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

Percentage Accurate: 99.8% → 99.9%

Time: 12.4s

Alternatives: 17

Speedup: 2.3×

• 28.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. pow3 N/A

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

   7. pow2 N/A

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

   8. pow-prod-up N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 93.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{if}\;\left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \frac{1}{21} + \left(t\_1 \cdot \frac{1}{5} + \left(t\_0 \cdot \frac{2}{3} + \left|x\right| \cdot 2\right)\right) \leq 40:\\ \;\;\;\;\left|\left(\mathsf{fma}\left(0.6666666666666666 \cdot x, x, 2\right) \cdot \left|x\right|\right) \cdot \frac{1}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(0.2 \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left|x\right| \cdot x\right) \cdot x\right)}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (if (<=
        (+
         (* (* (* t_1 (fabs x)) (fabs x)) (/ 1.0 21.0))
         (+ (* t_1 (/ 1.0 5.0)) (+ (* t_0 (/ 2.0 3.0)) (* (fabs x) 2.0))))
        40.0)
     (fabs
      (* (* (fma (* 0.6666666666666666 x) x 2.0) (fabs x)) (/ 1.0 (sqrt PI))))
     (fabs (/ (* (* 0.2 (* x x)) (* (* (fabs x) x) x)) (sqrt PI))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	double tmp;
	if (((((t_1 * fabs(x)) * fabs(x)) * (1.0 / 21.0)) + ((t_1 * (1.0 / 5.0)) + ((t_0 * (2.0 / 3.0)) + (fabs(x) * 2.0)))) <= 40.0) {
		tmp = fabs(((fma((0.6666666666666666 * x), x, 2.0) * fabs(x)) * (1.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((((0.2 * (x * x)) * ((fabs(x) * x) * x)) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(t_1 * abs(x)) * abs(x)) * Float64(1.0 / 21.0)) + Float64(Float64(t_1 * Float64(1.0 / 5.0)) + Float64(Float64(t_0 * Float64(2.0 / 3.0)) + Float64(abs(x) * 2.0)))) <= 40.0)
		tmp = abs(Float64(Float64(fma(Float64(0.6666666666666666 * x), x, 2.0) * abs(x)) * Float64(1.0 / sqrt(pi))));
	else
		tmp = abs(Float64(Float64(Float64(0.2 * Float64(x * x)) * Float64(Float64(abs(x) * x) * x)) / sqrt(pi)));
	end
	return tmp
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]}, If[LessEqual[N[(N[(N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / 21.0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 * N[(1.0 / 5.0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 * N[(2.0 / 3.0), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 40.0], N[Abs[N[(N[(N[(N[(0.6666666666666666 * x), $MachinePrecision] * x + 2.0), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(0.2 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[x], $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $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)))) < 40

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-fabs.f64 99.1

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

   7. Applied rewrites 99.1%

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

   9. Applied rewrites 99.1%

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

   if 40 < (+.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 rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fabs.f64 6.1

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

   7. Applied rewrites 6.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.1%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 83.9%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \frac{1}{21} + \left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \frac{1}{5} + \left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \frac{2}{3} + \left|x\right| \cdot 2\right)\right) \leq 40:\\ \;\;\;\;\left|\left(\mathsf{fma}\left(0.6666666666666666 \cdot x, x, 2\right) \cdot \left|x\right|\right) \cdot \frac{1}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(0.2 \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left|x\right| \cdot x\right) \cdot x\right)}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.8%

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

9. Applied rewrites 99.8%

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

10. Final simplification 99.8%

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

Alternative 4: 99.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.05:\\ \;\;\;\;\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2, x \cdot x, 0.6666666666666666\right) \cdot \left|x\right|, 2\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\mathsf{fma}\left(\mathsf{fma}\left(0.047619047619047616, x \cdot x, 0.2\right), x \cdot x, 0.6666666666666666\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.05)
   (fabs
    (*
     (fma (fabs x) (* (fma 0.2 (* x x) 0.6666666666666666) (fabs x)) 2.0)
     (* (sqrt (/ 1.0 PI)) (fabs x))))
   (/
    (fabs
     (*
      (*
       (fma (fma 0.047619047619047616 (* x x) 0.2) (* x x) 0.6666666666666666)
       (fabs x))
      (* x x)))
    (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.05) {
		tmp = fabs((fma(fabs(x), (fma(0.2, (x * x), 0.6666666666666666) * fabs(x)), 2.0) * (sqrt((1.0 / ((double) M_PI))) * fabs(x))));
	} else {
		tmp = fabs(((fma(fma(0.047619047619047616, (x * x), 0.2), (x * x), 0.6666666666666666) * fabs(x)) * (x * x))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.05)
		tmp = abs(Float64(fma(abs(x), Float64(fma(0.2, Float64(x * x), 0.6666666666666666) * abs(x)), 2.0) * Float64(sqrt(Float64(1.0 / pi)) * abs(x))));
	else
		tmp = Float64(abs(Float64(Float64(fma(fma(0.047619047619047616, Float64(x * x), 0.2), Float64(x * x), 0.6666666666666666) * abs(x)) * Float64(x * x))) / sqrt(pi));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.05], N[Abs[N[(N[(N[Abs[x], $MachinePrecision] * N[(N[(0.2 * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(N[(N[(N[(0.047619047619047616 * N[(x * x), $MachinePrecision] + 0.2), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.050000000000000003

   1. Initial program 99.8%

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.6%

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

   if 0.050000000000000003 < (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 rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 99.8%

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

      1. lift-fabs.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in x around inf

       \[\leadsto \frac{\left|\color{blue}{{x}^{6} \cdot \left(\frac{1}{21} \cdot \left|x\right| + \left(\frac{1}{5} \cdot \frac{\left|x\right|}{{x}^{2}} + \frac{2}{3} \cdot \frac{\left|x\right|}{{x}^{4}}\right)\right)}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

   12. Applied rewrites 98.4%

       \[\leadsto \frac{\left|\color{blue}{\left(\left|x\right| \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.047619047619047616, x \cdot x, 0.2\right), x \cdot x, 0.6666666666666666\right)\right) \cdot \left(x \cdot x\right)}\right|}{\sqrt{\pi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.05:\\ \;\;\;\;\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2, x \cdot x, 0.6666666666666666\right) \cdot \left|x\right|, 2\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\mathsf{fma}\left(\mathsf{fma}\left(0.047619047619047616, x \cdot x, 0.2\right), x \cdot x, 0.6666666666666666\right) \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)\right|}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 4:\\ \;\;\;\;\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2, x \cdot x, 0.6666666666666666\right) \cdot \left|x\right|, 2\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\left(\left(\left(\left|x\right| \cdot x\right) \cdot x\right) \cdot \mathsf{fma}\left(0.047619047619047616, x \cdot x, 0.2\right)\right) \cdot x\right) \cdot x}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 4.0)
   (fabs
    (*
     (fma (fabs x) (* (fma 0.2 (* x x) 0.6666666666666666) (fabs x)) 2.0)
     (* (sqrt (/ 1.0 PI)) (fabs x))))
   (fabs
    (/
     (*
      (* (* (* (* (fabs x) x) x) (fma 0.047619047619047616 (* x x) 0.2)) x)
      x)
     (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 4.0) {
		tmp = fabs((fma(fabs(x), (fma(0.2, (x * x), 0.6666666666666666) * fabs(x)), 2.0) * (sqrt((1.0 / ((double) M_PI))) * fabs(x))));
	} else {
		tmp = fabs(((((((fabs(x) * x) * x) * fma(0.047619047619047616, (x * x), 0.2)) * x) * x) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 4.0)
		tmp = abs(Float64(fma(abs(x), Float64(fma(0.2, Float64(x * x), 0.6666666666666666) * abs(x)), 2.0) * Float64(sqrt(Float64(1.0 / pi)) * abs(x))));
	else
		tmp = abs(Float64(Float64(Float64(Float64(Float64(Float64(abs(x) * x) * x) * fma(0.047619047619047616, Float64(x * x), 0.2)) * x) * x) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 4.0], N[Abs[N[(N[(N[Abs[x], $MachinePrecision] * N[(N[(0.2 * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(N[(N[(N[Abs[x], $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(0.047619047619047616 * N[(x * x), $MachinePrecision] + 0.2), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 4

   1. Initial program 99.8%

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.2%

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

   if 4 < (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 rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fabs.f64 6.1

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

   7. Applied rewrites 6.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.1%

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

   12. Applied rewrites 99.1%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 4:\\ \;\;\;\;\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2, x \cdot x, 0.6666666666666666\right) \cdot \left|x\right|, 2\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\left(\left(\left(\left|x\right| \cdot x\right) \cdot x\right) \cdot \mathsf{fma}\left(0.047619047619047616, x \cdot x, 0.2\right)\right) \cdot x\right) \cdot x}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 4:\\ \;\;\;\;\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2, x \cdot x, 0.6666666666666666\right) \cdot \left|x\right|, 2\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\mathsf{fma}\left(0.047619047619047616, x \cdot x, 0.2\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left|x\right| \cdot x\right) \cdot x\right)}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 4.0)
   (fabs
    (*
     (fma (fabs x) (* (fma 0.2 (* x x) 0.6666666666666666) (fabs x)) 2.0)
     (* (sqrt (/ 1.0 PI)) (fabs x))))
   (fabs
    (/
     (*
      (* (fma 0.047619047619047616 (* x x) 0.2) (* x x))
      (* (* (fabs x) x) x))
     (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 4.0) {
		tmp = fabs((fma(fabs(x), (fma(0.2, (x * x), 0.6666666666666666) * fabs(x)), 2.0) * (sqrt((1.0 / ((double) M_PI))) * fabs(x))));
	} else {
		tmp = fabs((((fma(0.047619047619047616, (x * x), 0.2) * (x * x)) * ((fabs(x) * x) * x)) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 4.0)
		tmp = abs(Float64(fma(abs(x), Float64(fma(0.2, Float64(x * x), 0.6666666666666666) * abs(x)), 2.0) * Float64(sqrt(Float64(1.0 / pi)) * abs(x))));
	else
		tmp = abs(Float64(Float64(Float64(fma(0.047619047619047616, Float64(x * x), 0.2) * Float64(x * x)) * Float64(Float64(abs(x) * x) * x)) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 4.0], N[Abs[N[(N[(N[Abs[x], $MachinePrecision] * N[(N[(0.2 * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(0.047619047619047616 * N[(x * x), $MachinePrecision] + 0.2), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[x], $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 4

   1. Initial program 99.8%

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.2%

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

   if 4 < (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 rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fabs.f64 6.1

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

   7. Applied rewrites 6.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.1%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 4:\\ \;\;\;\;\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2, x \cdot x, 0.6666666666666666\right) \cdot \left|x\right|, 2\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\mathsf{fma}\left(0.047619047619047616, x \cdot x, 0.2\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left|x\right| \cdot x\right) \cdot x\right)}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 4:\\ \;\;\;\;\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2, x \cdot x, 0.6666666666666666\right) \cdot \left|x\right|, 2\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\left(\left(\left(0.047619047619047616 \cdot \left|x\right|\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(x \cdot x\right)}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 4.0)
   (fabs
    (*
     (fma (fabs x) (* (fma 0.2 (* x x) 0.6666666666666666) (fabs x)) 2.0)
     (* (sqrt (/ 1.0 PI)) (fabs x))))
   (fabs
    (/
     (* (* (* (* (* 0.047619047619047616 (fabs x)) x) (* x x)) x) (* x x))
     (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 4.0) {
		tmp = fabs((fma(fabs(x), (fma(0.2, (x * x), 0.6666666666666666) * fabs(x)), 2.0) * (sqrt((1.0 / ((double) M_PI))) * fabs(x))));
	} else {
		tmp = fabs(((((((0.047619047619047616 * fabs(x)) * x) * (x * x)) * x) * (x * x)) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 4.0)
		tmp = abs(Float64(fma(abs(x), Float64(fma(0.2, Float64(x * x), 0.6666666666666666) * abs(x)), 2.0) * Float64(sqrt(Float64(1.0 / pi)) * abs(x))));
	else
		tmp = abs(Float64(Float64(Float64(Float64(Float64(Float64(0.047619047619047616 * abs(x)) * x) * Float64(x * x)) * x) * Float64(x * x)) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 4.0], N[Abs[N[(N[(N[Abs[x], $MachinePrecision] * N[(N[(0.2 * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(N[(N[(0.047619047619047616 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 4

   1. Initial program 99.8%

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.2%

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

   if 4 < (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 rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. metadata-eval N/A

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

      9. pow-plus N/A

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

      10. metadata-eval N/A

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

      11. pow-plus N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 98.2

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

   7. Applied rewrites 98.2%

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

   9. Applied rewrites 98.3%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 4:\\ \;\;\;\;\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2, x \cdot x, 0.6666666666666666\right) \cdot \left|x\right|, 2\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\left(\left(\left(0.047619047619047616 \cdot \left|x\right|\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(x \cdot x\right)}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 4:\\ \;\;\;\;\left|\mathsf{fma}\left(\left|x\right|, 2, \left(0.6666666666666666 \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\left(\left(\left(0.047619047619047616 \cdot \left|x\right|\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(x \cdot x\right)}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 4.0)
   (fabs
    (*
     (fma (fabs x) 2.0 (* (* 0.6666666666666666 x) (* (fabs x) x)))
     (/ 1.0 (sqrt PI))))
   (fabs
    (/
     (* (* (* (* (* 0.047619047619047616 (fabs x)) x) (* x x)) x) (* x x))
     (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 4.0) {
		tmp = fabs((fma(fabs(x), 2.0, ((0.6666666666666666 * x) * (fabs(x) * x))) * (1.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs(((((((0.047619047619047616 * fabs(x)) * x) * (x * x)) * x) * (x * x)) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 4.0)
		tmp = abs(Float64(fma(abs(x), 2.0, Float64(Float64(0.6666666666666666 * x) * Float64(abs(x) * x))) * Float64(1.0 / sqrt(pi))));
	else
		tmp = abs(Float64(Float64(Float64(Float64(Float64(Float64(0.047619047619047616 * abs(x)) * x) * Float64(x * x)) * x) * Float64(x * x)) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 4.0], N[Abs[N[(N[(N[Abs[x], $MachinePrecision] * 2.0 + N[(N[(0.6666666666666666 * x), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(N[(N[(0.047619047619047616 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 4

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. lower-*.f64 99.1

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

   7. Applied rewrites 99.1%

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

   if 4 < (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 rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. metadata-eval N/A

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

      9. pow-plus N/A

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

      10. metadata-eval N/A

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

      11. pow-plus N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 98.2

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

   7. Applied rewrites 98.2%

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

   9. Applied rewrites 98.3%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 4:\\ \;\;\;\;\left|\mathsf{fma}\left(\left|x\right|, 2, \left(0.6666666666666666 \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\left(\left(\left(0.047619047619047616 \cdot \left|x\right|\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(x \cdot x\right)}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \mathbf{if}\;\left|x\right| \leq 4:\\ \;\;\;\;\left|\mathsf{fma}\left(\left|x\right|, 2, \left(0.6666666666666666 \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(t\_0 \cdot t\_0\right) \cdot \left(0.047619047619047616 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (if (<= (fabs x) 4.0)
     (fabs
      (*
       (fma (fabs x) 2.0 (* (* 0.6666666666666666 x) (* (fabs x) x)))
       (/ 1.0 (sqrt PI))))
     (fabs (/ (* (* t_0 t_0) (* 0.047619047619047616 (fabs x))) (sqrt PI))))))

C

double code(double x) {
	double t_0 = (x * x) * x;
	double tmp;
	if (fabs(x) <= 4.0) {
		tmp = fabs((fma(fabs(x), 2.0, ((0.6666666666666666 * x) * (fabs(x) * x))) * (1.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((((t_0 * t_0) * (0.047619047619047616 * fabs(x))) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	tmp = 0.0
	if (abs(x) <= 4.0)
		tmp = abs(Float64(fma(abs(x), 2.0, Float64(Float64(0.6666666666666666 * x) * Float64(abs(x) * x))) * Float64(1.0 / sqrt(pi))));
	else
		tmp = abs(Float64(Float64(Float64(t_0 * t_0) * Float64(0.047619047619047616 * abs(x))) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 4.0], N[Abs[N[(N[(N[Abs[x], $MachinePrecision] * 2.0 + N[(N[(0.6666666666666666 * x), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(0.047619047619047616 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 4

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-fabs.f64 N/A

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

      12. lower-*.f64 99.1

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

   7. Applied rewrites 99.1%

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

   if 4 < (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 rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. metadata-eval N/A

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

      9. pow-plus N/A

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

      10. metadata-eval N/A

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

      11. pow-plus N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 98.2

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

   7. Applied rewrites 98.2%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 4:\\ \;\;\;\;\left|\mathsf{fma}\left(\left|x\right|, 2, \left(0.6666666666666666 \cdot x\right) \cdot \left(\left|x\right| \cdot x\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(0.047619047619047616 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-fabs.f64 91.0

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

7. Applied rewrites 91.0%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.8%

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

11. Final simplification 99.8%

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

Alternative 11: 99.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.8%

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

9. Applied rewrites 99.4%

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

10. Final simplification 99.4%

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

Alternative 12: 93.6% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

7. Applied rewrites 99.8%

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

   1. lift-fabs.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

9. Applied rewrites 99.4%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 93.7%

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

13. Final simplification 93.7%

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

Alternative 13: 93.6% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. sqr-abs N/A

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

   4. associate-*r* N/A

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

   5. associate-*r* N/A

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

   6. distribute-rgt-out N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. sqr-abs N/A

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

   11. unpow2 N/A

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

   12. *-commutative N/A

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

   13. distribute-rgt-in N/A

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

7. Applied rewrites 93.7%

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

Alternative 14: 89.5% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-fabs.f64 91.0

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

7. Applied rewrites 91.0%

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

9. Applied rewrites 91.0%

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

10. Final simplification 91.0%

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

Alternative 15: 89.1% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-fabs.f64 90.6

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

7. Applied rewrites 90.6%

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

Alternative 16: 68.3% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-fabs.f64 67.9

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

7. Applied rewrites 67.9%

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

8. Final simplification 67.9%

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

Alternative 17: 67.8% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-fabs.f64 67.5

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

7. Applied rewrites 67.5%

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

8. Final simplification 67.5%

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

Reproduce

herbie shell --seed 2024235 
(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)))))))