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

Percentage Accurate: 99.8% → 99.9%

Time: 13.4s

Alternatives: 11

Speedup: 2.9×

• 28.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 99.9

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

6. Applied egg-rr 99.9%

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

Alternative 2: 93.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := x \cdot t\_0\\ t_2 := \left|x\right| \cdot \left|t\_1\right|\\ \mathbf{if}\;\left(\left(\left|x\right| \cdot 2 + \frac{2}{3} \cdot \left|t\_0\right|\right) + \frac{1}{5} \cdot t\_2\right) + \frac{1}{21} \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot t\_2\right)\right) \leq 0.002:\\ \;\;\;\;\left|\left|x\right| \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.2 \cdot \left(t\_1 \cdot \frac{\left|x\right|}{\sqrt{\pi}}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))) (t_1 (* x t_0)) (t_2 (* (fabs x) (fabs t_1))))
   (if (<=
        (+
         (+
          (+ (* (fabs x) 2.0) (* (/ 2.0 3.0) (fabs t_0)))
          (* (/ 1.0 5.0) t_2))
         (* (/ 1.0 21.0) (* (fabs x) (* (fabs x) t_2))))
        0.002)
     (fabs
      (* (fabs x) (* (sqrt (/ 1.0 PI)) (fma x (* x 0.6666666666666666) 2.0))))
     (fabs (* 0.2 (* t_1 (/ (fabs x) (sqrt PI))))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = x * t_0;
	double t_2 = fabs(x) * fabs(t_1);
	double tmp;
	if (((((fabs(x) * 2.0) + ((2.0 / 3.0) * fabs(t_0))) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * (fabs(x) * (fabs(x) * t_2)))) <= 0.002) {
		tmp = fabs((fabs(x) * (sqrt((1.0 / ((double) M_PI))) * fma(x, (x * 0.6666666666666666), 2.0))));
	} else {
		tmp = fabs((0.2 * (t_1 * (fabs(x) / sqrt(((double) M_PI))))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(x * t_0)
	t_2 = Float64(abs(x) * abs(t_1))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(abs(x) * 2.0) + Float64(Float64(2.0 / 3.0) * abs(t_0))) + Float64(Float64(1.0 / 5.0) * t_2)) + Float64(Float64(1.0 / 21.0) * Float64(abs(x) * Float64(abs(x) * t_2)))) <= 0.002)
		tmp = abs(Float64(abs(x) * Float64(sqrt(Float64(1.0 / pi)) * fma(x, Float64(x * 0.6666666666666666), 2.0))));
	else
		tmp = abs(Float64(0.2 * Float64(t_1 * Float64(abs(x) / sqrt(pi)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[x], $MachinePrecision] * N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[Abs[x], $MachinePrecision] * 2.0), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.002], N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x * N[(x * 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.2 * N[(t$95$1 * N[(N[Abs[x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) < 2e-3

   1. Initial program 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 99.9

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. lower-*.f64 N/A

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

   9. Simplified 99.8%

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

   if 2e-3 < (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))))

   1. Initial program 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fabs.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-in N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-fabs.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

   7. Simplified 89.6%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{1}{5} \cdot \left(\left({x}^{4} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)}\right| \]

      2. *-commutative N/A

         \[\leadsto \left|\frac{1}{5} \cdot \color{blue}{\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left({x}^{4} \cdot \left|x\right|\right)\right)}\right| \]

      3. associate-*r* N/A

         \[\leadsto \left|\frac{1}{5} \cdot \color{blue}{\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot {x}^{4}\right) \cdot \left|x\right|\right)}\right| \]

      4. *-commutative N/A

         \[\leadsto \left|\frac{1}{5} \cdot \color{blue}{\left(\left|x\right| \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot {x}^{4}\right)\right)}\right| \]

      5. lower-*.f64 N/A

         \[\leadsto \left|\frac{1}{5} \cdot \color{blue}{\left(\left|x\right| \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot {x}^{4}\right)\right)}\right| \]

      6. lower-fabs.f64 N/A

         \[\leadsto \left|\frac{1}{5} \cdot \left(\color{blue}{\left|x\right|} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot {x}^{4}\right)\right)\right| \]

      7. *-commutative N/A

         \[\leadsto \left|\frac{1}{5} \cdot \left(\left|x\right| \cdot \color{blue}{\left({x}^{4} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)}\right)\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\frac{1}{5} \cdot \left(\left|x\right| \cdot \color{blue}{\left({x}^{4} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)}\right)\right| \]

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-PI.f64 89.6

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

   10. Simplified 89.6%

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

       1. lift-fabs.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-PI.f64 N/A

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

       6. frac-2neg N/A

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

       7. metadata-eval N/A

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

       8. metadata-eval N/A

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

       9. frac-2neg N/A

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

       10. lift-/.f64 N/A

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

       11. lift-sqrt.f64 N/A

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

       12. *-commutative N/A

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

       13. associate-*r* N/A

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

   12. Applied egg-rr 89.6%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left|x\right| \cdot 2 + \frac{2}{3} \cdot \left|x \cdot \left(x \cdot x\right)\right|\right) + \frac{1}{5} \cdot \left(\left|x\right| \cdot \left|x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right|\right)\right) + \frac{1}{21} \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left|x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right|\right)\right)\right) \leq 0.002:\\ \;\;\;\;\left|\left|x\right| \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.2 \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{\left|x\right|}{\sqrt{\pi}}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. associate-*l* N/A

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

   6. pow-sqr N/A

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

   7. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left({x}^{2} \cdot \left(\left(\frac{1}{21} \cdot {x}^{\color{blue}{4}} + \frac{1}{5} \cdot {x}^{2}\right) + \frac{2}{3}\right) + 2\right)\right)\right| \]

   8. distribute-lft-out N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right) + {x}^{2} \cdot \frac{2}{3}\right)} + 2\right)\right)\right| \]

   9. unpow2 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right) + {x}^{2} \cdot \frac{2}{3}\right) + 2\right)\right)\right| \]

   10. associate-*l* N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\left(\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right)\right)} + {x}^{2} \cdot \frac{2}{3}\right) + 2\right)\right)\right| \]

   11. unpow2 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\left(x \cdot \left(x \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right)\right) + \color{blue}{\left(x \cdot x\right)} \cdot \frac{2}{3}\right) + 2\right)\right)\right| \]

   12. associate-*l* N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\left(x \cdot \left(x \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right)\right) + \color{blue}{x \cdot \left(x \cdot \frac{2}{3}\right)}\right) + 2\right)\right)\right| \]

   13. distribute-lft-out N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right) + x \cdot \frac{2}{3}\right)} + 2\right)\right)\right| \]

7. Simplified 99.9%

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

9. Applied egg-rr 99.9%

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

11. Applied egg-rr 99.9%

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

12. Final simplification 99.9%

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

Alternative 4: 99.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. associate-*l* N/A

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

   6. pow-sqr N/A

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

   7. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left({x}^{2} \cdot \left(\left(\frac{1}{21} \cdot {x}^{\color{blue}{4}} + \frac{1}{5} \cdot {x}^{2}\right) + \frac{2}{3}\right) + 2\right)\right)\right| \]

   8. distribute-lft-out N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right) + {x}^{2} \cdot \frac{2}{3}\right)} + 2\right)\right)\right| \]

   9. unpow2 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right) + {x}^{2} \cdot \frac{2}{3}\right) + 2\right)\right)\right| \]

   10. associate-*l* N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\left(\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right)\right)} + {x}^{2} \cdot \frac{2}{3}\right) + 2\right)\right)\right| \]

   11. unpow2 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\left(x \cdot \left(x \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right)\right) + \color{blue}{\left(x \cdot x\right)} \cdot \frac{2}{3}\right) + 2\right)\right)\right| \]

   12. associate-*l* N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\left(x \cdot \left(x \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right)\right) + \color{blue}{x \cdot \left(x \cdot \frac{2}{3}\right)}\right) + 2\right)\right)\right| \]

   13. distribute-lft-out N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right) + x \cdot \frac{2}{3}\right)} + 2\right)\right)\right| \]

7. Simplified 99.9%

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

9. Applied egg-rr 99.9%

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

    1. lift-fabs.f64 N/A

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

    2. /-rgt-identity 99.9

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

11. Applied egg-rr 99.9%

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

12. Final simplification 99.9%

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

Alternative 5: 99.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. associate-*l* N/A

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

   6. pow-sqr N/A

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

   7. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left({x}^{2} \cdot \left(\left(\frac{1}{21} \cdot {x}^{\color{blue}{4}} + \frac{1}{5} \cdot {x}^{2}\right) + \frac{2}{3}\right) + 2\right)\right)\right| \]

   8. distribute-lft-out N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right) + {x}^{2} \cdot \frac{2}{3}\right)} + 2\right)\right)\right| \]

   9. unpow2 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right) + {x}^{2} \cdot \frac{2}{3}\right) + 2\right)\right)\right| \]

   10. associate-*l* N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\left(\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right)\right)} + {x}^{2} \cdot \frac{2}{3}\right) + 2\right)\right)\right| \]

   11. unpow2 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\left(x \cdot \left(x \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right)\right) + \color{blue}{\left(x \cdot x\right)} \cdot \frac{2}{3}\right) + 2\right)\right)\right| \]

   12. associate-*l* N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\left(x \cdot \left(x \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right)\right) + \color{blue}{x \cdot \left(x \cdot \frac{2}{3}\right)}\right) + 2\right)\right)\right| \]

   13. distribute-lft-out N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right) + x \cdot \frac{2}{3}\right)} + 2\right)\right)\right| \]

7. Simplified 99.9%

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

9. Applied egg-rr 99.9%

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

11. Applied egg-rr 99.5%

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

12. Final simplification 99.5%

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

Alternative 6: 93.6% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. associate-*l* N/A

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

   6. pow-sqr N/A

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

   7. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left({x}^{2} \cdot \left(\left(\frac{1}{21} \cdot {x}^{\color{blue}{4}} + \frac{1}{5} \cdot {x}^{2}\right) + \frac{2}{3}\right) + 2\right)\right)\right| \]

   8. distribute-lft-out N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right) + {x}^{2} \cdot \frac{2}{3}\right)} + 2\right)\right)\right| \]

   9. unpow2 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right) + {x}^{2} \cdot \frac{2}{3}\right) + 2\right)\right)\right| \]

   10. associate-*l* N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\left(\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right)\right)} + {x}^{2} \cdot \frac{2}{3}\right) + 2\right)\right)\right| \]

   11. unpow2 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\left(x \cdot \left(x \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right)\right) + \color{blue}{\left(x \cdot x\right)} \cdot \frac{2}{3}\right) + 2\right)\right)\right| \]

   12. associate-*l* N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\left(x \cdot \left(x \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right)\right) + \color{blue}{x \cdot \left(x \cdot \frac{2}{3}\right)}\right) + 2\right)\right)\right| \]

   13. distribute-lft-out N/A

       \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right) + x \cdot \frac{2}{3}\right)} + 2\right)\right)\right| \]

7. Simplified 99.9%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 96.0

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

10. Simplified 96.0%

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

Alternative 7: 93.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-fabs.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-in N/A

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

   12. +-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-fabs.f64 N/A

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

   15. +-commutative N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f64 N/A

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

7. Simplified 96.0%

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

9. Applied egg-rr 95.6%

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

10. Taylor expanded in x around 0

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

12. Simplified 95.6%

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

Alternative 8: 83.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 4 \cdot 10^{-27}:\\ \;\;\;\;\left|x\right| \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \frac{2}{\left|\sqrt{\pi} \cdot x\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 4e-27)
   (* (fabs x) (/ 2.0 (sqrt PI)))
   (* (* x x) (/ 2.0 (fabs (* (sqrt PI) x))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 4e-27) {
		tmp = fabs(x) * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = (x * x) * (2.0 / fabs((sqrt(((double) M_PI)) * x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 4e-27) {
		tmp = Math.abs(x) * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = (x * x) * (2.0 / Math.abs((Math.sqrt(Math.PI) * x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 4e-27:
		tmp = math.fabs(x) * (2.0 / math.sqrt(math.pi))
	else:
		tmp = (x * x) * (2.0 / math.fabs((math.sqrt(math.pi) * x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 4e-27)
		tmp = Float64(abs(x) * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(Float64(x * x) * Float64(2.0 / abs(Float64(sqrt(pi) * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 4e-27)
		tmp = abs(x) * (2.0 / sqrt(pi));
	else
		tmp = (x * x) * (2.0 / abs((sqrt(pi) * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 4e-27], N[(N[Abs[x], $MachinePrecision] * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(2.0 / N[Abs[N[(N[Sqrt[Pi], $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 4.0000000000000002e-27

   1. Initial program 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 99.9%

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

   9. Applied egg-rr 99.2%

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

       1. lift-fabs.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. metadata-eval N/A

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

       5. times-frac N/A

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

       6. *-commutative N/A

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

       7. times-frac N/A

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

       8. /-rgt-identity N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 99.9

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

   11. Applied egg-rr 99.9%

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

   if 4.0000000000000002e-27 < (fabs.f64 x)

   1. Initial program 99.9%

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

   4. Applied egg-rr 33.5%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot 0.4444444444444444 - 4\right), \frac{1}{\left|x\right| \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right) - 2\right)}, \left(\left|x\right| \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right)\right)}\right| \]

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-rgt-identity N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

   7. Simplified 63.7%

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

   9. Applied egg-rr 62.8%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 4 \cdot 10^{-27}:\\ \;\;\;\;\left|x\right| \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \frac{2}{\left|\sqrt{\pi} \cdot x\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.4% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 99.9

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-in N/A

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-fabs.f64 N/A

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

   12. lower-*.f64 N/A

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

9. Simplified 90.4%

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

Alternative 10: 89.0% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-fabs.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-in N/A

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

   12. +-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-fabs.f64 N/A

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

   15. +-commutative N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f64 N/A

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

7. Simplified 96.0%

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

9. Applied egg-rr 95.6%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. associate-*r* N/A

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

    3. distribute-rgt-out N/A

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

    4. lower-*.f64 N/A

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

    5. lower-fabs.f64 N/A

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

    6. +-commutative N/A

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

    7. unpow2 N/A

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

    8. associate-*r* N/A

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

    9. *-commutative N/A

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

    10. lower-fma.f64 N/A

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

    11. *-commutative N/A

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

    12. lower-*.f64 89.9

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

12. Simplified 89.9%

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

Alternative 11: 68.0% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

7. Simplified 64.4%

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

9. Applied egg-rr 64.0%

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

    1. lift-fabs.f64 N/A

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

    2. lift-PI.f64 N/A

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

    3. lift-sqrt.f64 N/A

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

    4. metadata-eval N/A

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

    5. times-frac N/A

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

    6. *-commutative N/A

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

    7. times-frac N/A

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

    8. /-rgt-identity N/A

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

    9. lower-*.f64 N/A

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

    10. lower-/.f64 64.4

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

11. Applied egg-rr 64.4%

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

12. Final simplification 64.4%

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

Reproduce

herbie shell --seed 2024208 
(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)))))))