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

Percentage Accurate: 99.8% → 99.8%

Time: 16.7s

Alternatives: 16

Speedup: 2.7×

• 27.7% 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: 99.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

Alternative 2: 89.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * Math.abs((x * x));
	double t_1 = Math.abs(x) * (Math.abs(x) * t_0);
	double tmp;
	if (((((Math.abs(x) * 2.0) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * (Math.abs(x) * (Math.abs(x) * t_1)))) <= 1.0) {
		tmp = Math.abs(x) * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.abs(((x * (Math.abs(x) / Math.sqrt(Math.PI))) * (x * 0.6666666666666666)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.fabs(x) * math.fabs((x * x))
	t_1 = math.fabs(x) * (math.fabs(x) * t_0)
	tmp = 0
	if ((((math.fabs(x) * 2.0) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * (math.fabs(x) * (math.fabs(x) * t_1)))) <= 1.0:
		tmp = math.fabs(x) * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.fabs(((x * (math.fabs(x) / math.sqrt(math.pi))) * (x * 0.6666666666666666)))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	t_0 = abs(x) * abs((x * x));
	t_1 = abs(x) * (abs(x) * t_0);
	tmp = 0.0;
	if (((((abs(x) * 2.0) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * (abs(x) * (abs(x) * t_1)))) <= 1.0)
		tmp = abs(x) * (2.0 / sqrt(pi));
	else
		tmp = abs(((x * (abs(x) / sqrt(pi))) * (x * 0.6666666666666666)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[Abs[x], $MachinePrecision] * 2.0), $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[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0], N[(N[Abs[x], $MachinePrecision] * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[(x * N[(N[Abs[x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * 0.6666666666666666), $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)))) < 1

   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%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 98.0%

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

      1. metadata-eval N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. sqrt-prod N/A

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

      6. rem-sqrt-square N/A

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

      7. fabs-div N/A

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

      8. lift-/.f64 N/A

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

      9. fabs-fabs N/A

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

      10. lift-fabs.f64 N/A

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

      11. metadata-eval N/A

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

      12. fabs-mul N/A

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

      13. lift-*.f64 N/A

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

      14. fabs-mul N/A

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

      15. lift-*.f64 N/A

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

      16. fabs-fabs N/A

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

   9. Applied rewrites 97.3%

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

       1. lift-fabs.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-PI.f64 N/A

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

       4. lift-sqrt.f64 N/A

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

       5. associate-/l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 98.0

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

   11. Applied rewrites 98.0%

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

   if 1 < (+.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 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 74.2

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

   10. Applied rewrites 74.2%

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

       1. lift-PI.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-fabs.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. associate-*r* N/A

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

       10. lift-*.f64 N/A

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

       11. associate-*r* N/A

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

       12. lower-*.f64 N/A

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

       13. lower-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lift-/.f64 N/A

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

       16. div-inv N/A

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

       17. lower-/.f64 74.2

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

       18. lift-*.f64 N/A

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

       19. *-commutative N/A

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

   12. Applied rewrites 74.2%

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

4. Final simplification 90.9%

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

Alternative 3: 99.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;\left|\left|x\right| \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right), 0.6666666666666666\right)\right)\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.5)
   (fabs
    (*
     (fabs x)
     (*
      (sqrt (/ 1.0 PI))
      (fma (* x x) (fma x (* x 0.2) 0.6666666666666666) 2.0))))
   (/
    (fabs
     (*
      x
      (*
       (* x x)
       (fma
        (* x x)
        (fma (* x x) 0.047619047619047616 0.2)
        0.6666666666666666))))
    (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.5) {
		tmp = fabs((fabs(x) * (sqrt((1.0 / ((double) M_PI))) * fma((x * x), fma(x, (x * 0.2), 0.6666666666666666), 2.0))));
	} else {
		tmp = fabs((x * ((x * x) * fma((x * x), fma((x * x), 0.047619047619047616, 0.2), 0.6666666666666666)))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.5)
		tmp = abs(Float64(abs(x) * Float64(sqrt(Float64(1.0 / pi)) * fma(Float64(x * x), fma(x, Float64(x * 0.2), 0.6666666666666666), 2.0))));
	else
		tmp = Float64(abs(Float64(x * Float64(Float64(x * x) * fma(Float64(x * x), fma(Float64(x * x), 0.047619047619047616, 0.2), 0.6666666666666666)))) / sqrt(pi));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.5], N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.047619047619047616 + 0.2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.5

   1. Initial program 99.8%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.1%

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

   if 0.5 < (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%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.1%

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

   9. Applied rewrites 99.1%

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

Alternative 4: 99.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.9

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

9. Applied rewrites 99.9%

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

Alternative 5: 99.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;\left|\left|x\right| \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x\right| \cdot \left(\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.5)
   (fabs
    (*
     (fabs x)
     (*
      (sqrt (/ 1.0 PI))
      (fma (* x x) (fma x (* x 0.2) 0.6666666666666666) 2.0))))
   (*
    (fabs x)
    (* (/ 0.047619047619047616 (sqrt PI)) (* (* x x) (* x (* x (* x x))))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.5) {
		tmp = fabs((fabs(x) * (sqrt((1.0 / ((double) M_PI))) * fma((x * x), fma(x, (x * 0.2), 0.6666666666666666), 2.0))));
	} else {
		tmp = fabs(x) * ((0.047619047619047616 / sqrt(((double) M_PI))) * ((x * x) * (x * (x * (x * x)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.5)
		tmp = abs(Float64(abs(x) * Float64(sqrt(Float64(1.0 / pi)) * fma(Float64(x * x), fma(x, Float64(x * 0.2), 0.6666666666666666), 2.0))));
	else
		tmp = Float64(abs(x) * Float64(Float64(0.047619047619047616 / sqrt(pi)) * Float64(Float64(x * x) * Float64(x * Float64(x * Float64(x * x))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.5], N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[x], $MachinePrecision] * N[(N[(0.047619047619047616 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.5

   1. Initial program 99.8%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.1%

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

   if 0.5 < (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%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. pow-plus N/A

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

      19. pow-plus N/A

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

      20. associate-*r* N/A

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

      21. unpow2 N/A

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

      22. *-commutative N/A

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

   7. Applied rewrites 98.3%

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

   9. Applied rewrites 98.2%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;\left|\left|x\right| \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x\right| \cdot \left(\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;\left|\left|x\right| \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.047619047619047616\right)\right)\right)\right)\right) \cdot \frac{\left|x\right|}{\sqrt{\pi}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.5)
   (fabs
    (*
     (fabs x)
     (*
      (sqrt (/ 1.0 PI))
      (fma (* x x) (fma x (* x 0.2) 0.6666666666666666) 2.0))))
   (*
    x
    (*
     (* x (* x (* x (* x (* x 0.047619047619047616)))))
     (/ (fabs x) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.5) {
		tmp = fabs((fabs(x) * (sqrt((1.0 / ((double) M_PI))) * fma((x * x), fma(x, (x * 0.2), 0.6666666666666666), 2.0))));
	} else {
		tmp = x * ((x * (x * (x * (x * (x * 0.047619047619047616))))) * (fabs(x) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.5)
		tmp = abs(Float64(abs(x) * Float64(sqrt(Float64(1.0 / pi)) * fma(Float64(x * x), fma(x, Float64(x * 0.2), 0.6666666666666666), 2.0))));
	else
		tmp = Float64(x * Float64(Float64(x * Float64(x * Float64(x * Float64(x * Float64(x * 0.047619047619047616))))) * Float64(abs(x) / sqrt(pi))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.5], N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(x * N[(N[(x * N[(x * N[(x * N[(x * N[(x * 0.047619047619047616), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.5

   1. Initial program 99.8%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.1%

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

   if 0.5 < (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%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. pow-plus N/A

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

      19. pow-plus N/A

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

      20. associate-*r* N/A

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

      21. unpow2 N/A

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

      22. *-commutative N/A

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

   7. Applied rewrites 98.3%

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

   9. Applied rewrites 98.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-PI.f64 N/A

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

       7. lift-sqrt.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. lift-fabs.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. associate-*l* N/A

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

       12. associate-*l* N/A

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

       13. lower-*.f64 N/A

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

   11. Applied rewrites 98.2%

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

Alternative 7: 99.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

8. Applied rewrites 99.8%

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

Alternative 8: 99.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.9

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

9. Applied rewrites 99.9%

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

11. Applied rewrites 99.4%

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

Alternative 9: 93.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;\left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, x \cdot \left(x \cdot \left(\left|x\right| \cdot 0.6666666666666666\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot x\right| \cdot \left|\frac{x \cdot \mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right)}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.5)
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (fma (fabs x) 2.0 (* x (* x (* (fabs x) 0.6666666666666666))))))
   (*
    (fabs (* x x))
    (fabs (/ (* x (fma (* x x) 0.2 0.6666666666666666)) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.5) {
		tmp = fabs(((1.0 / sqrt(((double) M_PI))) * fma(fabs(x), 2.0, (x * (x * (fabs(x) * 0.6666666666666666))))));
	} else {
		tmp = fabs((x * x)) * fabs(((x * fma((x * x), 0.2, 0.6666666666666666)) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.5)
		tmp = abs(Float64(Float64(1.0 / sqrt(pi)) * fma(abs(x), 2.0, Float64(x * Float64(x * Float64(abs(x) * 0.6666666666666666))))));
	else
		tmp = Float64(abs(Float64(x * x)) * abs(Float64(Float64(x * fma(Float64(x * x), 0.2, 0.6666666666666666)) / sqrt(pi))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.5], N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * 2.0 + N[(x * N[(x * N[(N[Abs[x], $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[Abs[N[(N[(x * N[(N[(x * x), $MachinePrecision] * 0.2 + 0.6666666666666666), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.5

   1. Initial program 99.8%

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

   4. Applied rewrites 99.9%

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

   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

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-fabs.f64 98.9

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

   7. Applied rewrites 98.9%

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

   if 0.5 < (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%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 89.1

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

   10. Applied rewrites 89.1%

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

       1. lift-PI.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-fabs.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-fma.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lift-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. associate-*r* N/A

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

       14. associate-*l* N/A

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

   12. Applied rewrites 89.1%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;\left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, x \cdot \left(x \cdot \left(\left|x\right| \cdot 0.6666666666666666\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot x\right| \cdot \left|\frac{x \cdot \mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right)}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

8. Applied rewrites 99.4%

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

Alternative 11: 93.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;\left|\left|x\right| \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot x\right| \cdot \left|\frac{x \cdot \mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right)}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.5)
   (fabs
    (* (fabs x) (* (sqrt (/ 1.0 PI)) (fma 0.6666666666666666 (* x x) 2.0))))
   (*
    (fabs (* x x))
    (fabs (/ (* x (fma (* x x) 0.2 0.6666666666666666)) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.5) {
		tmp = fabs((fabs(x) * (sqrt((1.0 / ((double) M_PI))) * fma(0.6666666666666666, (x * x), 2.0))));
	} else {
		tmp = fabs((x * x)) * fabs(((x * fma((x * x), 0.2, 0.6666666666666666)) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.5)
		tmp = abs(Float64(abs(x) * Float64(sqrt(Float64(1.0 / pi)) * fma(0.6666666666666666, Float64(x * x), 2.0))));
	else
		tmp = Float64(abs(Float64(x * x)) * abs(Float64(Float64(x * fma(Float64(x * x), 0.2, 0.6666666666666666)) / sqrt(pi))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.5], N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[Abs[N[(N[(x * N[(N[(x * x), $MachinePrecision] * 0.2 + 0.6666666666666666), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.5

   1. Initial program 99.8%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   7. Applied rewrites 98.9%

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

   if 0.5 < (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%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 89.1

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

   10. Applied rewrites 89.1%

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

       1. lift-PI.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-fabs.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-fma.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lift-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. associate-*r* N/A

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

       14. associate-*l* N/A

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

   12. Applied rewrites 89.1%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;\left|\left|x\right| \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot x\right| \cdot \left|\frac{x \cdot \mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right)}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 93.5% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 96.1%

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

Alternative 13: 89.4% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-in N/A

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-in N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

7. Applied rewrites 91.5%

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

Alternative 14: 88.9% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-fabs.f64 91.5

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

7. Applied rewrites 91.5%

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

   1. lift-PI.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-fabs.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-/.f64 N/A

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

   12. un-div-inv N/A

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

9. Applied rewrites 91.1%

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

10. Final simplification 91.1%

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

Alternative 15: 68.5% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 71.0%

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

   1. metadata-eval N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. rem-square-sqrt N/A

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

   5. sqrt-prod N/A

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

   6. rem-sqrt-square N/A

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

   7. fabs-div N/A

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

   8. lift-/.f64 N/A

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

   9. fabs-fabs N/A

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

   10. lift-fabs.f64 N/A

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

   11. metadata-eval N/A

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

   12. fabs-mul N/A

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

   13. lift-*.f64 N/A

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

   14. fabs-mul N/A

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

   15. lift-*.f64 N/A

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

   16. fabs-fabs N/A

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

9. Applied rewrites 71.0%

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

10. Final simplification 71.0%

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

Alternative 16: 68.5% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 71.0%

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

   1. metadata-eval N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. rem-square-sqrt N/A

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

   5. sqrt-prod N/A

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

   6. rem-sqrt-square N/A

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

   7. fabs-div N/A

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

   8. lift-/.f64 N/A

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

   9. fabs-fabs N/A

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

   10. lift-fabs.f64 N/A

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

   11. metadata-eval N/A

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

   12. fabs-mul N/A

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

   13. lift-*.f64 N/A

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

   14. fabs-mul N/A

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

   15. lift-*.f64 N/A

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

   16. fabs-fabs N/A

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

9. Applied rewrites 70.6%

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

    1. lift-fabs.f64 N/A

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

    2. *-commutative N/A

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

    3. lift-PI.f64 N/A

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

    4. lift-sqrt.f64 N/A

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

    5. associate-/l* N/A

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

    6. lower-*.f64 N/A

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

    7. lower-/.f64 70.7

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

11. Applied rewrites 70.7%

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

Reproduce

herbie shell --seed 2024214 
(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)))))))