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

Percentage Accurate: 99.8% → 99.8%

Time: 14.3s

Alternatives: 10

Speedup: 3.0×

• 27.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Simplified 99.8%

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

Alternative 2: 99.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 3: 99.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.02:\\ \;\;\;\;\left|x\right| \cdot \frac{1}{\sqrt{\pi} \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.047619047619047616 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.02)
   (* (fabs x) (/ 1.0 (* (sqrt PI) (fma x (* x -0.16666666666666666) 0.5))))
   (fabs
    (/
     (* (* (* x x) (* (* x x) (* x x))) (* 0.047619047619047616 (fabs x)))
     (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.02) {
		tmp = fabs(x) * (1.0 / (sqrt(((double) M_PI)) * fma(x, (x * -0.16666666666666666), 0.5)));
	} else {
		tmp = fabs(((((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) <= 0.02)
		tmp = Float64(abs(x) * Float64(1.0 / Float64(sqrt(pi) * fma(x, Float64(x * -0.16666666666666666), 0.5))));
	else
		tmp = abs(Float64(Float64(Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(x * x))) * Float64(0.047619047619047616 * abs(x))) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.02], N[(N[Abs[x], $MachinePrecision] * N[(1.0 / N[(N[Sqrt[Pi], $MachinePrecision] * N[(x * N[(x * -0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.047619047619047616 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0200000000000000004

   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 egg-rr 99.2%

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

   5. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. distribute-rgt-out N/A

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

      8. associate-*r/ N/A

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

      9. *-rgt-identity N/A

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

      10. associate-*l/ N/A

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

      11. lower-/.f64 N/A

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

   7. Simplified 99.0%

      \[\leadsto \left|\frac{1}{\color{blue}{\frac{\sqrt{\pi} \cdot \mathsf{fma}\left(-0.16666666666666666, x \cdot x, 0.5\right)}{\left|x\right|}}}\right| \]
   8. Step-by-step derivation

      1. lift-PI.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fabs.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. /-rgt-identity N/A

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

      9. clear-num N/A

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

      10. inv-pow N/A

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

      11. sqr-pow N/A

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

   9. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\pi} \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 0.5\right)} \cdot \left|x\right|} \]

   if 0.0200000000000000004 < (fabs.f64 x)

   1. Initial program 99.7%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. cube-prod N/A

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

      8. unpow2 N/A

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

      9. cube-unmult N/A

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

      10. pow-sqr N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. pow-sqr N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

      23. lower-fabs.f64 98.0

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

   7. Simplified 98.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.02:\\ \;\;\;\;\left|x\right| \cdot \frac{1}{\sqrt{\pi} \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.047619047619047616 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 3.0× speedup

Math

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

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(x) * Float64(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[x], $MachinePrecision] * N[(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] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 99.4%

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

6. Applied egg-rr 99.8%

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

8. Applied egg-rr 99.4%

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

10. Applied egg-rr 99.8%

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

Alternative 5: 93.7% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 99.5%

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

5. Taylor expanded in x around 0

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

7. Simplified 99.5%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 94.2

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

10. Simplified 94.2%

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

Alternative 6: 93.4% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 99.4%

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

6. Applied egg-rr 99.8%

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

8. Applied egg-rr 99.4%

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

9. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. *-commutative N/A

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

    3. +-commutative N/A

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

    4. associate-*r* N/A

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

    5. distribute-rgt-out N/A

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

    6. associate-*r* N/A

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

    7. *-commutative N/A

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

    8. distribute-lft-in N/A

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

    9. lower-*.f64 N/A

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

    10. lower-fabs.f64 N/A

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

    11. +-commutative N/A

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

    12. unpow2 N/A

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

    13. associate-*l* N/A

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

    14. lower-fma.f64 N/A

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

11. Simplified 94.0%

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

Alternative 7: 89.8% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-in N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-in N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-fabs.f64 N/A

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

   12. lower-*.f64 N/A

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

7. Simplified 90.2%

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

Alternative 8: 89.8% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-fabs.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 89.8

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

7. Simplified 89.8%

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

   1. lift-fabs.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. fabs-mul N/A

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

   9. lift-fabs.f64 N/A

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

   10. fabs-fabs N/A

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

   11. lift-fabs.f64 N/A

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

   12. lower-*.f64 N/A

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

9. Applied egg-rr 90.2%

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

10. Final simplification 90.2%

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

Alternative 9: 89.3% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 99.4%

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

6. Applied egg-rr 99.8%

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

8. Applied egg-rr 99.4%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. associate-*r* N/A

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

    3. distribute-rgt-in N/A

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

    4. lower-*.f64 N/A

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

    5. lower-fabs.f64 N/A

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

    6. +-commutative N/A

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

    7. *-commutative N/A

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

    8. lower-fma.f64 N/A

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

    9. unpow2 N/A

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

    10. lower-*.f64 89.8

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

11. Simplified 89.8%

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

Alternative 10: 68.7% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-fabs.f64 72.8

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

7. Simplified 72.8%

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

   1. lift-fabs.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. fabs-div N/A

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

   8. rem-sqrt-square N/A

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

   9. sqrt-prod N/A

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

   10. rem-square-sqrt N/A

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

   11. lift-*.f64 N/A

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

   12. fabs-mul N/A

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

   13. lift-fabs.f64 N/A

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

   14. fabs-fabs N/A

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

   15. metadata-eval N/A

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

   16. lift-fabs.f64 N/A

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

   17. associate-/l* N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

9. Applied egg-rr 73.3%

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

10. Final simplification 73.3%

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

Reproduce

herbie shell --seed 2024210 
(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)))))))