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

Percentage Accurate: 99.8% → 99.8%

Time: 10.7s

Alternatives: 19

Speedup: 1.5×

• 27.9% 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 99.5%

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

2. Final simplification 99.5%

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

Alternative 2: 99.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 3: 99.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.3%

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

   1. *-un-lft-identity 99.3%

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

5. Applied egg-rr 99.3%

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

   1. *-lft-identity 99.3%

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

   2. unpow1 99.3%

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

   3. sqr-pow 32.1%

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

   4. fabs-sqr 32.1%

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

   5. sqr-pow 99.3%

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

   6. unpow1 99.3%

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

7. Simplified 99.3%

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

8. Taylor expanded in x around 0 99.3%

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

9. Final simplification 99.3%

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

Alternative 4: 98.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \left|\frac{\mathsf{fma}\left(2, x, 0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)}{\sqrt{\pi}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (/
   (fma 2.0 x (+ (* 0.2 (pow x 5.0)) (* 0.047619047619047616 (pow x 7.0))))
   (sqrt PI))))

C

double code(double x) {
	return fabs((fma(2.0, x, ((0.2 * pow(x, 5.0)) + (0.047619047619047616 * pow(x, 7.0)))) / sqrt(((double) M_PI))));
}

Julia

function code(x)
	return abs(Float64(fma(2.0, x, Float64(Float64(0.2 * (x ^ 5.0)) + Float64(0.047619047619047616 * (x ^ 7.0)))) / sqrt(pi)))
end

Wolfram

code[x_] := N[Abs[N[(N[(2.0 * x + N[(N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Simplified 99.0%

   \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

4. Taylor expanded in x around inf 98.2%

   \[\leadsto \left|\frac{\mathsf{fma}\left(2, x, \color{blue}{0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}}\right)}{\sqrt{\pi}}\right| \]

5. Final simplification 98.2%

   \[\leadsto \left|\frac{\mathsf{fma}\left(2, x, 0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)}{\sqrt{\pi}}\right| \]

Alternative 5: 98.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \left|\frac{\mathsf{fma}\left(2, x, 0.047619047619047616 \cdot {x}^{7}\right)}{\sqrt{\pi}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs (/ (fma 2.0 x (* 0.047619047619047616 (pow x 7.0))) (sqrt PI))))

C

double code(double x) {
	return fabs((fma(2.0, x, (0.047619047619047616 * pow(x, 7.0))) / sqrt(((double) M_PI))));
}

Julia

function code(x)
	return abs(Float64(fma(2.0, x, Float64(0.047619047619047616 * (x ^ 7.0))) / sqrt(pi)))
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.0%

   \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

4. Taylor expanded in x around inf 98.1%

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

5. Final simplification 98.1%

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

Alternative 6: 89.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 2 \cdot x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{0.047619047619047616 \cdot {x}^{6}}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (fabs
    (* (sqrt (/ 1.0 PI)) (+ (* x (* 0.6666666666666666 (* x x))) (* 2.0 x))))
   (fabs (* x (/ (* 0.047619047619047616 (pow x 6.0)) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * ((x * (0.6666666666666666 * (x * x))) + (2.0 * x))));
	} else {
		tmp = fabs((x * ((0.047619047619047616 * pow(x, 6.0)) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * ((x * (0.6666666666666666 * (x * x))) + (2.0 * x))));
	} else {
		tmp = Math.abs((x * ((0.047619047619047616 * Math.pow(x, 6.0)) / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.2:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * ((x * (0.6666666666666666 * (x * x))) + (2.0 * x))))
	else:
		tmp = math.fabs((x * ((0.047619047619047616 * math.pow(x, 6.0)) / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(x * Float64(0.6666666666666666 * Float64(x * x))) + Float64(2.0 * x))));
	else
		tmp = abs(Float64(x * Float64(Float64(0.047619047619047616 * (x ^ 6.0)) / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.2)
		tmp = abs((sqrt((1.0 / pi)) * ((x * (0.6666666666666666 * (x * x))) + (2.0 * x))));
	else
		tmp = abs((x * ((0.047619047619047616 * (x ^ 6.0)) / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(x * N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 87.6%

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

      1. associate-*r* 87.6%

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

      2. unpow2 87.6%

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

      3. associate-*r* 87.6%

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

      4. distribute-rgt-out 87.6%

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

      5. +-commutative 87.6%

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

      6. *-commutative 87.6%

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

      7. associate-*l* 87.6%

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

      8. *-commutative 87.6%

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

      9. *-commutative 87.6%

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

      10. distribute-lft-in 87.6%

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

      11. fma-udef 87.6%

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

   6. Simplified 87.6%

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

      1. fma-udef 87.6%

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

      2. distribute-rgt-in 87.6%

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

   8. Applied egg-rr 87.6%

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

   if 2.2000000000000002 < x

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 34.7%

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

      1. *-commutative 34.7%

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

      2. associate-*l* 34.8%

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

      3. associate-*l* 35.1%

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

      4. *-commutative 35.1%

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

   6. Simplified 35.1%

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

      1. expm1-log1p-u 34.6%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left(\left(\sqrt{\frac{1}{\pi}} \cdot {x}^{6}\right) \cdot 0.047619047619047616\right)\right)\right)}\right| \]

      2. expm1-udef 34.3%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left(\left(\sqrt{\frac{1}{\pi}} \cdot {x}^{6}\right) \cdot 0.047619047619047616\right)\right)} - 1}\right| \]

      3. add-sqr-sqrt 1.8%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left(\left(\sqrt{\frac{1}{\pi}} \cdot {x}^{6}\right) \cdot 0.047619047619047616\right)\right)} - 1\right| \]

      4. fabs-sqr 1.8%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left(\left(\sqrt{\frac{1}{\pi}} \cdot {x}^{6}\right) \cdot 0.047619047619047616\right)\right)} - 1\right| \]

      5. add-sqr-sqrt 3.7%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{x} \cdot \left(\left(\sqrt{\frac{1}{\pi}} \cdot {x}^{6}\right) \cdot 0.047619047619047616\right)\right)} - 1\right| \]

      6. *-commutative 3.7%

         \[\leadsto \left|e^{\mathsf{log1p}\left(x \cdot \color{blue}{\left(0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{6}\right)\right)}\right)} - 1\right| \]

      7. *-commutative 3.7%

         \[\leadsto \left|e^{\mathsf{log1p}\left(x \cdot \left(0.047619047619047616 \cdot \color{blue}{\left({x}^{6} \cdot \sqrt{\frac{1}{\pi}}\right)}\right)\right)} - 1\right| \]

      8. sqrt-div 3.7%

         \[\leadsto \left|e^{\mathsf{log1p}\left(x \cdot \left(0.047619047619047616 \cdot \left({x}^{6} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)\right)\right)} - 1\right| \]

      9. metadata-eval 3.7%

         \[\leadsto \left|e^{\mathsf{log1p}\left(x \cdot \left(0.047619047619047616 \cdot \left({x}^{6} \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)\right)\right)} - 1\right| \]

      10. un-div-inv 3.7%

          \[\leadsto \left|e^{\mathsf{log1p}\left(x \cdot \left(0.047619047619047616 \cdot \color{blue}{\frac{{x}^{6}}{\sqrt{\pi}}}\right)\right)} - 1\right| \]

   8. Applied egg-rr 3.7%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(x \cdot \left(0.047619047619047616 \cdot \frac{{x}^{6}}{\sqrt{\pi}}\right)\right)} - 1}\right| \]
   9. Step-by-step derivation

      1. expm1-def 4.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \left(0.047619047619047616 \cdot \frac{{x}^{6}}{\sqrt{\pi}}\right)\right)\right)}\right| \]

      2. expm1-log1p 35.1%

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

      3. associate-*r/ 35.1%

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

   10. Simplified 35.1%

       \[\leadsto \left|\color{blue}{x \cdot \frac{0.047619047619047616 \cdot {x}^{6}}{\sqrt{\pi}}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 2 \cdot x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{0.047619047619047616 \cdot {x}^{6}}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 7: 89.3% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(x * N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[Power[x, 7.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 87.6%

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

      1. associate-*r* 87.6%

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

      2. unpow2 87.6%

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

      3. associate-*r* 87.6%

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

      4. distribute-rgt-out 87.6%

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

      5. +-commutative 87.6%

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

      6. *-commutative 87.6%

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

      7. associate-*l* 87.6%

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

      8. *-commutative 87.6%

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

      9. *-commutative 87.6%

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

      10. distribute-lft-in 87.6%

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

      11. fma-udef 87.6%

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

   6. Simplified 87.6%

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

      1. fma-udef 87.6%

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

      2. distribute-rgt-in 87.6%

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

   8. Applied egg-rr 87.6%

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

   if 2.2000000000000002 < x

   1. Initial program 99.5%

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

   3. Simplified 99.0%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around inf 34.7%

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

      1. associate-*r* 34.7%

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

   6. Simplified 34.7%

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

      1. expm1-log1p-u 4.0%

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

      2. expm1-udef 3.7%

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

      3. associate-*l* 3.7%

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

      4. sqrt-div 3.7%

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

      5. metadata-eval 3.7%

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

      6. un-div-inv 3.7%

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

   8. Applied egg-rr 3.7%

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

      1. expm1-def 4.0%

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

      2. expm1-log1p 34.8%

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

   10. Simplified 34.8%

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

4. Final simplification 87.6%

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

Alternative 8: 89.3% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(x * N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 87.6%

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

      1. associate-*r* 87.6%

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

      2. unpow2 87.6%

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

      3. associate-*r* 87.6%

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

      4. distribute-rgt-out 87.6%

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

      5. +-commutative 87.6%

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

      6. *-commutative 87.6%

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

      7. associate-*l* 87.6%

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

      8. *-commutative 87.6%

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

      9. *-commutative 87.6%

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

      10. distribute-lft-in 87.6%

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

      11. fma-udef 87.6%

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

   6. Simplified 87.6%

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

      1. fma-udef 87.6%

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

      2. distribute-rgt-in 87.6%

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

   8. Applied egg-rr 87.6%

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

   if 2.2000000000000002 < x

   1. Initial program 99.5%

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

   3. Simplified 99.0%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around inf 34.7%

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

      1. associate-*r* 34.7%

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

   6. Simplified 34.7%

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

      1. sqrt-div 34.7%

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

      2. metadata-eval 34.7%

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

      3. un-div-inv 34.8%

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

   8. Applied egg-rr 34.8%

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

4. Final simplification 87.6%

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

Alternative 9: 89.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ t_1 := x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq 5 \cdot 10^{+28}:\\ \;\;\;\;\left|t_0 \cdot \left(t_1 + 2 \cdot x\right)\right|\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+102}:\\ \;\;\;\;\left|t_0 \cdot \frac{t_1 \cdot t_1 - \left(2 \cdot x\right) \cdot \left(2 \cdot x\right)}{t_1 - 2 \cdot x}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{t_1}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))) (t_1 (* x (* 0.6666666666666666 (* x x)))))
   (if (<= x 5e+28)
     (fabs (* t_0 (+ t_1 (* 2.0 x))))
     (if (<= x 6.5e+102)
       (fabs
        (* t_0 (/ (- (* t_1 t_1) (* (* 2.0 x) (* 2.0 x))) (- t_1 (* 2.0 x)))))
       (fabs (/ t_1 (sqrt PI)))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double t_1 = x * (0.6666666666666666 * (x * x));
	double tmp;
	if (x <= 5e+28) {
		tmp = fabs((t_0 * (t_1 + (2.0 * x))));
	} else if (x <= 6.5e+102) {
		tmp = fabs((t_0 * (((t_1 * t_1) - ((2.0 * x) * (2.0 * x))) / (t_1 - (2.0 * x)))));
	} else {
		tmp = fabs((t_1 / sqrt(((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	double t_1 = x * (0.6666666666666666 * (x * x));
	double tmp;
	if (x <= 5e+28) {
		tmp = Math.abs((t_0 * (t_1 + (2.0 * x))));
	} else if (x <= 6.5e+102) {
		tmp = Math.abs((t_0 * (((t_1 * t_1) - ((2.0 * x) * (2.0 * x))) / (t_1 - (2.0 * x)))));
	} else {
		tmp = Math.abs((t_1 / Math.sqrt(Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / math.pi))
	t_1 = x * (0.6666666666666666 * (x * x))
	tmp = 0
	if x <= 5e+28:
		tmp = math.fabs((t_0 * (t_1 + (2.0 * x))))
	elif x <= 6.5e+102:
		tmp = math.fabs((t_0 * (((t_1 * t_1) - ((2.0 * x) * (2.0 * x))) / (t_1 - (2.0 * x)))))
	else:
		tmp = math.fabs((t_1 / math.sqrt(math.pi)))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / pi))
	t_1 = Float64(x * Float64(0.6666666666666666 * Float64(x * x)))
	tmp = 0.0
	if (x <= 5e+28)
		tmp = abs(Float64(t_0 * Float64(t_1 + Float64(2.0 * x))));
	elseif (x <= 6.5e+102)
		tmp = abs(Float64(t_0 * Float64(Float64(Float64(t_1 * t_1) - Float64(Float64(2.0 * x) * Float64(2.0 * x))) / Float64(t_1 - Float64(2.0 * x)))));
	else
		tmp = abs(Float64(t_1 / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 / pi));
	t_1 = x * (0.6666666666666666 * (x * x));
	tmp = 0.0;
	if (x <= 5e+28)
		tmp = abs((t_0 * (t_1 + (2.0 * x))));
	elseif (x <= 6.5e+102)
		tmp = abs((t_0 * (((t_1 * t_1) - ((2.0 * x) * (2.0 * x))) / (t_1 - (2.0 * x)))));
	else
		tmp = abs((t_1 / sqrt(pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5e+28], N[Abs[N[(t$95$0 * N[(t$95$1 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 6.5e+102], N[Abs[N[(t$95$0 * N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(N[(2.0 * x), $MachinePrecision] * N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$1 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.99999999999999957e28

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 87.6%

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

      1. associate-*r* 87.6%

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

      2. unpow2 87.6%

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

      3. associate-*r* 87.6%

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

      4. distribute-rgt-out 87.6%

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

      5. +-commutative 87.6%

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

      6. *-commutative 87.6%

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

      7. associate-*l* 87.6%

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

      8. *-commutative 87.6%

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

      9. *-commutative 87.6%

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

      10. distribute-lft-in 87.6%

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

      11. fma-udef 87.6%

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

   6. Simplified 87.6%

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

      1. fma-udef 87.6%

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

      2. distribute-rgt-in 87.6%

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

   8. Applied egg-rr 87.6%

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

   if 4.99999999999999957e28 < x < 6.5000000000000004e102

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 87.6%

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

      1. associate-*r* 87.6%

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

      2. unpow2 87.6%

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

      3. associate-*r* 87.6%

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

      4. distribute-rgt-out 87.6%

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

      5. +-commutative 87.6%

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

      6. *-commutative 87.6%

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

      7. associate-*l* 87.6%

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

      8. *-commutative 87.6%

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

      9. *-commutative 87.6%

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

      10. distribute-lft-in 87.6%

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

      11. fma-udef 87.6%

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

   6. Simplified 87.6%

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

      1. fma-udef 87.0%

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

   8. Applied egg-rr 87.6%

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

      1. distribute-rgt-in 87.6%

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

      2. flip-+ 43.1%

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

   10. Applied egg-rr 43.1%

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

   if 6.5000000000000004e102 < x

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 87.6%

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

      1. associate-*r* 87.6%

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

      2. unpow2 87.6%

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

      3. associate-*r* 87.6%

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

      4. distribute-rgt-out 87.6%

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

      5. +-commutative 87.6%

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

      6. *-commutative 87.6%

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

      7. associate-*l* 87.6%

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

      8. *-commutative 87.6%

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

      9. *-commutative 87.6%

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

      10. distribute-lft-in 87.6%

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

      11. fma-udef 87.6%

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

   6. Simplified 87.6%

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

      1. *-commutative 87.6%

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

      2. sqrt-div 87.6%

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

      3. metadata-eval 87.6%

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

      4. un-div-inv 87.0%

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

   8. Applied egg-rr 87.0%

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

   9. Taylor expanded in x around inf 23.7%

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

       1. unpow2 23.7%

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

   11. Simplified 23.7%

       \[\leadsto \left|\frac{x \cdot \color{blue}{\left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)}}{\sqrt{\pi}}\right| \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+28}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 2 \cdot x\right)\right|\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+102}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \frac{\left(x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right) - \left(2 \cdot x\right) \cdot \left(2 \cdot x\right)}{x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) - 2 \cdot x}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 10: 75.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.6666666666666666 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq 4 \cdot 10^{+102}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \frac{t_0 \cdot t_0 - 4}{t_0 - 2}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot t_0}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.6666666666666666 (* x x))))
   (if (<= x 4e+102)
     (fabs (* (sqrt (/ 1.0 PI)) (* x (/ (- (* t_0 t_0) 4.0) (- t_0 2.0)))))
     (fabs (/ (* x t_0) (sqrt PI))))))

C

double code(double x) {
	double t_0 = 0.6666666666666666 * (x * x);
	double tmp;
	if (x <= 4e+102) {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * (x * (((t_0 * t_0) - 4.0) / (t_0 - 2.0)))));
	} else {
		tmp = fabs(((x * t_0) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 0.6666666666666666 * (x * x);
	double tmp;
	if (x <= 4e+102) {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * (x * (((t_0 * t_0) - 4.0) / (t_0 - 2.0)))));
	} else {
		tmp = Math.abs(((x * t_0) / Math.sqrt(Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.6666666666666666 * (x * x)
	tmp = 0
	if x <= 4e+102:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * (x * (((t_0 * t_0) - 4.0) / (t_0 - 2.0)))))
	else:
		tmp = math.fabs(((x * t_0) / math.sqrt(math.pi)))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.6666666666666666 * Float64(x * x))
	tmp = 0.0
	if (x <= 4e+102)
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(x * Float64(Float64(Float64(t_0 * t_0) - 4.0) / Float64(t_0 - 2.0)))));
	else
		tmp = abs(Float64(Float64(x * t_0) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.6666666666666666 * (x * x);
	tmp = 0.0;
	if (x <= 4e+102)
		tmp = abs((sqrt((1.0 / pi)) * (x * (((t_0 * t_0) - 4.0) / (t_0 - 2.0)))));
	else
		tmp = abs(((x * t_0) / sqrt(pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 4e+102], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x * N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - 4.0), $MachinePrecision] / N[(t$95$0 - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x * t$95$0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.99999999999999991e102

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 87.6%

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

      1. associate-*r* 87.6%

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

      2. unpow2 87.6%

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

      3. associate-*r* 87.6%

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

      4. distribute-rgt-out 87.6%

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

      5. +-commutative 87.6%

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

      6. *-commutative 87.6%

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

      7. associate-*l* 87.6%

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

      8. *-commutative 87.6%

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

      9. *-commutative 87.6%

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

      10. distribute-lft-in 87.6%

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

      11. fma-udef 87.6%

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

   6. Simplified 87.6%

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

      1. fma-udef 87.0%

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

      2. flip-+ 76.1%

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

      3. metadata-eval 76.1%

         \[\leadsto \left|\frac{x \cdot \frac{\left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) - \color{blue}{4}}{0.6666666666666666 \cdot \left(x \cdot x\right) - 2}}{\sqrt{\pi}}\right| \]

   8. Applied egg-rr 76.7%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \color{blue}{\frac{\left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) - 4}{0.6666666666666666 \cdot \left(x \cdot x\right) - 2}}\right)\right| \]

   if 3.99999999999999991e102 < x

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 87.6%

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

      1. associate-*r* 87.6%

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

      2. unpow2 87.6%

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

      3. associate-*r* 87.6%

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

      4. distribute-rgt-out 87.6%

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

      5. +-commutative 87.6%

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

      6. *-commutative 87.6%

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

      7. associate-*l* 87.6%

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

      8. *-commutative 87.6%

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

      9. *-commutative 87.6%

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

      10. distribute-lft-in 87.6%

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

      11. fma-udef 87.6%

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

   6. Simplified 87.6%

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

      1. *-commutative 87.6%

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

      2. sqrt-div 87.6%

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

      3. metadata-eval 87.6%

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

      4. un-div-inv 87.0%

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

   8. Applied egg-rr 87.0%

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

   9. Taylor expanded in x around inf 23.7%

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

       1. unpow2 23.7%

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

   11. Simplified 23.7%

       \[\leadsto \left|\frac{x \cdot \color{blue}{\left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)}}{\sqrt{\pi}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+102}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \frac{\left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) - 4}{0.6666666666666666 \cdot \left(x \cdot x\right) - 2}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 11: 74.6% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.6666666666666666 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq 4 \cdot 10^{+102}:\\ \;\;\;\;\left|\frac{x \cdot \frac{t_0 \cdot t_0 - 4}{t_0 - 2}}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot t_0}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.6666666666666666 (* x x))))
   (if (<= x 4e+102)
     (fabs (/ (* x (/ (- (* t_0 t_0) 4.0) (- t_0 2.0))) (sqrt PI)))
     (fabs (/ (* x t_0) (sqrt PI))))))

C

double code(double x) {
	double t_0 = 0.6666666666666666 * (x * x);
	double tmp;
	if (x <= 4e+102) {
		tmp = fabs(((x * (((t_0 * t_0) - 4.0) / (t_0 - 2.0))) / sqrt(((double) M_PI))));
	} else {
		tmp = fabs(((x * t_0) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 0.6666666666666666 * (x * x);
	double tmp;
	if (x <= 4e+102) {
		tmp = Math.abs(((x * (((t_0 * t_0) - 4.0) / (t_0 - 2.0))) / Math.sqrt(Math.PI)));
	} else {
		tmp = Math.abs(((x * t_0) / Math.sqrt(Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.6666666666666666 * (x * x)
	tmp = 0
	if x <= 4e+102:
		tmp = math.fabs(((x * (((t_0 * t_0) - 4.0) / (t_0 - 2.0))) / math.sqrt(math.pi)))
	else:
		tmp = math.fabs(((x * t_0) / math.sqrt(math.pi)))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.6666666666666666 * Float64(x * x))
	tmp = 0.0
	if (x <= 4e+102)
		tmp = abs(Float64(Float64(x * Float64(Float64(Float64(t_0 * t_0) - 4.0) / Float64(t_0 - 2.0))) / sqrt(pi)));
	else
		tmp = abs(Float64(Float64(x * t_0) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.6666666666666666 * (x * x);
	tmp = 0.0;
	if (x <= 4e+102)
		tmp = abs(((x * (((t_0 * t_0) - 4.0) / (t_0 - 2.0))) / sqrt(pi)));
	else
		tmp = abs(((x * t_0) / sqrt(pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 4e+102], N[Abs[N[(N[(x * N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - 4.0), $MachinePrecision] / N[(t$95$0 - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x * t$95$0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.99999999999999991e102

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 87.6%

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

      1. associate-*r* 87.6%

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

      2. unpow2 87.6%

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

      3. associate-*r* 87.6%

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

      4. distribute-rgt-out 87.6%

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

      5. +-commutative 87.6%

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

      6. *-commutative 87.6%

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

      7. associate-*l* 87.6%

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

      8. *-commutative 87.6%

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

      9. *-commutative 87.6%

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

      10. distribute-lft-in 87.6%

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

      11. fma-udef 87.6%

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

   6. Simplified 87.6%

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

      1. *-commutative 87.6%

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

      2. sqrt-div 87.6%

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

      3. metadata-eval 87.6%

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

      4. un-div-inv 87.0%

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

   8. Applied egg-rr 87.0%

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

      1. fma-udef 87.0%

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

      2. flip-+ 76.1%

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

      3. metadata-eval 76.1%

         \[\leadsto \left|\frac{x \cdot \frac{\left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) - \color{blue}{4}}{0.6666666666666666 \cdot \left(x \cdot x\right) - 2}}{\sqrt{\pi}}\right| \]

   10. Applied egg-rr 76.1%

       \[\leadsto \left|\frac{x \cdot \color{blue}{\frac{\left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) - 4}{0.6666666666666666 \cdot \left(x \cdot x\right) - 2}}}{\sqrt{\pi}}\right| \]

   if 3.99999999999999991e102 < x

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 87.6%

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

      1. associate-*r* 87.6%

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

      2. unpow2 87.6%

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

      3. associate-*r* 87.6%

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

      4. distribute-rgt-out 87.6%

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

      5. +-commutative 87.6%

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

      6. *-commutative 87.6%

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

      7. associate-*l* 87.6%

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

      8. *-commutative 87.6%

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

      9. *-commutative 87.6%

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

      10. distribute-lft-in 87.6%

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

      11. fma-udef 87.6%

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

   6. Simplified 87.6%

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

      1. *-commutative 87.6%

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

      2. sqrt-div 87.6%

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

      3. metadata-eval 87.6%

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

      4. un-div-inv 87.0%

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

   8. Applied egg-rr 87.0%

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

   9. Taylor expanded in x around inf 23.7%

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

       1. unpow2 23.7%

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

   11. Simplified 23.7%

       \[\leadsto \left|\frac{x \cdot \color{blue}{\left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)}}{\sqrt{\pi}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+102}:\\ \;\;\;\;\left|\frac{x \cdot \frac{\left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) - 4}{0.6666666666666666 \cdot \left(x \cdot x\right) - 2}}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 12: 89.3% accurate, 6.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (* (sqrt (/ 1.0 PI)) (+ (* x (* 0.6666666666666666 (* x x))) (* 2.0 x)))))

C

double code(double x) {
	return fabs((sqrt((1.0 / ((double) M_PI))) * ((x * (0.6666666666666666 * (x * x))) + (2.0 * x))));
}

Java

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

Python

def code(x):
	return math.fabs((math.sqrt((1.0 / math.pi)) * ((x * (0.6666666666666666 * (x * x))) + (2.0 * x))))

Julia

function code(x)
	return abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(x * Float64(0.6666666666666666 * Float64(x * x))) + Float64(2.0 * x))))
end

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(x * N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around 0 87.6%

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

   1. associate-*r* 87.6%

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

   2. unpow2 87.6%

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

   3. associate-*r* 87.6%

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

   4. distribute-rgt-out 87.6%

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

   5. +-commutative 87.6%

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

   6. *-commutative 87.6%

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

   7. associate-*l* 87.6%

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

   8. *-commutative 87.6%

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

   9. *-commutative 87.6%

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

   10. distribute-lft-in 87.6%

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

   11. fma-udef 87.6%

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

6. Simplified 87.6%

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

   1. fma-udef 87.6%

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

   2. distribute-rgt-in 87.6%

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

8. Applied egg-rr 87.6%

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

9. Final simplification 87.6%

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

Alternative 13: 89.3% accurate, 6.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (fabs (* (sqrt (/ 1.0 PI)) (* x (+ 2.0 (* 0.6666666666666666 (* x x)))))))

C

double code(double x) {
	return fabs((sqrt((1.0 / ((double) M_PI))) * (x * (2.0 + (0.6666666666666666 * (x * x))))));
}

Java

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

Python

def code(x):
	return math.fabs((math.sqrt((1.0 / math.pi)) * (x * (2.0 + (0.6666666666666666 * (x * x))))))

Julia

function code(x)
	return abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(x * Float64(2.0 + Float64(0.6666666666666666 * Float64(x * x))))))
end

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x * N[(2.0 + N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around 0 87.6%

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

   1. associate-*r* 87.6%

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

   2. unpow2 87.6%

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

   3. associate-*r* 87.6%

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

   4. distribute-rgt-out 87.6%

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

   5. +-commutative 87.6%

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

   6. *-commutative 87.6%

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

   7. associate-*l* 87.6%

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

   8. *-commutative 87.6%

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

   9. *-commutative 87.6%

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

   10. distribute-lft-in 87.6%

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

   11. fma-udef 87.6%

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

6. Simplified 87.6%

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

   1. fma-udef 87.0%

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

8. Applied egg-rr 87.6%

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

9. Final simplification 87.6%

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

Alternative 14: 68.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;\left|2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.75)
   (fabs (* 2.0 (* x (pow PI -0.5))))
   (fabs (/ (* x (* 0.6666666666666666 (* x x))) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.75) {
		tmp = fabs((2.0 * (x * pow(((double) M_PI), -0.5))));
	} else {
		tmp = fabs(((x * (0.6666666666666666 * (x * x))) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.75) {
		tmp = Math.abs((2.0 * (x * Math.pow(Math.PI, -0.5))));
	} else {
		tmp = Math.abs(((x * (0.6666666666666666 * (x * x))) / Math.sqrt(Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.75:
		tmp = math.fabs((2.0 * (x * math.pow(math.pi, -0.5))))
	else:
		tmp = math.fabs(((x * (0.6666666666666666 * (x * x))) / math.sqrt(math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.75)
		tmp = abs(Float64(2.0 * Float64(x * (pi ^ -0.5))));
	else
		tmp = abs(Float64(Float64(x * Float64(0.6666666666666666 * Float64(x * x))) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.75)
		tmp = abs((2.0 * (x * (pi ^ -0.5))));
	else
		tmp = abs(((x * (0.6666666666666666 * (x * x))) / sqrt(pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.75], N[Abs[N[(2.0 * N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x * N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.75

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 69.6%

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

      1. *-commutative 69.6%

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

      2. unpow1 69.6%

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

      3. sqr-pow 31.9%

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

      4. fabs-sqr 31.9%

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

      5. sqr-pow 69.6%

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

      6. unpow1 69.6%

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

      7. *-commutative 69.6%

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

   6. Simplified 69.6%

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

      1. sqrt-div 69.6%

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

      2. metadata-eval 69.6%

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

      3. div-inv 69.0%

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

      4. clear-num 69.1%

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

   8. Applied egg-rr 69.1%

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

      1. associate-/r/ 69.6%

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

      2. metadata-eval 69.6%

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

      3. sqrt-div 69.6%

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

      4. inv-pow 69.6%

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

      5. sqrt-pow1 69.6%

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

      6. metadata-eval 69.6%

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

   10. Applied egg-rr 69.6%

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

   if 1.75 < x

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 87.6%

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

      1. associate-*r* 87.6%

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

      2. unpow2 87.6%

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

      3. associate-*r* 87.6%

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

      4. distribute-rgt-out 87.6%

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

      5. +-commutative 87.6%

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

      6. *-commutative 87.6%

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

      7. associate-*l* 87.6%

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

      8. *-commutative 87.6%

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

      9. *-commutative 87.6%

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

      10. distribute-lft-in 87.6%

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

      11. fma-udef 87.6%

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

   6. Simplified 87.6%

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

      1. *-commutative 87.6%

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

      2. sqrt-div 87.6%

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

      3. metadata-eval 87.6%

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

      4. un-div-inv 87.0%

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

   8. Applied egg-rr 87.0%

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

   9. Taylor expanded in x around inf 23.7%

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

       1. unpow2 23.7%

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

   11. Simplified 23.7%

       \[\leadsto \left|\frac{x \cdot \color{blue}{\left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)}}{\sqrt{\pi}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;\left|2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 15: 88.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \left|\frac{x \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)}{\sqrt{\pi}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs (/ (* x (+ 2.0 (* 0.6666666666666666 (* x x)))) (sqrt PI))))

C

double code(double x) {
	return fabs(((x * (2.0 + (0.6666666666666666 * (x * x)))) / sqrt(((double) M_PI))));
}

Java

public static double code(double x) {
	return Math.abs(((x * (2.0 + (0.6666666666666666 * (x * x)))) / Math.sqrt(Math.PI)));
}

Python

def code(x):
	return math.fabs(((x * (2.0 + (0.6666666666666666 * (x * x)))) / math.sqrt(math.pi)))

Julia

function code(x)
	return abs(Float64(Float64(x * Float64(2.0 + Float64(0.6666666666666666 * Float64(x * x)))) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = abs(((x * (2.0 + (0.6666666666666666 * (x * x)))) / sqrt(pi)));
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around 0 87.6%

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

   1. associate-*r* 87.6%

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

   2. unpow2 87.6%

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

   3. associate-*r* 87.6%

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

   4. distribute-rgt-out 87.6%

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

   5. +-commutative 87.6%

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

   6. *-commutative 87.6%

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

   7. associate-*l* 87.6%

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

   8. *-commutative 87.6%

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

   9. *-commutative 87.6%

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

   10. distribute-lft-in 87.6%

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

   11. fma-udef 87.6%

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

6. Simplified 87.6%

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

   1. *-commutative 87.6%

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

   2. sqrt-div 87.6%

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

   3. metadata-eval 87.6%

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

   4. un-div-inv 87.0%

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

8. Applied egg-rr 87.0%

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

   1. fma-udef 87.0%

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

10. Applied egg-rr 87.0%

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

11. Final simplification 87.0%

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

Alternative 16: 68.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\left|2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|2 \cdot \sqrt{\frac{x \cdot x}{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e-14)
   (fabs (* 2.0 (* x (pow PI -0.5))))
   (fabs (* 2.0 (sqrt (/ (* x x) PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 5e-14) {
		tmp = fabs((2.0 * (x * pow(((double) M_PI), -0.5))));
	} else {
		tmp = fabs((2.0 * sqrt(((x * x) / ((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5e-14) {
		tmp = Math.abs((2.0 * (x * Math.pow(Math.PI, -0.5))));
	} else {
		tmp = Math.abs((2.0 * Math.sqrt(((x * x) / Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5e-14:
		tmp = math.fabs((2.0 * (x * math.pow(math.pi, -0.5))))
	else:
		tmp = math.fabs((2.0 * math.sqrt(((x * x) / math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e-14)
		tmp = abs(Float64(2.0 * Float64(x * (pi ^ -0.5))));
	else
		tmp = abs(Float64(2.0 * sqrt(Float64(Float64(x * x) / pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5e-14)
		tmp = abs((2.0 * (x * (pi ^ -0.5))));
	else
		tmp = abs((2.0 * sqrt(((x * x) / pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5e-14], N[Abs[N[(2.0 * N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(2.0 * N[Sqrt[N[(N[(x * x), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000002e-14

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 69.5%

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

      1. *-commutative 69.5%

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

      2. unpow1 69.5%

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

      3. sqr-pow 31.4%

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

      4. fabs-sqr 31.4%

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

      5. sqr-pow 69.5%

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

      6. unpow1 69.5%

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

      7. *-commutative 69.5%

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

   6. Simplified 69.5%

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

      1. sqrt-div 69.5%

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

      2. metadata-eval 69.5%

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

      3. div-inv 69.0%

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

      4. clear-num 69.0%

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

   8. Applied egg-rr 69.0%

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

      1. associate-/r/ 69.5%

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

      2. metadata-eval 69.5%

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

      3. sqrt-div 69.5%

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

      4. inv-pow 69.5%

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

      5. sqrt-pow1 69.5%

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

      6. metadata-eval 69.5%

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

   10. Applied egg-rr 69.5%

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

   if 5.0000000000000002e-14 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 74.2%

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

      1. *-commutative 74.2%

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

      2. unpow1 74.2%

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

      3. sqr-pow 74.2%

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

      4. fabs-sqr 74.2%

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

      5. sqr-pow 74.2%

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

      6. unpow1 74.2%

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

      7. *-commutative 74.2%

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

   6. Simplified 74.2%

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

      1. sqrt-div 74.2%

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

      2. metadata-eval 74.2%

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

      3. div-inv 74.2%

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

      4. clear-num 73.7%

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

   8. Applied egg-rr 73.7%

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

      1. expm1-log1p-u 73.7%

         \[\leadsto \left|2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{\sqrt{\pi}}{x}}\right)\right)}\right| \]

      2. expm1-udef 43.1%

         \[\leadsto \left|2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\frac{\sqrt{\pi}}{x}}\right)} - 1\right)}\right| \]

      3. clear-num 43.1%

         \[\leadsto \left|2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\sqrt{\pi}}}\right)} - 1\right)\right| \]

   10. Applied egg-rr 43.1%

       \[\leadsto \left|2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}}\right)} - 1\right)}\right| \]
   11. Step-by-step derivation

       1. expm1-def 74.2%

          \[\leadsto \left|2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}}\right)\right)}\right| \]

       2. expm1-log1p 74.2%

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

   12. Simplified 74.2%

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

       1. add-sqr-sqrt 74.2%

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

       2. sqrt-unprod 74.2%

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

       3. frac-times 74.7%

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

       4. add-sqr-sqrt 74.2%

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

   14. Applied egg-rr 74.2%

       \[\leadsto \left|2 \cdot \color{blue}{\sqrt{\frac{x \cdot x}{\pi}}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\left|2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|2 \cdot \sqrt{\frac{x \cdot x}{\pi}}\right|\\ \end{array} \]

Alternative 17: 68.4% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around 0 69.6%

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

   1. *-commutative 69.6%

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

   2. unpow1 69.6%

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

   3. sqr-pow 31.9%

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

   4. fabs-sqr 31.9%

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

   5. sqr-pow 69.6%

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

   6. unpow1 69.6%

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

   7. *-commutative 69.6%

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

6. Simplified 69.6%

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

   1. sqrt-div 69.6%

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

   2. metadata-eval 69.6%

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

   3. div-inv 69.0%

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

   4. clear-num 69.1%

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

8. Applied egg-rr 69.1%

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

   1. associate-/r/ 69.6%

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

   2. metadata-eval 69.6%

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

   3. sqrt-div 69.6%

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

   4. inv-pow 69.6%

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

   5. sqrt-pow1 69.6%

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

   6. metadata-eval 69.6%

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

10. Applied egg-rr 69.6%

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

11. Final simplification 69.6%

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

Alternative 18: 68.0% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around 0 69.6%

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

   1. *-commutative 69.6%

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

   2. unpow1 69.6%

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

   3. sqr-pow 31.9%

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

   4. fabs-sqr 31.9%

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

   5. sqr-pow 69.6%

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

   6. unpow1 69.6%

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

   7. *-commutative 69.6%

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

6. Simplified 69.6%

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

   1. sqrt-div 69.6%

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

   2. metadata-eval 69.6%

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

   3. div-inv 69.0%

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

   4. clear-num 69.1%

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

8. Applied egg-rr 69.1%

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

   1. expm1-log1p-u 67.3%

      \[\leadsto \left|2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{\sqrt{\pi}}{x}}\right)\right)}\right| \]

   2. expm1-udef 5.9%

      \[\leadsto \left|2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\frac{\sqrt{\pi}}{x}}\right)} - 1\right)}\right| \]

   3. clear-num 5.9%

      \[\leadsto \left|2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\sqrt{\pi}}}\right)} - 1\right)\right| \]

10. Applied egg-rr 5.9%

    \[\leadsto \left|2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}}\right)} - 1\right)}\right| \]
11. Step-by-step derivation

    1. expm1-def 67.3%

       \[\leadsto \left|2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}}\right)\right)}\right| \]

    2. expm1-log1p 69.0%

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

12. Simplified 69.0%

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

13. Final simplification 69.0%

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

Alternative 19: 68.0% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around 0 87.6%

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

   1. associate-*r* 87.6%

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

   2. unpow2 87.6%

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

   3. associate-*r* 87.6%

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

   4. distribute-rgt-out 87.6%

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

   5. +-commutative 87.6%

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

   6. *-commutative 87.6%

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

   7. associate-*l* 87.6%

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

   8. *-commutative 87.6%

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

   9. *-commutative 87.6%

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

   10. distribute-lft-in 87.6%

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

   11. fma-udef 87.6%

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

6. Simplified 87.6%

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

   1. *-commutative 87.6%

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

   2. sqrt-div 87.6%

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

   3. metadata-eval 87.6%

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

   4. un-div-inv 87.0%

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

8. Applied egg-rr 87.0%

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

9. Taylor expanded in x around 0 69.4%

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

    1. *-commutative 69.4%

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

11. Simplified 69.4%

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

12. Final simplification 69.4%

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

Reproduce

herbie shell --seed 2023189 
(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)))))))