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

Percentage Accurate: 99.8% → 99.8%

Time: 14.4s

Alternatives: 13

Speedup: 3.0×

• 28.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
      (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x))))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x))))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
      (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x))))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x))))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \left|\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right), 0.2 \cdot {\left(\left|x\right|\right)}^{5}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (fma
    0.047619047619047616
    (pow (fabs x) 7.0)
    (fma
     (fabs x)
     (fma x (* x 0.6666666666666666) 2.0)
     (* 0.2 (pow (fabs x) 5.0))))
   (sqrt (/ 1.0 PI)))))

C

double code(double x) {
	return fabs((fma(0.047619047619047616, pow(fabs(x), 7.0), fma(fabs(x), fma(x, (x * 0.6666666666666666), 2.0), (0.2 * pow(fabs(x), 5.0)))) * sqrt((1.0 / ((double) M_PI)))));
}

Julia

function code(x)
	return abs(Float64(fma(0.047619047619047616, (abs(x) ^ 7.0), fma(abs(x), fma(x, Float64(x * 0.6666666666666666), 2.0), Float64(0.2 * (abs(x) ^ 5.0)))) * sqrt(Float64(1.0 / pi))))
end

Wolfram

code[x_] := N[Abs[N[(N[(0.047619047619047616 * N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] * N[(x * N[(x * 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision] + N[(0.2 * N[Power[N[Abs[x], $MachinePrecision], 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / Pi), $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. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)\right)}\right| \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.9%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right), 0.2 \cdot {\left(\left|x\right|\right)}^{5}\right)\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]
6. Add Preprocessing

Alternative 2: 99.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 3: 99.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left(\left(t\_0 \cdot t\_0\right) \cdot \left|\frac{x}{\sqrt{\pi}}\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= (fabs x) 0.4)
     (fabs
      (*
       (/ 1.0 (sqrt PI))
       (* (fabs x) (fma (* x x) (fma x (* x 0.2) 0.6666666666666666) 2.0))))
     (* 0.047619047619047616 (* (* t_0 t_0) (fabs (/ x (sqrt PI))))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double tmp;
	if (fabs(x) <= 0.4) {
		tmp = fabs(((1.0 / sqrt(((double) M_PI))) * (fabs(x) * fma((x * x), fma(x, (x * 0.2), 0.6666666666666666), 2.0))));
	} else {
		tmp = 0.047619047619047616 * ((t_0 * t_0) * fabs((x / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (abs(x) <= 0.4)
		tmp = abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(abs(x) * fma(Float64(x * x), fma(x, Float64(x * 0.2), 0.6666666666666666), 2.0))));
	else
		tmp = Float64(0.047619047619047616 * Float64(Float64(t_0 * t_0) * abs(Float64(x / sqrt(pi)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.4], N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(0.047619047619047616 * N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[Abs[N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.40000000000000002

   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. Add Preprocessing

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(2 \cdot \left|x\right| + {x}^{2} \cdot \left(\frac{1}{5} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{2}{3} \cdot \left|x\right|\right)\right)}\right| \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

   7. Applied rewrites 99.3%

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

   if 0.40000000000000002 < (fabs.f64 x)

   1. Initial program 99.8%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r* N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(\left(\frac{1}{21} \cdot {x}^{6}\right) \cdot \left|x\right|\right)}\right| \]

      2. *-commutative N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(\left|x\right| \cdot \left(\frac{1}{21} \cdot {x}^{6}\right)\right)}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(\left|x\right| \cdot \left(\frac{1}{21} \cdot {x}^{6}\right)\right)}\right| \]

      4. lower-fabs.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\color{blue}{\left|x\right|} \cdot \left(\frac{1}{21} \cdot {x}^{6}\right)\right)\right| \]

      5. lower-*.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \color{blue}{\left(\frac{1}{21} \cdot {x}^{6}\right)}\right)\right| \]

      6. metadata-eval N/A

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

      7. pow-plus N/A

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

      8. metadata-eval N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\frac{1}{21} \cdot \left({x}^{\color{blue}{\left(4 + 1\right)}} \cdot x\right)\right)\right)\right| \]

      9. pow-plus N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\frac{1}{21} \cdot \left(\color{blue}{\left({x}^{4} \cdot x\right)} \cdot x\right)\right)\right)\right| \]

      10. associate-*r* N/A

          \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\frac{1}{21} \cdot \color{blue}{\left({x}^{4} \cdot \left(x \cdot x\right)\right)}\right)\right)\right| \]

      11. unpow2 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\frac{1}{21} \cdot \left({x}^{4} \cdot \color{blue}{{x}^{2}}\right)\right)\right)\right| \]

      12. *-commutative N/A

          \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\frac{1}{21} \cdot \color{blue}{\left({x}^{2} \cdot {x}^{4}\right)}\right)\right)\right| \]

      13. lower-*.f64 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\frac{1}{21} \cdot \color{blue}{\left({x}^{2} \cdot {x}^{4}\right)}\right)\right)\right| \]

      14. unpow2 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\frac{1}{21} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{4}\right)\right)\right)\right| \]

      15. lower-*.f64 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\frac{1}{21} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{4}\right)\right)\right)\right| \]

      16. metadata-eval N/A

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

      17. pow-sqr N/A

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

      18. unpow2 N/A

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

      19. associate-*l* N/A

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

      20. unpow2 N/A

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

      21. cube-unmult N/A

          \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\frac{1}{21} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{{x}^{3}}\right)\right)\right)\right)\right| \]

      22. metadata-eval N/A

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

      23. pow-plus N/A

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

      24. *-rgt-identity N/A

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

      25. *-inverses N/A

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

      26. associate-/l* N/A

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

      27. unpow2 N/A

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

      28. associate-/l* N/A

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

      29. pow-sqr N/A

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

      30. metadata-eval N/A

          \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\frac{1}{21} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{{x}^{\color{blue}{4}}}{x}\right)\right)\right)\right)\right| \]

      31. lower-*.f64 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\frac{1}{21} \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot \frac{{x}^{4}}{x}\right)}\right)\right)\right)\right| \]

   7. Applied rewrites 98.6%

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

   9. Applied rewrites 98.6%

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

       1. lift-PI.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-fabs.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lift-*.f64 N/A

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

       12. associate-*r* N/A

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

       13. lift-*.f64 N/A

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

       14. cube-unmult N/A

           \[\leadsto \left(\left|\frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right| \cdot \color{blue}{{\left(x \cdot x\right)}^{3}}\right) \cdot \frac{1}{21} \]

       15. *-commutative N/A

           \[\leadsto \color{blue}{\left({\left(x \cdot x\right)}^{3} \cdot \left|\frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right|\right)} \cdot \frac{1}{21} \]

       16. lower-*.f64 N/A

           \[\leadsto \color{blue}{\left({\left(x \cdot x\right)}^{3} \cdot \left|\frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right|\right)} \cdot \frac{1}{21} \]

   11. Applied rewrites 98.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left|\frac{x}{\sqrt{\pi}}\right|\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left(\left|\frac{x}{\sqrt{\pi}}\right| \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.4)
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (* (fabs x) (fma (* x x) (fma x (* x 0.2) 0.6666666666666666) 2.0))))
   (*
    0.047619047619047616
    (* (fabs (/ x (sqrt PI))) (* x (* x (* x (* x (* x x)))))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.4) {
		tmp = fabs(((1.0 / sqrt(((double) M_PI))) * (fabs(x) * fma((x * x), fma(x, (x * 0.2), 0.6666666666666666), 2.0))));
	} else {
		tmp = 0.047619047619047616 * (fabs((x / sqrt(((double) M_PI)))) * (x * (x * (x * (x * (x * x))))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.4)
		tmp = abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(abs(x) * fma(Float64(x * x), fma(x, Float64(x * 0.2), 0.6666666666666666), 2.0))));
	else
		tmp = Float64(0.047619047619047616 * Float64(abs(Float64(x / sqrt(pi))) * Float64(x * Float64(x * Float64(x * Float64(x * Float64(x * x)))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.4], N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(0.047619047619047616 * N[(N[Abs[N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.40000000000000002

   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. Add Preprocessing

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(2 \cdot \left|x\right| + {x}^{2} \cdot \left(\frac{1}{5} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{2}{3} \cdot \left|x\right|\right)\right)}\right| \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

   7. Applied rewrites 99.3%

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

   if 0.40000000000000002 < (fabs.f64 x)

   1. Initial program 99.8%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r* N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(\left(\frac{1}{21} \cdot {x}^{6}\right) \cdot \left|x\right|\right)}\right| \]

      2. *-commutative N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(\left|x\right| \cdot \left(\frac{1}{21} \cdot {x}^{6}\right)\right)}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(\left|x\right| \cdot \left(\frac{1}{21} \cdot {x}^{6}\right)\right)}\right| \]

      4. lower-fabs.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\color{blue}{\left|x\right|} \cdot \left(\frac{1}{21} \cdot {x}^{6}\right)\right)\right| \]

      5. lower-*.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \color{blue}{\left(\frac{1}{21} \cdot {x}^{6}\right)}\right)\right| \]

      6. metadata-eval N/A

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

      7. pow-plus N/A

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

      8. metadata-eval N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\frac{1}{21} \cdot \left({x}^{\color{blue}{\left(4 + 1\right)}} \cdot x\right)\right)\right)\right| \]

      9. pow-plus N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\frac{1}{21} \cdot \left(\color{blue}{\left({x}^{4} \cdot x\right)} \cdot x\right)\right)\right)\right| \]

      10. associate-*r* N/A

          \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\frac{1}{21} \cdot \color{blue}{\left({x}^{4} \cdot \left(x \cdot x\right)\right)}\right)\right)\right| \]

      11. unpow2 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\frac{1}{21} \cdot \left({x}^{4} \cdot \color{blue}{{x}^{2}}\right)\right)\right)\right| \]

      12. *-commutative N/A

          \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\frac{1}{21} \cdot \color{blue}{\left({x}^{2} \cdot {x}^{4}\right)}\right)\right)\right| \]

      13. lower-*.f64 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\frac{1}{21} \cdot \color{blue}{\left({x}^{2} \cdot {x}^{4}\right)}\right)\right)\right| \]

      14. unpow2 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\frac{1}{21} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{4}\right)\right)\right)\right| \]

      15. lower-*.f64 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\frac{1}{21} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{4}\right)\right)\right)\right| \]

      16. metadata-eval N/A

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

      17. pow-sqr N/A

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

      18. unpow2 N/A

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

      19. associate-*l* N/A

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

      20. unpow2 N/A

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

      21. cube-unmult N/A

          \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\frac{1}{21} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{{x}^{3}}\right)\right)\right)\right)\right| \]

      22. metadata-eval N/A

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

      23. pow-plus N/A

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

      24. *-rgt-identity N/A

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

      25. *-inverses N/A

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

      26. associate-/l* N/A

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

      27. unpow2 N/A

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

      28. associate-/l* N/A

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

      29. pow-sqr N/A

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

      30. metadata-eval N/A

          \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\frac{1}{21} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{{x}^{\color{blue}{4}}}{x}\right)\right)\right)\right)\right| \]

      31. lower-*.f64 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\frac{1}{21} \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot \frac{{x}^{4}}{x}\right)}\right)\right)\right)\right| \]

   7. Applied rewrites 98.6%

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

   9. Applied rewrites 98.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left(\left|\frac{x}{\sqrt{\pi}}\right| \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.4%

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

6. Applied rewrites 99.8%

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

8. Applied rewrites 99.8%

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

9. Final simplification 99.8%

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

Alternative 6: 99.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.4%

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

6. Applied rewrites 99.8%

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

8. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. distribute-rgt-in N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 99.8

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

   12. lift-fma.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-*.f64 99.8

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

10. Applied rewrites 99.8%

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

11. Final simplification 99.8%

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

Alternative 7: 93.6% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(2 \cdot \left|x\right| + {x}^{2} \cdot \left(\frac{1}{5} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{2}{3} \cdot \left|x\right|\right)\right)}\right| \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. +-commutative N/A

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

   8. associate-*r* N/A

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

   9. distribute-rgt-in N/A

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

   10. associate-*r* N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

7. Applied rewrites 93.1%

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

Alternative 8: 93.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|\mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right) \cdot \left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{x \cdot \left(0.2 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.4)
   (fabs
    (* (fma x (* x 0.6666666666666666) 2.0) (* (fabs x) (sqrt (/ 1.0 PI)))))
   (fabs (* x (/ (* x (* 0.2 (* x (* x x)))) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.4) {
		tmp = fabs((fma(x, (x * 0.6666666666666666), 2.0) * (fabs(x) * sqrt((1.0 / ((double) M_PI))))));
	} else {
		tmp = fabs((x * ((x * (0.2 * (x * (x * x)))) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.4)
		tmp = abs(Float64(fma(x, Float64(x * 0.6666666666666666), 2.0) * Float64(abs(x) * sqrt(Float64(1.0 / pi)))));
	else
		tmp = abs(Float64(x * Float64(Float64(x * Float64(0.2 * Float64(x * Float64(x * x)))) / sqrt(pi))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.4], N[Abs[N[(N[(x * N[(x * 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(N[(x * N[(0.2 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.40000000000000002

   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. Add Preprocessing

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

   7. Applied rewrites 99.0%

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

   if 0.40000000000000002 < (fabs.f64 x)

   1. Initial program 99.8%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left|\frac{\color{blue}{2 \cdot \left|x\right| + {x}^{2} \cdot \left(\frac{1}{5} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{2}{3} \cdot \left|x\right|\right)}}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-in N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

   7. Applied rewrites 81.8%

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

      1. lift-fabs.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. fabs-mul N/A

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

   9. Applied rewrites 81.8%

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

   10. Taylor expanded in x around inf

       \[\leadsto \left|x \cdot \frac{\color{blue}{\frac{1}{5} \cdot {x}^{4}}}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]
   11. Step-by-step derivation

       1. metadata-eval N/A

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

       2. pow-sqr N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. unpow2 N/A

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

       6. associate-*l* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

       10. unpow3 N/A

           \[\leadsto \left|x \cdot \frac{x \cdot \left(\frac{1}{5} \cdot \color{blue}{{x}^{3}}\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

       11. lower-*.f64 N/A

           \[\leadsto \left|x \cdot \frac{\color{blue}{x \cdot \left(\frac{1}{5} \cdot {x}^{3}\right)}}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

       12. lower-*.f64 N/A

           \[\leadsto \left|x \cdot \frac{x \cdot \color{blue}{\left(\frac{1}{5} \cdot {x}^{3}\right)}}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

       13. cube-mult N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 N/A

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

       16. unpow2 N/A

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

       17. lower-*.f64 81.8

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

   12. Applied rewrites 81.8%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|\mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right) \cdot \left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{x \cdot \left(0.2 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.6% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[Abs[N[(x * N[(N[(x * N[(x * N[(N[(x * 0.2), $MachinePrecision] * x + 0.6666666666666666), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $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. Add Preprocessing

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

   \[\leadsto \left|\frac{\color{blue}{2 \cdot \left|x\right| + {x}^{2} \cdot \left(\frac{1}{5} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{2}{3} \cdot \left|x\right|\right)}}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. +-commutative N/A

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

   8. associate-*r* N/A

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

   9. distribute-rgt-in N/A

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

   10. associate-*r* N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

7. Applied rewrites 92.6%

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

   1. lift-fabs.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/l* N/A

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

   11. fabs-mul N/A

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

9. Applied rewrites 93.0%

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

    1. associate-*l* N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. lower-*.f64 93.0

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

11. Applied rewrites 93.0%

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

Alternative 10: 93.1% accurate, 3.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (fabs (* x (/ (fma x (* x (* 0.2 (* x x))) 2.0) (sqrt PI)))))

C

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

Julia

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

Wolfram

code[x_] := N[Abs[N[(x * N[(N[(x * N[(x * N[(0.2 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $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. Add Preprocessing

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

   \[\leadsto \left|\frac{\color{blue}{2 \cdot \left|x\right| + {x}^{2} \cdot \left(\frac{1}{5} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{2}{3} \cdot \left|x\right|\right)}}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. +-commutative N/A

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

   8. associate-*r* N/A

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

   9. distribute-rgt-in N/A

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

   10. associate-*r* N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

7. Applied rewrites 92.6%

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

   1. lift-fabs.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/l* N/A

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

   11. fabs-mul N/A

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

9. Applied rewrites 93.0%

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

10. Taylor expanded in x around inf

    \[\leadsto \left|x \cdot \frac{\mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{5} \cdot {x}^{2}\right)}, 2\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]
11. Step-by-step derivation

    1. lower-*.f64 N/A

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

    2. unpow2 N/A

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

    3. lower-*.f64 92.3

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

12. Applied rewrites 92.3%

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

Alternative 11: 89.5% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-in N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-in N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. lower-*.f64 N/A

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

7. Applied rewrites 88.8%

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

8. Final simplification 88.8%

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

Alternative 12: 89.5% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

   \[\leadsto \left|\frac{\color{blue}{2 \cdot \left|x\right| + {x}^{2} \cdot \left(\frac{1}{5} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{2}{3} \cdot \left|x\right|\right)}}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. +-commutative N/A

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

   8. associate-*r* N/A

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

   9. distribute-rgt-in N/A

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

   10. associate-*r* N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

7. Applied rewrites 92.6%

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

   1. lift-fabs.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/l* N/A

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

   11. fabs-mul N/A

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

9. Applied rewrites 93.0%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. unpow2 N/A

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

    4. lower-*.f64 88.8

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

12. Applied rewrites 88.8%

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

Alternative 13: 68.2% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-fabs.f64 64.9

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

7. Applied rewrites 64.9%

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

   1. lift-fabs.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. fabs-div N/A

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

   6. lift-*.f64 N/A

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

   7. fabs-mul N/A

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

   8. lift-fabs.f64 N/A

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

   9. fabs-fabs N/A

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

   10. lift-fabs.f64 N/A

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

   11. metadata-eval N/A

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

   12. rem-sqrt-square N/A

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

   13. sqrt-prod N/A

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

   14. rem-square-sqrt N/A

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

   15. associate-/l* N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-/.f64 65.3

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

9. Applied rewrites 65.3%

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

10. Final simplification 65.3%

    \[\leadsto \left|x\right| \cdot \frac{2}{\sqrt{\pi}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(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)))))))