   7. Applied egg-rr 35.8%

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

      1. *-lft-identity 99.9%

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

   9. Simplified 35.8%

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

   if 2.2000000000000002 < 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 99.4%

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

   5. Taylor expanded in x around 0 98.6%

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

      1. rem-square-sqrt 33.9%

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

      2. fabs-sqr 33.9%

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

      3. rem-square-sqrt 98.6%

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

      4. +-commutative 98.6%

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

      5. fma-define 98.6%

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

      6. rem-square-sqrt 34.0%

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

      7. fabs-sqr 34.0%

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

      8. rem-square-sqrt 98.6%

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

      9. rem-square-sqrt 34.0%

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

      10. fabs-sqr 34.0%

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

      11. rem-square-sqrt 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in x around inf 36.6%

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

      1. unpow-1 36.6%

         \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot \sqrt{\color{blue}{{\pi}^{-1}}} + 0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right| \]

      2. metadata-eval 36.6%

         \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot \sqrt{{\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}} + 0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right| \]

      3. pow-sqr 36.6%

         \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot \sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}} + 0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right| \]

      4. rem-sqrt-square 36.6%

         \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot \color{blue}{\left|{\pi}^{-0.5}\right|} + 0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right| \]

      5. rem-square-sqrt 36.6%

         \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot \left|\color{blue}{\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}}\right| + 0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right| \]

      6. fabs-sqr 36.6%

         \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot \color{blue}{\left(\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}\right)} + 0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right| \]

      7. rem-square-sqrt 36.6%

         \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot \color{blue}{{\pi}^{-0.5}} + 0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right| \]

      8. associate-*r* 36.6%

         \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5} + \color{blue}{\left(0.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \sqrt{\frac{1}{\pi}}}\right)\right| \]

      9. unpow-1 36.6%

         \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5} + \left(0.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \sqrt{\color{blue}{{\pi}^{-1}}}\right)\right| \]

      10. metadata-eval 36.6%

          \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5} + \left(0.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \sqrt{{\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right)\right| \]

      11. pow-sqr 36.6%

          \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5} + \left(0.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}}\right)\right| \]

      12. rem-sqrt-square 36.6%

          \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5} + \left(0.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \color{blue}{\left|{\pi}^{-0.5}\right|}\right)\right| \]

      13. rem-square-sqrt 36.6%

          \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5} + \left(0.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \left|\color{blue}{\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}}\right|\right)\right| \]

      14. fabs-sqr 36.6%

          \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5} + \left(0.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \color{blue}{\left(\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}\right)}\right)\right| \]

      15. rem-square-sqrt 36.6%

          \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5} + \left(0.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \color{blue}{{\pi}^{-0.5}}\right)\right| \]

   10. Simplified 36.6%

       \[\leadsto \left|\color{blue}{{x}^{7} \cdot \left({\pi}^{-0.5} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 34.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{7} \cdot \left({\pi}^{-0.5} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.6)
   (* x (/ 2.0 (sqrt PI)))
   (fabs
    (*
     (pow x 7.0)
     (* (pow PI -0.5) (+ 0.047619047619047616 (/ 0.2 (pow x 2.0))))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = fabs((pow(x, 7.0) * (pow(((double) M_PI), -0.5) * (0.047619047619047616 + (0.2 / pow(x, 2.0))))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.abs((Math.pow(x, 7.0) * (Math.pow(Math.PI, -0.5) * (0.047619047619047616 + (0.2 / Math.pow(x, 2.0))))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.6:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.fabs((math.pow(x, 7.0) * (math.pow(math.pi, -0.5) * (0.047619047619047616 + (0.2 / math.pow(x, 2.0))))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.6)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = abs(Float64((x ^ 7.0) * Float64((pi ^ -0.5) * Float64(0.047619047619047616 + Float64(0.2 / (x ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.6)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = abs(((x ^ 7.0) * ((pi ^ -0.5) * (0.047619047619047616 + (0.2 / (x ^ 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.6], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[Power[x, 7.0], $MachinePrecision] * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(0.047619047619047616 + N[(0.2 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001

   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 99.4%

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

   5. Taylor expanded in x around 0 98.6%

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

      1. rem-square-sqrt 33.9%

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

      2. fabs-sqr 33.9%

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

      3. rem-square-sqrt 98.6%

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

      4. +-commutative 98.6%

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

      5. fma-define 98.6%

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

      6. rem-square-sqrt 34.0%

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

      7. fabs-sqr 34.0%

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

      8. rem-square-sqrt 98.6%

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

      9. rem-square-sqrt 34.0%

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

      10. fabs-sqr 34.0%

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

      11. rem-square-sqrt 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in x around 0 65.8%

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

      1. *-commutative 65.8%

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

      2. unpow1/2 65.8%

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

      3. rem-exp-log 65.8%

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

      4. exp-neg 65.8%

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

      5. exp-prod 65.8%

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

      6. distribute-lft-neg-out 65.8%

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

      7. exp-neg 65.8%

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

      8. exp-to-pow 65.8%

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

      9. unpow1/2 65.8%

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

      10. associate-/l* 65.4%

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

      11. *-rgt-identity 65.4%

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

      12. associate-*l/ 65.4%

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

   10. Simplified 65.4%

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

       1. add-sqr-sqrt 33.9%

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

       2. fabs-sqr 33.9%

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

       3. add-sqr-sqrt 35.4%

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

       4. associate-/l* 35.7%

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

   12. Applied egg-rr 35.7%

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

   if 1.6000000000000001 < 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 99.4%

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

   5. Taylor expanded in x around 0 98.6%

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

      1. rem-square-sqrt 33.9%

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

      2. fabs-sqr 33.9%

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

      3. rem-square-sqrt 98.6%

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

      4. +-commutative 98.6%

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

      5. fma-define 98.6%

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

      6. rem-square-sqrt 34.0%

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

      7. fabs-sqr 34.0%

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

      8. rem-square-sqrt 98.6%

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

      9. rem-square-sqrt 34.0%

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

      10. fabs-sqr 34.0%

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

      11. rem-square-sqrt 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in x around inf 36.6%

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

      1. unpow-1 36.6%

         \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot \sqrt{\color{blue}{{\pi}^{-1}}} + 0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right| \]

      2. metadata-eval 36.6%

         \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot \sqrt{{\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}} + 0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right| \]

      3. pow-sqr 36.6%

         \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot \sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}} + 0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right| \]

      4. rem-sqrt-square 36.6%

         \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot \color{blue}{\left|{\pi}^{-0.5}\right|} + 0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right| \]

      5. rem-square-sqrt 36.6%

         \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot \left|\color{blue}{\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}}\right| + 0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right| \]

      6. fabs-sqr 36.6%

         \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot \color{blue}{\left(\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}\right)} + 0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right| \]

      7. rem-square-sqrt 36.6%

         \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot \color{blue}{{\pi}^{-0.5}} + 0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right| \]

      8. associate-*r* 36.6%

         \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5} + \color{blue}{\left(0.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \sqrt{\frac{1}{\pi}}}\right)\right| \]

      9. unpow-1 36.6%

         \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5} + \left(0.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \sqrt{\color{blue}{{\pi}^{-1}}}\right)\right| \]

      10. metadata-eval 36.6%

          \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5} + \left(0.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \sqrt{{\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right)\right| \]

      11. pow-sqr 36.6%

          \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5} + \left(0.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}}\right)\right| \]

      12. rem-sqrt-square 36.6%

          \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5} + \left(0.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \color{blue}{\left|{\pi}^{-0.5}\right|}\right)\right| \]

      13. rem-square-sqrt 36.6%

          \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5} + \left(0.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \left|\color{blue}{\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}}\right|\right)\right| \]

      14. fabs-sqr 36.6%

          \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5} + \left(0.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \color{blue}{\left(\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}\right)}\right)\right| \]

      15. rem-square-sqrt 36.6%

          \[\leadsto \left|{x}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5} + \left(0.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \color{blue}{{\pi}^{-0.5}}\right)\right| \]

   10. Simplified 36.6%

       \[\leadsto \left|\color{blue}{{x}^{7} \cdot \left({\pi}^{-0.5} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.5% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(N[(x * N[(2.0 + N[(N[Power[x, 4.0], $MachinePrecision] * N[(0.2 + N[(0.047619047619047616 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $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 99.4%

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

5. Taylor expanded in x around 0 98.6%

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

   1. rem-square-sqrt 33.9%

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

   2. fabs-sqr 33.9%

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

   3. rem-square-sqrt 98.6%

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

   4. +-commutative 98.6%

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

   5. fma-define 98.6%

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

   6. rem-square-sqrt 34.0%

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

   7. fabs-sqr 34.0%

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

   8. rem-square-sqrt 98.6%

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

   9. rem-square-sqrt 34.0%

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

   10. fabs-sqr 34.0%

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

   11. rem-square-sqrt 98.6%

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

7. Simplified 98.6%

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

8. Taylor expanded in x around 0 98.6%

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

   1. *-commutative 98.6%

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

10. Simplified 98.6%

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

11. Final simplification 98.6%

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

Alternative 6: 34.5% accurate, 8.7× speedup

Math

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

FPCore

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

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) * pow(((double) M_PI), -0.5));
	}
	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.pow(Math.PI, -0.5));
	}
	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.pow(math.pi, -0.5))
	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) * (pi ^ -0.5)));
	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) * (pi ^ -0.5));
	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[Power[Pi, -0.5], $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 99.4%

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

   5. Taylor expanded in x around 0 98.6%

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

      1. rem-square-sqrt 33.9%

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

      2. fabs-sqr 33.9%

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

      3. rem-square-sqrt 98.6%

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

      4. +-commutative 98.6%

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

      5. fma-define 98.6%

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

      6. rem-square-sqrt 34.0%

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

      7. fabs-sqr 34.0%

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

      8. rem-square-sqrt 98.6%

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

      9. rem-square-sqrt 34.0%

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

      10. fabs-sqr 34.0%

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

      11. rem-square-sqrt 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in x around 0 65.8%

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

      1. *-commutative 65.8%

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

      2. unpow1/2 65.8%

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

      3. rem-exp-log 65.8%

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

      4. exp-neg 65.8%

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

      5. exp-prod 65.8%

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

      6. distribute-lft-neg-out 65.8%

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

      7. exp-neg 65.8%

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

      8. exp-to-pow 65.8%

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

      9. unpow1/2 65.8%

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

      10. associate-/l* 65.4%

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

      11. *-rgt-identity 65.4%

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

      12. associate-*l/ 65.4%

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

   10. Simplified 65.4%

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

       1. add-sqr-sqrt 33.9%

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

       2. fabs-sqr 33.9%

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

       3. add-sqr-sqrt 35.4%

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

       4. associate-/l* 35.7%

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

   12. Applied egg-rr 35.7%

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

   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 99.4%

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

   5. Taylor expanded in x around 0 99.1%

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

      1. associate-*r* 99.1%

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

      2. *-commutative 99.1%

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

      3. rem-square-sqrt 34.0%

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

      4. fabs-sqr 34.0%

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

      5. rem-square-sqrt 99.1%

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

      6. associate-*l* 99.1%

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

      7. +-commutative 99.1%

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

      8. fma-define 99.1%

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

      9. rem-square-sqrt 34.3%

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

      10. fabs-sqr 34.3%

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

      11. rem-square-sqrt 99.1%

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

      12. rem-square-sqrt 34.3%

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

      13. fabs-sqr 34.3%

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

      14. rem-square-sqrt 99.1%

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

   7. Simplified 99.1%

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

   8. Taylor expanded in x around inf 38.8%

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

      1. associate-*r* 38.8%

         \[\leadsto \left|x \cdot \color{blue}{\left(\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

      2. unpow-1 38.8%

         \[\leadsto \left|x \cdot \left(\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot \sqrt{\color{blue}{{\pi}^{-1}}}\right)\right| \]

      3. metadata-eval 38.8%

         \[\leadsto \left|x \cdot \left(\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot \sqrt{{\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right)\right| \]

      4. pow-sqr 38.8%

         \[\leadsto \left|x \cdot \left(\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot \sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}}\right)\right| \]

      5. rem-sqrt-square 38.8%

         \[\leadsto \left|x \cdot \left(\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot \color{blue}{\left|{\pi}^{-0.5}\right|}\right)\right| \]

      6. rem-square-sqrt 38.8%

         \[\leadsto \left|x \cdot \left(\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot \left|\color{blue}{\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}}\right|\right)\right| \]

      7. fabs-sqr 38.8%

         \[\leadsto \left|x \cdot \left(\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot \color{blue}{\left(\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}\right)}\right)\right| \]

      8. rem-square-sqrt 38.8%

         \[\leadsto \left|x \cdot \left(\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot \color{blue}{{\pi}^{-0.5}}\right)\right| \]

      9. *-commutative 38.8%

         \[\leadsto \left|x \cdot \color{blue}{\left({\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)}\right| \]

   10. Simplified 38.8%

       \[\leadsto \left|x \cdot \color{blue}{\left({\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)}\right| \]

   11. Taylor expanded in x around 0 38.8%

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

   13. Simplified 3.7%

       \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.5% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.04 \cdot \frac{{x}^{10}}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.75) (* x (/ 2.0 (sqrt PI))) (sqrt (* 0.04 (/ (pow x 10.0) PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.75) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = sqrt((0.04 * (pow(x, 10.0) / ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.75) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.sqrt((0.04 * (Math.pow(x, 10.0) / Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.75:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.sqrt((0.04 * (math.pow(x, 10.0) / math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.75)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = sqrt(Float64(0.04 * Float64((x ^ 10.0) / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.75)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = sqrt((0.04 * ((x ^ 10.0) / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.75], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.04 * N[(N[Power[x, 10.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.75

   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 99.4%

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

   5. Taylor expanded in x around 0 98.6%

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

      1. rem-square-sqrt 33.9%

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

      2. fabs-sqr 33.9%

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

      3. rem-square-sqrt 98.6%

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

      4. +-commutative 98.6%

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

      5. fma-define 98.6%

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

      6. rem-square-sqrt 34.0%

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

      7. fabs-sqr 34.0%

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

      8. rem-square-sqrt 98.6%

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

      9. rem-square-sqrt 34.0%

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

      10. fabs-sqr 34.0%

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

      11. rem-square-sqrt 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in x around 0 65.8%

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

      1. *-commutative 65.8%

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

      2. unpow1/2 65.8%

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

      3. rem-exp-log 65.8%

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

      4. exp-neg 65.8%

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

      5. exp-prod 65.8%

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

      6. distribute-lft-neg-out 65.8%

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

      7. exp-neg 65.8%

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

      8. exp-to-pow 65.8%

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

      9. unpow1/2 65.8%

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

      10. associate-/l* 65.4%

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

      11. *-rgt-identity 65.4%

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

      12. associate-*l/ 65.4%

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

   10. Simplified 65.4%

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

       1. add-sqr-sqrt 33.9%

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

       2. fabs-sqr 33.9%

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

       3. add-sqr-sqrt 35.4%

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

       4. associate-/l* 35.7%

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

   12. Applied egg-rr 35.7%

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

   if 1.75 < 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 99.4%

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

   5. Taylor expanded in x around inf 34.8%

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

      1. *-commutative 34.8%

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

      2. *-commutative 34.8%

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

      3. unpow2 34.8%

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

      4. associate-*r* 34.8%

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

      5. rem-square-sqrt 2.0%

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

      6. fabs-sqr 2.0%

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

      7. rem-square-sqrt 34.8%

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

      8. pow-plus 34.8%

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

      9. metadata-eval 34.8%

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

      10. pow-plus 34.8%

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

      11. metadata-eval 34.8%

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

   7. Simplified 34.8%

      \[\leadsto \left|\color{blue}{0.2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{5}\right)}\right| \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 34.8%

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

      2. sqrt-unprod 36.1%

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

      3. sqr-abs 36.1%

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

      4. *-commutative 36.1%

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

      5. *-commutative 36.1%

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

      6. swap-sqr 36.1%

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

      7. swap-sqr 36.1%

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

      8. add-sqr-sqrt 36.1%

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

      9. pow-prod-up 36.1%

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

      10. metadata-eval 36.1%

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

      11. metadata-eval 36.1%

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

   9. Applied egg-rr 36.1%

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

       1. *-commutative 36.1%

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

       2. associate-*l/ 36.1%

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

       3. *-lft-identity 36.1%

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

   11. Simplified 36.1%

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

Alternative 8: 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 99.4%

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

5. Taylor expanded in x around 0 98.6%

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

   1. rem-square-sqrt 33.9%

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

   2. fabs-sqr 33.9%

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

   3. rem-square-sqrt 98.6%

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

   4. +-commutative 98.6%

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

   5. fma-define 98.6%

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

   6. rem-square-sqrt 34.0%

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

   7. fabs-sqr 34.0%

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

   8. rem-square-sqrt 98.6%

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

   9. rem-square-sqrt 34.0%

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

   10. fabs-sqr 34.0%

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

   11. rem-square-sqrt 98.6%

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

7. Simplified 98.6%

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

8. Taylor expanded in x around 0 65.8%

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

   1. *-commutative 65.8%

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

   2. unpow1/2 65.8%

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

   3. rem-exp-log 65.8%

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

   4. exp-neg 65.8%

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

   5. exp-prod 65.8%

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

   6. distribute-lft-neg-out 65.8%

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

   7. exp-neg 65.8%

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

   8. exp-to-pow 65.8%

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

   9. unpow1/2 65.8%

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

   10. associate-/l* 65.4%

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

   11. *-rgt-identity 65.4%

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

   12. associate-*l/ 65.4%

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

10. Simplified 65.4%

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

    1. add-sqr-sqrt 33.9%

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

    2. fabs-sqr 33.9%

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

    3. add-sqr-sqrt 35.4%

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

    4. associate-/l* 35.7%

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

12. Applied egg-rr 35.7%

    \[\leadsto \color{blue}{x \cdot \frac{2}{\sqrt{\pi}}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024167 
(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)))))))