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

Percentage Accurate: 99.8% → 99.8%

Time: 5.3s

Alternatives: 14

Speedup: N/A×

• 28.4% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x \cdot x\right) \cdot x\right) \cdot x\\ \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(t\_0 \cdot \left|x\right|, \left(x \cdot x\right) \cdot 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, t\_0, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)\right| \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right), \left|x\right| \cdot \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(t\_0 \cdot t\_0\right) \cdot 0.047619047619047616\right)\right)\right| \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

Alternative 3: 99.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right)\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (fabs (* (/ 1.0 (sqrt PI)) (* x (fma 0.6666666666666666 (* x x) 2.0))))
   (/
    (fabs (* (fma (* x x) 0.047619047619047616 0.2) (* (* (* x x) x) (* x x))))
    (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = fabs(((1.0 / sqrt(((double) M_PI))) * (x * fma(0.6666666666666666, (x * x), 2.0))));
	} else {
		tmp = fabs((fma((x * x), 0.047619047619047616, 0.2) * (((x * x) * x) * (x * x)))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(x * fma(0.6666666666666666, Float64(x * x), 2.0))));
	else
		tmp = Float64(abs(Float64(fma(Float64(x * x), 0.047619047619047616, 0.2) * Float64(Float64(Float64(x * x) * x) * Float64(x * x)))) / sqrt(pi));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(x * N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(N[(N[(x * x), $MachinePrecision] * 0.047619047619047616 + 0.2), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 89.5%

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

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, \color{blue}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      2. pow2 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, \color{blue}{{\left(\left(x \cdot x\right) \cdot x\right)}^{2}} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}^{2} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left(\color{blue}{\left(x \cdot x\right)} \cdot x\right)}^{2} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      5. pow3 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left({x}^{3}\right)}}^{2} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      6. pow-pow N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, \color{blue}{{x}^{\left(3 \cdot 2\right)}} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      7. unpow1 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left({x}^{1}\right)}}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left({x}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      9. sqrt-pow1 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left(\sqrt{{x}^{2}}\right)}}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      10. pow2 N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left(\sqrt{\color{blue}{x \cdot x}}\right)}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      11. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left(\left|x\right|\right)}}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left(\left|x\right|\right)}^{\color{blue}{6}} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, \color{blue}{{\left(\left|x\right|\right)}^{6}} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      14. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left(\sqrt{x \cdot x}\right)}}^{6} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      15. pow2 N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left(\sqrt{\color{blue}{{x}^{2}}}\right)}^{6} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      16. sqrt-pow1 N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left({x}^{\left(\frac{2}{2}\right)}\right)}}^{6} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      17. metadata-eval N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left({x}^{\color{blue}{1}}\right)}^{6} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      18. unpow1 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\left|\color{blue}{{x}^{6} \cdot \left(\frac{1}{21} \cdot \left|x\right| + \frac{1}{5} \cdot \frac{\left|x\right|}{{x}^{2}}\right)}\right|}{\sqrt{\pi}} \]

   8. Applied rewrites 34.5%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\left|{x}^{5} \cdot \color{blue}{\left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)}\right|}{\sqrt{\pi}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

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

       2. metadata-eval N/A

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

       3. *-commutative N/A

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

       4. sqr-pow N/A

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

       5. pow-prod-down N/A

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

       6. sqrt-pow2 N/A

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

       7. rem-sqrt-square-rev N/A

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

       8. metadata-eval N/A

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

       9. pow-prod-up N/A

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

       10. pow3 N/A

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

       11. pow2 N/A

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

       12. associate-*l* N/A

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

       13. lower-*.f64 N/A

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

   11. Applied rewrites 36.7%

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

Alternative 4: 99.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, \color{blue}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   2. pow2 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, \color{blue}{{\left(\left(x \cdot x\right) \cdot x\right)}^{2}} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}^{2} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left(\color{blue}{\left(x \cdot x\right)} \cdot x\right)}^{2} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   5. pow3 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left({x}^{3}\right)}}^{2} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   6. pow-pow N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, \color{blue}{{x}^{\left(3 \cdot 2\right)}} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   7. unpow1 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left({x}^{1}\right)}}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left({x}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   9. sqrt-pow1 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left(\sqrt{{x}^{2}}\right)}}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   10. pow2 N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left(\sqrt{\color{blue}{x \cdot x}}\right)}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   11. rem-sqrt-square-rev N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left(\left|x\right|\right)}}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left(\left|x\right|\right)}^{\color{blue}{6}} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   13. lower-pow.f64 N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, \color{blue}{{\left(\left|x\right|\right)}^{6}} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   14. rem-sqrt-square-rev N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left(\sqrt{x \cdot x}\right)}}^{6} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   15. pow2 N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left(\sqrt{\color{blue}{{x}^{2}}}\right)}^{6} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   16. sqrt-pow1 N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left({x}^{\left(\frac{2}{2}\right)}\right)}}^{6} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   17. metadata-eval N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left({x}^{\color{blue}{1}}\right)}^{6} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   18. unpow1 99.4

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

5. Applied rewrites 99.4%

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

Alternative 5: 98.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)\right)}\right| \]

4. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)}\right| \]

5. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \color{blue}{\left(\frac{2}{3} \cdot \left({x}^{2} \cdot \left|x\right|\right) + 2 \cdot \left|x\right|\right)}\right)\right| \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left({\left(\left|x\right|\right)}^{7} \cdot \frac{1}{21} + \left(\frac{2}{3} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \color{blue}{2} \cdot \left|x\right|\right)\right)\right| \]

   2. *-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left({\left(\left|x\right|\right)}^{7} \cdot \frac{1}{21} + \left(\left({x}^{2} \cdot \left|x\right|\right) \cdot \frac{2}{3} + 2 \cdot \left|\color{blue}{x}\right|\right)\right)\right| \]

   3. pow2 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left({\left(\left|x\right|\right)}^{7} \cdot \frac{1}{21} + \left(\left(\left(x \cdot x\right) \cdot \left|x\right|\right) \cdot \frac{2}{3} + 2 \cdot \left|x\right|\right)\right)\right| \]

   4. sqr-abs-rev N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left({\left(\left|x\right|\right)}^{7} \cdot \frac{1}{21} + \left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \frac{2}{3} + 2 \cdot \left|x\right|\right)\right)\right| \]

   5. *-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left({\left(\left|x\right|\right)}^{7} \cdot \frac{1}{21} + \left(\frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) + 2 \cdot \left|\color{blue}{x}\right|\right)\right)\right| \]

   6. sqr-abs-rev N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left({\left(\left|x\right|\right)}^{7} \cdot \frac{1}{21} + \left(\frac{2}{3} \cdot \left(\left(x \cdot x\right) \cdot \left|x\right|\right) + 2 \cdot \left|x\right|\right)\right)\right| \]

   7. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left({\left(\left|x\right|\right)}^{7} \cdot \frac{1}{21} + \left(\frac{2}{3} \cdot \left(\left(x \cdot x\right) \cdot \left|x\right|\right) + 2 \cdot \left|x\right|\right)\right)\right| \]

   8. associate-*r* N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left({\left(\left|x\right|\right)}^{7} \cdot \frac{1}{21} + \left(\left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot \left|x\right| + 2 \cdot \left|\color{blue}{x}\right|\right)\right)\right| \]

   9. *-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left({\left(\left|x\right|\right)}^{7} \cdot \frac{1}{21} + \left(\left(\left(x \cdot x\right) \cdot \frac{2}{3}\right) \cdot \left|x\right| + 2 \cdot \left|x\right|\right)\right)\right| \]

   10. distribute-rgt-in N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left({\left(\left|x\right|\right)}^{7} \cdot \frac{1}{21} + \left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{3} + \color{blue}{2}\right)\right)\right| \]

7. Applied rewrites 99.1%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \color{blue}{0.047619047619047616}, x \cdot \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

8. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(x \cdot \left(2 + \color{blue}{{x}^{2} \cdot \left(\frac{2}{3} + \frac{1}{21} \cdot {x}^{4}\right)}\right)\right)\right| \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{1}{21} \cdot {x}^{4}\right)\right) \cdot x\right)\right| \]

   2. unpow1 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{1}{21} \cdot {x}^{4}\right)\right) \cdot {x}^{1}\right)\right| \]

   3. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{1}{21} \cdot {x}^{4}\right)\right) \cdot {x}^{\left(\frac{2}{2}\right)}\right)\right| \]

   4. sqrt-pow1 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{1}{21} \cdot {x}^{4}\right)\right) \cdot \sqrt{{x}^{2}}\right)\right| \]

   5. pow2 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{1}{21} \cdot {x}^{4}\right)\right) \cdot \sqrt{x \cdot x}\right)\right| \]

   6. rem-sqrt-square-rev N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{1}{21} \cdot {x}^{4}\right)\right) \cdot \left|x\right|\right)\right| \]

   7. lower-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{1}{21} \cdot {x}^{4}\right)\right) \cdot \left|x\right|\right)\right| \]

10. Applied rewrites 99.1%

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

Alternative 6: 89.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 89.5%

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

   if 2.2000000000000002 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, \color{blue}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      2. pow2 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, \color{blue}{{\left(\left(x \cdot x\right) \cdot x\right)}^{2}} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}^{2} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left(\color{blue}{\left(x \cdot x\right)} \cdot x\right)}^{2} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      5. pow3 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left({x}^{3}\right)}}^{2} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      6. pow-pow N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, \color{blue}{{x}^{\left(3 \cdot 2\right)}} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      7. unpow1 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left({x}^{1}\right)}}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left({x}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      9. sqrt-pow1 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left(\sqrt{{x}^{2}}\right)}}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      10. pow2 N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left(\sqrt{\color{blue}{x \cdot x}}\right)}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      11. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left(\left|x\right|\right)}}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left(\left|x\right|\right)}^{\color{blue}{6}} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, \color{blue}{{\left(\left|x\right|\right)}^{6}} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      14. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left(\sqrt{x \cdot x}\right)}}^{6} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      15. pow2 N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left(\sqrt{\color{blue}{{x}^{2}}}\right)}^{6} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      16. sqrt-pow1 N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left({x}^{\left(\frac{2}{2}\right)}\right)}}^{6} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      17. metadata-eval N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left({x}^{\color{blue}{1}}\right)}^{6} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      18. unpow1 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\left|\color{blue}{{x}^{6} \cdot \left(\frac{1}{21} \cdot \left|x\right| + \frac{1}{5} \cdot \frac{\left|x\right|}{{x}^{2}}\right)}\right|}{\sqrt{\pi}} \]

   8. Applied rewrites 34.5%

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

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot x\right) \cdot \left(\color{blue}{\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right)} \cdot \left(x \cdot x\right)\right)\right|}{\sqrt{\pi}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot x\right) \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(x \cdot x\right)\right)\right|}{\sqrt{\pi}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\left|\left(\frac{1}{21} \cdot x\right) \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{x}\right) \cdot \left(x \cdot x\right)\right)\right|}{\sqrt{\pi}} \]

       3. metadata-eval 36.4

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

   11. Applied rewrites 36.4%

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

Alternative 7: 89.5% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 89.5%

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

   if 2.2000000000000002 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, \color{blue}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      2. pow2 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, \color{blue}{{\left(\left(x \cdot x\right) \cdot x\right)}^{2}} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}^{2} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left(\color{blue}{\left(x \cdot x\right)} \cdot x\right)}^{2} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      5. pow3 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left({x}^{3}\right)}}^{2} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      6. pow-pow N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, \color{blue}{{x}^{\left(3 \cdot 2\right)}} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      7. unpow1 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left({x}^{1}\right)}}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left({x}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      9. sqrt-pow1 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left(\sqrt{{x}^{2}}\right)}}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      10. pow2 N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left(\sqrt{\color{blue}{x \cdot x}}\right)}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      11. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left(\left|x\right|\right)}}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left(\left|x\right|\right)}^{\color{blue}{6}} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, \color{blue}{{\left(\left|x\right|\right)}^{6}} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      14. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left(\sqrt{x \cdot x}\right)}}^{6} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      15. pow2 N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left(\sqrt{\color{blue}{{x}^{2}}}\right)}^{6} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      16. sqrt-pow1 N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left({x}^{\left(\frac{2}{2}\right)}\right)}}^{6} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      17. metadata-eval N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left({x}^{\color{blue}{1}}\right)}^{6} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      18. unpow1 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\left|\color{blue}{{x}^{6} \cdot \left(\frac{1}{21} \cdot \left|x\right| + \frac{1}{5} \cdot \frac{\left|x\right|}{{x}^{2}}\right)}\right|}{\sqrt{\pi}} \]

   8. Applied rewrites 34.5%

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

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\left|\frac{1}{21} \cdot \color{blue}{{x}^{7}}\right|}{\sqrt{\pi}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{\left|\frac{1}{21} \cdot {x}^{7}\right|}{\sqrt{\pi}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\left|{x}^{7} \cdot \frac{1}{\color{blue}{21}}\right|}{\sqrt{\pi}} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\left|{x}^{7} \cdot \frac{1}{\color{blue}{21}}\right|}{\sqrt{\pi}} \]

       4. lift-pow.f64 N/A

          \[\leadsto \frac{\left|{x}^{7} \cdot \frac{1}{21}\right|}{\sqrt{\pi}} \]

       5. metadata-eval 36.5

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

   11. Applied rewrites 36.5%

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

Alternative 8: 89.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, 0.047619047619047616, x + x\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs (* (/ 1.0 (sqrt PI)) (fma (pow x 7.0) 0.047619047619047616 (+ x x)))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * fma(pow(x, 7.0), 0.047619047619047616, (x + x))));
}

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * fma((x ^ 7.0), 0.047619047619047616, Float64(x + x))))
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)\right)}\right| \]

4. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)}\right| \]

5. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \color{blue}{2 \cdot \left|x\right|}\right)\right| \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left({\left(\left|x\right|\right)}^{7} \cdot \frac{1}{21} + 2 \cdot \left|\color{blue}{x}\right|\right)\right| \]

   2. lower-fma.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

   3. lift-pow.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

   4. rem-sqrt-square-rev N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\sqrt{x \cdot x}\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

   5. pow2 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\sqrt{{x}^{2}}\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

   6. sqrt-pow1 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left({x}^{\left(\frac{2}{2}\right)}\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

   7. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left({x}^{1}\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

   8. unpow1 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

   9. count-2-rev N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \left|x\right| + \left|x\right|\right)\right| \]

   10. lower-+.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \left|x\right| + \left|x\right|\right)\right| \]

   11. rem-sqrt-square-rev N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \sqrt{x \cdot x} + \left|x\right|\right)\right| \]

   12. pow2 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \sqrt{{x}^{2}} + \left|x\right|\right)\right| \]

   13. sqrt-pow1 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, {x}^{\left(\frac{2}{2}\right)} + \left|x\right|\right)\right| \]

   14. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, {x}^{1} + \left|x\right|\right)\right| \]

   15. unpow1 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + \left|x\right|\right)\right| \]

   16. rem-sqrt-square-rev N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + \sqrt{x \cdot x}\right)\right| \]

   17. pow2 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + \sqrt{{x}^{2}}\right)\right| \]

   18. sqrt-pow1 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + {x}^{\left(\frac{2}{2}\right)}\right)\right| \]

   19. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + {x}^{1}\right)\right| \]

   20. unpow1 98.8

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, 0.047619047619047616, x + x\right)\right| \]

7. Applied rewrites 98.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \color{blue}{0.047619047619047616}, x + x\right)\right| \]
8. Add Preprocessing

Alternative 9: 89.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.25:\\ \;\;\;\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.2\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.25)
   (fabs (* (/ 1.0 (sqrt PI)) (* x (fma 0.6666666666666666 (* x x) 2.0))))
   (/ (fabs (* (* (* (* x x) x) (* x x)) 0.2)) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (x <= 2.25) {
		tmp = fabs(((1.0 / sqrt(((double) M_PI))) * (x * fma(0.6666666666666666, (x * x), 2.0))));
	} else {
		tmp = fabs(((((x * x) * x) * (x * x)) * 0.2)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.25)
		tmp = abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(x * fma(0.6666666666666666, Float64(x * x), 2.0))));
	else
		tmp = Float64(abs(Float64(Float64(Float64(Float64(x * x) * x) * Float64(x * x)) * 0.2)) / sqrt(pi));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.25], N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(x * N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.2), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.25

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 89.5%

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

   if 2.25 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, \color{blue}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      2. pow2 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, \color{blue}{{\left(\left(x \cdot x\right) \cdot x\right)}^{2}} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}^{2} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left(\color{blue}{\left(x \cdot x\right)} \cdot x\right)}^{2} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      5. pow3 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left({x}^{3}\right)}}^{2} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      6. pow-pow N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, \color{blue}{{x}^{\left(3 \cdot 2\right)}} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      7. unpow1 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left({x}^{1}\right)}}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left({x}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      9. sqrt-pow1 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left(\sqrt{{x}^{2}}\right)}}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      10. pow2 N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left(\sqrt{\color{blue}{x \cdot x}}\right)}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      11. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left(\left|x\right|\right)}}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left(\left|x\right|\right)}^{\color{blue}{6}} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      13. lower-pow.f64 N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, \color{blue}{{\left(\left|x\right|\right)}^{6}} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      14. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left(\sqrt{x \cdot x}\right)}}^{6} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      15. pow2 N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left(\sqrt{\color{blue}{{x}^{2}}}\right)}^{6} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      16. sqrt-pow1 N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left({x}^{\left(\frac{2}{2}\right)}\right)}}^{6} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      17. metadata-eval N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left({x}^{\color{blue}{1}}\right)}^{6} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      18. unpow1 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\left|\color{blue}{{x}^{6} \cdot \left(\frac{1}{21} \cdot \left|x\right| + \frac{1}{5} \cdot \frac{\left|x\right|}{{x}^{2}}\right)}\right|}{\sqrt{\pi}} \]

   8. Applied rewrites 34.5%

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

   9. Taylor expanded in x around 0

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

       1. metadata-eval N/A

          \[\leadsto \frac{\left|\frac{1}{5} \cdot {x}^{5}\right|}{\sqrt{\pi}} \]

       2. *-commutative N/A

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

       3. sqr-pow N/A

          \[\leadsto \frac{\left|\left({x}^{\left(\frac{5}{2}\right)} \cdot {x}^{\left(\frac{5}{2}\right)}\right) \cdot \frac{1}{5}\right|}{\sqrt{\pi}} \]

       4. pow-prod-down N/A

          \[\leadsto \frac{\left|{\left(x \cdot x\right)}^{\left(\frac{5}{2}\right)} \cdot \frac{1}{5}\right|}{\sqrt{\pi}} \]

       5. sqrt-pow2 N/A

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

       6. rem-sqrt-square-rev N/A

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

       7. metadata-eval N/A

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

       8. pow-prod-up N/A

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

       9. pow3 N/A

          \[\leadsto \frac{\left|\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot {\left(\left|x\right|\right)}^{2}\right) \cdot \frac{1}{5}\right|}{\sqrt{\pi}} \]

       10. pow2 N/A

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

       11. associate-*l* N/A

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

       12. lower-*.f64 N/A

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

   11. Applied rewrites 31.1%

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

Alternative 10: 89.5% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 89.5%

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

Alternative 11: 89.0% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, \color{blue}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   2. pow2 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, \color{blue}{{\left(\left(x \cdot x\right) \cdot x\right)}^{2}} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}^{2} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left(\color{blue}{\left(x \cdot x\right)} \cdot x\right)}^{2} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   5. pow3 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left({x}^{3}\right)}}^{2} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   6. pow-pow N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, \color{blue}{{x}^{\left(3 \cdot 2\right)}} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   7. unpow1 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left({x}^{1}\right)}}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left({x}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   9. sqrt-pow1 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left(\sqrt{{x}^{2}}\right)}}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   10. pow2 N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left(\sqrt{\color{blue}{x \cdot x}}\right)}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   11. rem-sqrt-square-rev N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left(\left|x\right|\right)}}^{\left(3 \cdot 2\right)} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left(\left|x\right|\right)}^{\color{blue}{6}} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   13. lower-pow.f64 N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, \color{blue}{{\left(\left|x\right|\right)}^{6}} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   14. rem-sqrt-square-rev N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left(\sqrt{x \cdot x}\right)}}^{6} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   15. pow2 N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left(\sqrt{\color{blue}{{x}^{2}}}\right)}^{6} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   16. sqrt-pow1 N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\color{blue}{\left({x}^{\left(\frac{2}{2}\right)}\right)}}^{6} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   17. metadata-eval N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, {\left({x}^{\color{blue}{1}}\right)}^{6} \cdot \frac{1}{21}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   18. unpow1 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 89.0%

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

Alternative 12: 68.1% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5000:\\ \;\;\;\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(x + x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(x + x\right) \cdot \left(x + x\right)}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5000.0)
   (fabs (* (/ 1.0 (sqrt PI)) (+ x x)))
   (sqrt (/ (* (+ x x) (+ x x)) PI))))

C

double code(double x) {
	double tmp;
	if (x <= 5000.0) {
		tmp = fabs(((1.0 / sqrt(((double) M_PI))) * (x + x)));
	} else {
		tmp = sqrt((((x + x) * (x + x)) / ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5000.0) {
		tmp = Math.abs(((1.0 / Math.sqrt(Math.PI)) * (x + x)));
	} else {
		tmp = Math.sqrt((((x + x) * (x + x)) / Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5000.0:
		tmp = math.fabs(((1.0 / math.sqrt(math.pi)) * (x + x)))
	else:
		tmp = math.sqrt((((x + x) * (x + x)) / math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5000.0)
		tmp = abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(x + x)));
	else
		tmp = sqrt(Float64(Float64(Float64(x + x) * Float64(x + x)) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5000.0)
		tmp = abs(((1.0 / sqrt(pi)) * (x + x)));
	else
		tmp = sqrt((((x + x) * (x + x)) / pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5000.0], N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(x + x), $MachinePrecision] * N[(x + x), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e3

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 68.1%

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

   if 5e3 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 68.1%

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

      1. lift-fabs.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. fabs-div N/A

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

   8. Applied rewrites 67.6%

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

   10. Applied rewrites 53.3%

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

Alternative 13: 68.1% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 68.1%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(x + x\right)}\right| \]
7. Add Preprocessing

Alternative 14: 67.6% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 68.1%

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

   1. lift-fabs.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. fabs-div N/A

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

8. Applied rewrites 67.6%

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

   1. metadata-eval 67.6

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

   2. metadata-eval 67.6

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

   3. metadata-eval 67.6

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

10. Applied rewrites 67.6%

    \[\leadsto \frac{\color{blue}{\left|x + x\right|}}{\sqrt{\pi}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025120 
(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)))))))