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

Percentage Accurate: 99.8% → 99.9%

Time: 3.2s

Alternatives: 7

Speedup: 1.3×

• 28.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := 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] * N[(N[(x * x), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * N[Abs[N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.047619047619047616 * N[Power[N[Abs[x], $MachinePrecision], 5.0], $MachinePrecision]), $MachinePrecision] * N[(x * x), $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
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 77.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision] * x), $MachinePrecision] + N[(0.2 * N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.047619047619047616 * N[Power[N[Abs[x], $MachinePrecision], 5.0], $MachinePrecision]), $MachinePrecision] * N[(x * x), $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
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

4. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \color{blue}{\left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{5}\right) \cdot \left(x \cdot x\right)}\right)\right| \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. lift-fabs.f64 N/A

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

   10. sqr-abs-rev N/A

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

   11. pow2 N/A

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

   12. associate-*r* N/A

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

   13. distribute-rgt-in N/A

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

   14. *-commutative N/A

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

   15. +-commutative N/A

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

   16. *-commutative N/A

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

   17. pow2 N/A

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

   18. lower-*.f64 N/A

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

6. Applied rewrites 76.1%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval 76.1

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

8. Applied rewrites 76.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. lift-fabs.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. sqr-abs-rev N/A

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

   8. lift-*.f64 N/A

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

   9. lift-fabs.f64 N/A

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

   10. associate-*l* N/A

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

   11. sqr-abs-rev N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 76.1

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

10. Applied rewrites 76.1%

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

Alternative 3: 99.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right|\\ \left|\frac{-1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(\left(x \cdot x\right) \cdot 0.6666666666666666\right) \cdot \left|x\right|\right) + 0.2 \cdot t\_0\right) - \frac{-1}{21} \cdot \left(\left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fabs (* (* (* (* x x) x) x) x))))
   (fabs
    (*
     (/ -1.0 (sqrt PI))
     (-
      (+
       (+ (* 2.0 (fabs x)) (* (* (* x x) 0.6666666666666666) (fabs x)))
       (* 0.2 t_0))
      (* (/ -1.0 21.0) (* (* t_0 (fabs x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = fabs(((((x * x) * x) * x) * x));
	return fabs(((-1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + (((x * x) * 0.6666666666666666) * fabs(x))) + (0.2 * t_0)) - ((-1.0 / 21.0) * ((t_0 * fabs(x)) * fabs(x))))));
}

Java

public static double code(double x) {
	double t_0 = Math.abs(((((x * x) * x) * x) * x));
	return Math.abs(((-1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + (((x * x) * 0.6666666666666666) * Math.abs(x))) + (0.2 * t_0)) - ((-1.0 / 21.0) * ((t_0 * Math.abs(x)) * Math.abs(x))))));
}

Python

def code(x):
	t_0 = math.fabs(((((x * x) * x) * x) * x))
	return math.fabs(((-1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + (((x * x) * 0.6666666666666666) * math.fabs(x))) + (0.2 * t_0)) - ((-1.0 / 21.0) * ((t_0 * math.fabs(x)) * math.fabs(x))))))

Julia

function code(x)
	t_0 = abs(Float64(Float64(Float64(Float64(x * x) * x) * x) * x))
	return abs(Float64(Float64(-1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(Float64(x * x) * 0.6666666666666666) * abs(x))) + Float64(0.2 * t_0)) - Float64(Float64(-1.0 / 21.0) * Float64(Float64(t_0 * abs(x)) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = abs(((((x * x) * x) * x) * x));
	tmp = abs(((-1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + (((x * x) * 0.6666666666666666) * abs(x))) + (0.2 * t_0)) - ((-1.0 / 21.0) * ((t_0 * abs(x)) * abs(x))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[Abs[N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 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[(N[(x * x), $MachinePrecision] * 0.6666666666666666), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2 * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $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
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.8

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

   7. lift-*.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. lift-fabs.f64 N/A

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

   10. sqr-abs N/A

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

   11. lower-*.f64 99.8

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

   12. lift-/.f64 N/A

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

   13. metadata-eval 99.8

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(\left(x \cdot x\right) \cdot \color{blue}{0.6666666666666666}\right) \cdot \left|x\right|\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

4. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \color{blue}{\left(\left(x \cdot x\right) \cdot 0.6666666666666666\right) \cdot \left|x\right|}\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. metadata-eval 99.8

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(\left(x \cdot x\right) \cdot 0.6666666666666666\right) \cdot \left|x\right|\right) + \color{blue}{0.2} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

6. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(\left(x \cdot x\right) \cdot 0.6666666666666666\right) \cdot \left|x\right|\right) + \color{blue}{0.2} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

7. Final simplification 99.8%

   \[\leadsto \left|\frac{-1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \left(\left(x \cdot x\right) \cdot 0.6666666666666666\right) \cdot \left|x\right|\right) + 0.2 \cdot \left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right|\right) - \frac{-1}{21} \cdot \left(\left(\left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
8. Add Preprocessing

Alternative 4: 99.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right|\\ \left|\frac{-1}{\sqrt{\pi}} \cdot \left(\left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) + 0.2 \cdot t\_0\right) - \frac{-1}{21} \cdot \left(\left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fabs (* (* (* (* x x) x) x) x))))
   (fabs
    (*
     (/ -1.0 (sqrt PI))
     (-
      (+ (* (fabs x) (fma (* x x) 0.6666666666666666 2.0)) (* 0.2 t_0))
      (* (/ -1.0 21.0) (* (* t_0 (fabs x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = fabs(((((x * x) * x) * x) * x));
	return fabs(((-1.0 / sqrt(((double) M_PI))) * (((fabs(x) * fma((x * x), 0.6666666666666666, 2.0)) + (0.2 * t_0)) - ((-1.0 / 21.0) * ((t_0 * fabs(x)) * fabs(x))))));
}

Julia

function code(x)
	t_0 = abs(Float64(Float64(Float64(Float64(x * x) * x) * x) * x))
	return abs(Float64(Float64(-1.0 / sqrt(pi)) * Float64(Float64(Float64(abs(x) * fma(Float64(x * x), 0.6666666666666666, 2.0)) + Float64(0.2 * t_0)) - Float64(Float64(-1.0 / 21.0) * Float64(Float64(t_0 * abs(x)) * abs(x))))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Abs[N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(-1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision]), $MachinePrecision] + N[(0.2 * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $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
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. lift-fabs.f64 N/A

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

   9. distribute-rgt-out N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\color{blue}{\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)} + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. metadata-eval 99.8

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) + \color{blue}{0.2} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

6. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) + \color{blue}{0.2} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

7. Final simplification 99.8%

   \[\leadsto \left|\frac{-1}{\sqrt{\pi}} \cdot \left(\left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) + 0.2 \cdot \left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right|\right) - \frac{-1}{21} \cdot \left(\left(\left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
8. Add Preprocessing

Alternative 5: 99.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \left|\frac{-1}{\sqrt{\pi}} \cdot \left(\left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) + 0.2 \cdot \left(t\_0 \cdot \left|x\right|\right)\right) - \frac{-1}{21} \cdot \left(\left|\left(t\_0 \cdot x\right) \cdot x\right| \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) (* x x))))
   (fabs
    (*
     (/ -1.0 (sqrt PI))
     (-
      (+
       (* (fabs x) (fma (* x x) 0.6666666666666666 2.0))
       (* 0.2 (* t_0 (fabs x))))
      (* (/ -1.0 21.0) (* (fabs (* (* t_0 x) x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = (x * x) * (x * x);
	return fabs(((-1.0 / sqrt(((double) M_PI))) * (((fabs(x) * fma((x * x), 0.6666666666666666, 2.0)) + (0.2 * (t_0 * fabs(x)))) - ((-1.0 / 21.0) * (fabs(((t_0 * x) * x)) * fabs(x))))));
}

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * Float64(x * x))
	return abs(Float64(Float64(-1.0 / sqrt(pi)) * Float64(Float64(Float64(abs(x) * fma(Float64(x * x), 0.6666666666666666, 2.0)) + Float64(0.2 * Float64(t_0 * abs(x)))) - Float64(Float64(-1.0 / 21.0) * Float64(abs(Float64(Float64(t_0 * x) * x)) * abs(x))))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(-1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.0 / 21.0), $MachinePrecision] * N[(N[Abs[N[(N[(t$95$0 * x), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[Abs[x], $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
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. lift-fabs.f64 N/A

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

   9. distribute-rgt-out N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\color{blue}{\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)} + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. metadata-eval 99.8

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) + \color{blue}{0.2} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

6. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) + \color{blue}{0.2} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. lift-fabs.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. sqr-abs-rev N/A

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

   9. pow2 N/A

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

   10. associate-*l* N/A

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

   11. sqr-abs-rev N/A

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

   12. pow2 N/A

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

   13. lower-*.f64 N/A

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

   14. pow2 N/A

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

   15. lift-*.f64 N/A

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

   16. pow2 N/A

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

   17. lift-*.f64 99.8

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) + 0.2 \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

8. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) + 0.2 \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. lift-fabs.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. sqr-abs-rev N/A

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

   8. lift-*.f64 N/A

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

   9. lift-fabs.f64 N/A

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

   10. associate-*l* N/A

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

   11. sqr-abs-rev N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 99.8

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) + 0.2 \cdot \left(\color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

10. Applied rewrites 99.8%

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) + 0.2 \cdot \left(\color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

11. Final simplification 99.8%

    \[\leadsto \left|\frac{-1}{\sqrt{\pi}} \cdot \left(\left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) + 0.2 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right)\right) - \frac{-1}{21} \cdot \left(\left|\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x\right| \cdot \left|x\right|\right)\right)\right| \]
12. Add Preprocessing

Alternative 6: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left|\frac{-1}{\sqrt{\pi}} \cdot \left(\left(\left|x\right| \cdot 2 + 0.2 \cdot \left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right|\right) - \frac{-1}{21} \cdot \left(\left|\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x\right| \cdot \left|x\right|\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ -1.0 (sqrt PI))
   (-
    (+ (* (fabs x) 2.0) (* 0.2 (fabs (* (* (* (* x x) x) x) x))))
    (* (/ -1.0 21.0) (* (fabs (* (* (* (* x x) (* x x)) x) x)) (fabs x)))))))

C

double code(double x) {
	return fabs(((-1.0 / sqrt(((double) M_PI))) * (((fabs(x) * 2.0) + (0.2 * fabs(((((x * x) * x) * x) * x)))) - ((-1.0 / 21.0) * (fabs(((((x * x) * (x * x)) * x) * x)) * fabs(x))))));
}

Java

public static double code(double x) {
	return Math.abs(((-1.0 / Math.sqrt(Math.PI)) * (((Math.abs(x) * 2.0) + (0.2 * Math.abs(((((x * x) * x) * x) * x)))) - ((-1.0 / 21.0) * (Math.abs(((((x * x) * (x * x)) * x) * x)) * Math.abs(x))))));
}

Python

def code(x):
	return math.fabs(((-1.0 / math.sqrt(math.pi)) * (((math.fabs(x) * 2.0) + (0.2 * math.fabs(((((x * x) * x) * x) * x)))) - ((-1.0 / 21.0) * (math.fabs(((((x * x) * (x * x)) * x) * x)) * math.fabs(x))))))

Julia

function code(x)
	return abs(Float64(Float64(-1.0 / sqrt(pi)) * Float64(Float64(Float64(abs(x) * 2.0) + Float64(0.2 * abs(Float64(Float64(Float64(Float64(x * x) * x) * x) * x)))) - Float64(Float64(-1.0 / 21.0) * Float64(abs(Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * x) * x)) * abs(x))))))
end

MATLAB

function tmp = code(x)
	tmp = abs(((-1.0 / sqrt(pi)) * (((abs(x) * 2.0) + (0.2 * abs(((((x * x) * x) * x) * x)))) - ((-1.0 / 21.0) * (abs(((((x * x) * (x * x)) * x) * x)) * abs(x))))));
end

Wolfram

code[x_] := N[Abs[N[(N[(-1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Abs[x], $MachinePrecision] * 2.0), $MachinePrecision] + N[(0.2 * N[Abs[N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.0 / 21.0), $MachinePrecision] * N[(N[Abs[N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[Abs[x], $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
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. lift-fabs.f64 N/A

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

   9. distribute-rgt-out N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\color{blue}{\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)} + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. metadata-eval 99.8

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) + \color{blue}{0.2} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

6. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) + \color{blue}{0.2} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. lift-fabs.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. sqr-abs-rev N/A

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

   9. pow2 N/A

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

   10. associate-*l* N/A

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

   11. sqr-abs-rev N/A

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

   12. pow2 N/A

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

   13. lower-*.f64 N/A

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

   14. pow2 N/A

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

   15. lift-*.f64 N/A

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

   16. pow2 N/A

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

   17. lift-*.f64 99.8

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) + 0.2 \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

8. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) + 0.2 \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.3%

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left|x\right| \cdot \color{blue}{2} + 0.2 \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

12. Final simplification 98.3%

    \[\leadsto \left|\frac{-1}{\sqrt{\pi}} \cdot \left(\left(\left|x\right| \cdot 2 + 0.2 \cdot \left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right|\right) - \frac{-1}{21} \cdot \left(\left|\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x\right| \cdot \left|x\right|\right)\right)\right| \]
13. Add Preprocessing

Alternative 7: 27.2% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around inf

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

   1. lift-fabs.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. sqr-abs-rev N/A

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

   5. lift-fabs.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lower-*.f64 27.7

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. pow3 N/A

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

   12. lower-pow.f64 27.7

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

   13. lift-fabs.f64 N/A

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

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

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

   15. sqrt-unprod N/A

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

   16. rem-square-sqrt 27.7

       \[\leadsto \frac{\left|0.6666666666666666 \cdot {x}^{3}\right|}{\sqrt{\pi}} \]

7. Applied rewrites 27.7%

   \[\leadsto \frac{\left|\color{blue}{0.6666666666666666 \cdot {x}^{3}}\right|}{\sqrt{\pi}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. sqr-pow N/A

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

   4. pow-prod-down N/A

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

   5. sqr-abs-rev N/A

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

   6. pow-prod-down N/A

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

   7. sqr-pow N/A

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

   8. pow3 N/A

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

   9. sqr-abs-rev N/A

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

   10. pow2 N/A

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

   11. associate-*r* N/A

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

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

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

   13. pow2 N/A

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

   14. sqrt-pow1 N/A

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

   15. metadata-eval N/A

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

   16. unpow1 N/A

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

   17. lower-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. pow2 N/A

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

   21. lift-*.f64 27.7

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

9. Applied rewrites 27.7%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-*l* N/A

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

    4. unpow1 N/A

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

    5. metadata-eval N/A

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

    6. sqrt-pow1 N/A

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

    7. pow2 N/A

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

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

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

    9. lower-*.f64 N/A

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

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

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

    11. pow2 N/A

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

    12. sqrt-pow1 N/A

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

    13. metadata-eval N/A

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

    14. unpow1 N/A

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

    15. lower-*.f64 27.7

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

11. Applied rewrites 27.7%

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

Reproduce

herbie shell --seed 2025083 
(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)))))))