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

Percentage Accurate: 99.8% → 99.8%

Time: 10.2s

Alternatives: 7

Speedup: 4.4×

• 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, 4.4× speedup

Math

\[\begin{array}{l} \\ {\pi}^{-0.5} \cdot \left|x \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (pow PI -0.5)
  (fabs
   (*
    x
    (+
     2.0
     (+
      (* 0.047619047619047616 (pow x 6.0))
      (+ (* 0.2 (pow x 4.0)) (* 0.6666666666666666 (* x x)))))))))

C

double code(double x) {
	return pow(((double) M_PI), -0.5) * fabs((x * (2.0 + ((0.047619047619047616 * pow(x, 6.0)) + ((0.2 * pow(x, 4.0)) + (0.6666666666666666 * (x * x)))))));
}

Java

public static double code(double x) {
	return Math.pow(Math.PI, -0.5) * Math.abs((x * (2.0 + ((0.047619047619047616 * Math.pow(x, 6.0)) + ((0.2 * Math.pow(x, 4.0)) + (0.6666666666666666 * (x * x)))))));
}

Python

def code(x):
	return math.pow(math.pi, -0.5) * math.fabs((x * (2.0 + ((0.047619047619047616 * math.pow(x, 6.0)) + ((0.2 * math.pow(x, 4.0)) + (0.6666666666666666 * (x * x)))))))

Julia

function code(x)
	return Float64((pi ^ -0.5) * abs(Float64(x * Float64(2.0 + Float64(Float64(0.047619047619047616 * (x ^ 6.0)) + Float64(Float64(0.2 * (x ^ 4.0)) + Float64(0.6666666666666666 * Float64(x * x))))))))
end

MATLAB

function tmp = code(x)
	tmp = (pi ^ -0.5) * abs((x * (2.0 + ((0.047619047619047616 * (x ^ 6.0)) + ((0.2 * (x ^ 4.0)) + (0.6666666666666666 * (x * x)))))));
end

Wolfram

code[x_] := N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Abs[N[(x * N[(2.0 + N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. pow1 99.9%

      \[\leadsto \color{blue}{{\left(\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)}^{1}} \]

   2. mul-fabs 99.9%

      \[\leadsto {\color{blue}{\left(\left|x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)}}^{1} \]

   3. +-commutative 99.9%

      \[\leadsto {\left(\left|x \cdot \frac{\color{blue}{\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}}{\sqrt{\pi}}\right|\right)}^{1} \]

   4. pow2 99.9%

      \[\leadsto {\left(\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, \color{blue}{{x}^{2}}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|\right)}^{1} \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{{\left(\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|\right)}^{1}} \]
7. Step-by-step derivation

   1. unpow1 99.9%

      \[\leadsto \color{blue}{\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|} \]

   2. associate-*r/ 99.4%

      \[\leadsto \left|\color{blue}{\frac{x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)}{\sqrt{\pi}}}\right| \]

   3. fabs-div 99.4%

      \[\leadsto \color{blue}{\frac{\left|x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|}{\left|\sqrt{\pi}\right|}} \]

8. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{\left|x \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|}{\sqrt{\pi}}} \]

9. Taylor expanded in x around 0 99.9%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left|x \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|} \]
10. Step-by-step derivation

    1. pow2 99.9%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left|x \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right| \]

11. Applied egg-rr 99.9%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left|x \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right| \]
12. Step-by-step derivation

    1. *-un-lft-identity 99.9%

       \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{1}{\pi}}\right)} \cdot \left|x \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right| \]

    2. pow1/2 99.9%

       \[\leadsto \left(1 \cdot \color{blue}{{\left(\frac{1}{\pi}\right)}^{0.5}}\right) \cdot \left|x \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right| \]

    3. inv-pow 99.9%

       \[\leadsto \left(1 \cdot {\color{blue}{\left({\pi}^{-1}\right)}}^{0.5}\right) \cdot \left|x \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right| \]

    4. pow-pow 99.9%

       \[\leadsto \left(1 \cdot \color{blue}{{\pi}^{\left(-1 \cdot 0.5\right)}}\right) \cdot \left|x \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right| \]

    5. metadata-eval 99.9%

       \[\leadsto \left(1 \cdot {\pi}^{\color{blue}{-0.5}}\right) \cdot \left|x \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right| \]

13. Applied egg-rr 99.9%

    \[\leadsto \color{blue}{\left(1 \cdot {\pi}^{-0.5}\right)} \cdot \left|x \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right| \]
14. Step-by-step derivation

    1. *-lft-identity 99.9%

       \[\leadsto \color{blue}{{\pi}^{-0.5}} \cdot \left|x \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right| \]

15. Simplified 99.9%

    \[\leadsto \color{blue}{{\pi}^{-0.5}} \cdot \left|x \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right| \]
16. Add Preprocessing

Alternative 2: 35.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.005:\\ \;\;\;\;x \cdot \left({\pi}^{-0.5} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.005)
   (* x (* (pow PI -0.5) 2.0))
   (* (pow PI -0.5) (* 0.047619047619047616 (pow x 7.0)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.005) {
		tmp = x * (pow(((double) M_PI), -0.5) * 2.0);
	} else {
		tmp = pow(((double) M_PI), -0.5) * (0.047619047619047616 * pow(x, 7.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.005) {
		tmp = x * (Math.pow(Math.PI, -0.5) * 2.0);
	} else {
		tmp = Math.pow(Math.PI, -0.5) * (0.047619047619047616 * Math.pow(x, 7.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.005:
		tmp = x * (math.pow(math.pi, -0.5) * 2.0)
	else:
		tmp = math.pow(math.pi, -0.5) * (0.047619047619047616 * math.pow(x, 7.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.005)
		tmp = Float64(x * Float64((pi ^ -0.5) * 2.0));
	else
		tmp = Float64((pi ^ -0.5) * Float64(0.047619047619047616 * (x ^ 7.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.005)
		tmp = x * ((pi ^ -0.5) * 2.0);
	else
		tmp = (pi ^ -0.5) * (0.047619047619047616 * (x ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.005], N[(x * N[(N[Power[Pi, -0.5], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0050000000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 99.9%

         \[\leadsto \color{blue}{{\left(\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)}^{1}} \]

      2. mul-fabs 99.9%

         \[\leadsto {\color{blue}{\left(\left|x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)}}^{1} \]

      3. +-commutative 99.9%

         \[\leadsto {\left(\left|x \cdot \frac{\color{blue}{\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}}{\sqrt{\pi}}\right|\right)}^{1} \]

      4. pow2 99.9%

         \[\leadsto {\left(\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, \color{blue}{{x}^{2}}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|\right)}^{1} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{{\left(\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 99.9%

         \[\leadsto \color{blue}{\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|} \]

      2. associate-*r/ 99.2%

         \[\leadsto \left|\color{blue}{\frac{x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)}{\sqrt{\pi}}}\right| \]

      3. fabs-div 99.2%

         \[\leadsto \color{blue}{\frac{\left|x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|}{\left|\sqrt{\pi}\right|}} \]

   8. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{\left|x \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|}{\sqrt{\pi}}} \]

   9. Taylor expanded in x around inf 98.2%

      \[\leadsto \frac{\left|x \cdot \left(2 + \color{blue}{0.047619047619047616 \cdot {x}^{6}}\right)\right|}{\sqrt{\pi}} \]

   10. Taylor expanded in x around 0 98.2%

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

       1. *-commutative 98.2%

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

   12. Simplified 98.2%

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

   13. Taylor expanded in x around 0 98.9%

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

       1. *-commutative 98.9%

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

       2. rem-square-sqrt 54.6%

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

       3. fabs-sqr 54.6%

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

       4. rem-square-sqrt 56.5%

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

       5. associate-*l* 56.5%

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

       6. *-commutative 56.5%

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

       7. associate-*r* 56.5%

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

       8. rem-exp-log 56.5%

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

       9. exp-neg 56.5%

          \[\leadsto x \cdot \left(\sqrt{\color{blue}{e^{-\log \pi}}} \cdot 2\right) \]

       10. unpow1/2 56.5%

           \[\leadsto x \cdot \left(\color{blue}{{\left(e^{-\log \pi}\right)}^{0.5}} \cdot 2\right) \]

       11. exp-prod 56.5%

           \[\leadsto x \cdot \left(\color{blue}{e^{\left(-\log \pi\right) \cdot 0.5}} \cdot 2\right) \]

       12. distribute-lft-neg-out 56.5%

           \[\leadsto x \cdot \left(e^{\color{blue}{-\log \pi \cdot 0.5}} \cdot 2\right) \]

       13. distribute-rgt-neg-in 56.5%

           \[\leadsto x \cdot \left(e^{\color{blue}{\log \pi \cdot \left(-0.5\right)}} \cdot 2\right) \]

       14. metadata-eval 56.5%

           \[\leadsto x \cdot \left(e^{\log \pi \cdot \color{blue}{-0.5}} \cdot 2\right) \]

       15. exp-to-pow 56.5%

           \[\leadsto x \cdot \left(\color{blue}{{\pi}^{-0.5}} \cdot 2\right) \]

       16. *-commutative 56.5%

           \[\leadsto x \cdot \color{blue}{\left(2 \cdot {\pi}^{-0.5}\right)} \]

   15. Simplified 56.5%

       \[\leadsto \color{blue}{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)} \]

   if 0.0050000000000000001 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 99.8%

         \[\leadsto \color{blue}{{\left(\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)}^{1}} \]

      2. mul-fabs 99.8%

         \[\leadsto {\color{blue}{\left(\left|x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)}}^{1} \]

      3. +-commutative 99.8%

         \[\leadsto {\left(\left|x \cdot \frac{\color{blue}{\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}}{\sqrt{\pi}}\right|\right)}^{1} \]

      4. pow2 99.8%

         \[\leadsto {\left(\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, \color{blue}{{x}^{2}}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|\right)}^{1} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{{\left(\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 99.8%

         \[\leadsto \color{blue}{\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|} \]

      2. associate-*r/ 99.9%

         \[\leadsto \left|\color{blue}{\frac{x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)}{\sqrt{\pi}}}\right| \]

      3. fabs-div 99.9%

         \[\leadsto \color{blue}{\frac{\left|x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|}{\left|\sqrt{\pi}\right|}} \]

   8. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\left|x \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|}{\sqrt{\pi}}} \]

   9. Taylor expanded in x around inf 98.6%

      \[\leadsto \frac{\left|x \cdot \left(2 + \color{blue}{0.047619047619047616 \cdot {x}^{6}}\right)\right|}{\sqrt{\pi}} \]
   10. Step-by-step derivation

       1. div-inv 98.7%

          \[\leadsto \color{blue}{\left|x \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \cdot \frac{1}{\sqrt{\pi}}} \]

       2. +-commutative 98.7%

          \[\leadsto \left|x \cdot \color{blue}{\left(0.047619047619047616 \cdot {x}^{6} + 2\right)}\right| \cdot \frac{1}{\sqrt{\pi}} \]

       3. fma-define 98.7%

          \[\leadsto \left|x \cdot \color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}\right| \cdot \frac{1}{\sqrt{\pi}} \]

       4. pow1/2 98.7%

          \[\leadsto \left|x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right| \cdot \frac{1}{\color{blue}{{\pi}^{0.5}}} \]

       5. pow-flip 98.7%

          \[\leadsto \left|x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right| \cdot \color{blue}{{\pi}^{\left(-0.5\right)}} \]

       6. metadata-eval 98.7%

          \[\leadsto \left|x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right| \cdot {\pi}^{\color{blue}{-0.5}} \]

   11. Applied egg-rr 98.7%

       \[\leadsto \color{blue}{\left|x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right| \cdot {\pi}^{-0.5}} \]
   12. Step-by-step derivation

       1. *-commutative 98.7%

          \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left|x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right|} \]

       2. rem-square-sqrt 0.0%

          \[\leadsto {\pi}^{-0.5} \cdot \left|\color{blue}{\sqrt{x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)} \cdot \sqrt{x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}}\right| \]

       3. fabs-sqr 0.0%

          \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(\sqrt{x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)} \cdot \sqrt{x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}\right)} \]

       4. rem-square-sqrt 0.1%

          \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right)} \]

       5. fma-undefine 0.1%

          \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \color{blue}{\left(0.047619047619047616 \cdot {x}^{6} + 2\right)}\right) \]

       6. +-commutative 0.1%

          \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \color{blue}{\left(2 + 0.047619047619047616 \cdot {x}^{6}\right)}\right) \]

       7. distribute-lft-in 0.1%

          \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(x \cdot 2 + x \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)} \]

       8. fma-define 0.1%

          \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\mathsf{fma}\left(x, 2, x \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)} \]

       9. *-commutative 0.1%

          \[\leadsto {\pi}^{-0.5} \cdot \mathsf{fma}\left(x, 2, \color{blue}{\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot x}\right) \]

       10. associate-*l* 0.1%

           \[\leadsto {\pi}^{-0.5} \cdot \mathsf{fma}\left(x, 2, \color{blue}{0.047619047619047616 \cdot \left({x}^{6} \cdot x\right)}\right) \]

       11. pow-plus 0.1%

           \[\leadsto {\pi}^{-0.5} \cdot \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot \color{blue}{{x}^{\left(6 + 1\right)}}\right) \]

       12. metadata-eval 0.1%

           \[\leadsto {\pi}^{-0.5} \cdot \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {x}^{\color{blue}{7}}\right) \]

   13. Simplified 0.1%

       \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {x}^{7}\right)} \]

   14. Taylor expanded in x around inf 0.1%

       \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.005:\\ \;\;\;\;x \cdot \left({\pi}^{-0.5} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 35.7% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left({\pi}^{-0.5} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (* (pow PI -0.5) (fma 0.047619047619047616 (pow x 6.0) 2.0))))

C

double code(double x) {
	return x * (pow(((double) M_PI), -0.5) * fma(0.047619047619047616, pow(x, 6.0), 2.0));
}

Julia

function code(x)
	return Float64(x * Float64((pi ^ -0.5) * fma(0.047619047619047616, (x ^ 6.0), 2.0)))
end

Wolfram

code[x_] := N[(x * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. pow1 99.9%

      \[\leadsto \color{blue}{{\left(\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)}^{1}} \]

   2. mul-fabs 99.9%

      \[\leadsto {\color{blue}{\left(\left|x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)}}^{1} \]

   3. +-commutative 99.9%

      \[\leadsto {\left(\left|x \cdot \frac{\color{blue}{\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}}{\sqrt{\pi}}\right|\right)}^{1} \]

   4. pow2 99.9%

      \[\leadsto {\left(\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, \color{blue}{{x}^{2}}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|\right)}^{1} \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{{\left(\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|\right)}^{1}} \]
7. Step-by-step derivation

   1. unpow1 99.9%

      \[\leadsto \color{blue}{\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|} \]

   2. associate-*r/ 99.4%

      \[\leadsto \left|\color{blue}{\frac{x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)}{\sqrt{\pi}}}\right| \]

   3. fabs-div 99.4%

      \[\leadsto \color{blue}{\frac{\left|x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|}{\left|\sqrt{\pi}\right|}} \]

8. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{\left|x \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|}{\sqrt{\pi}}} \]

9. Taylor expanded in x around inf 98.4%

   \[\leadsto \frac{\left|x \cdot \left(2 + \color{blue}{0.047619047619047616 \cdot {x}^{6}}\right)\right|}{\sqrt{\pi}} \]
10. Step-by-step derivation

    1. div-inv 98.8%

       \[\leadsto \color{blue}{\left|x \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \cdot \frac{1}{\sqrt{\pi}}} \]

    2. +-commutative 98.8%

       \[\leadsto \left|x \cdot \color{blue}{\left(0.047619047619047616 \cdot {x}^{6} + 2\right)}\right| \cdot \frac{1}{\sqrt{\pi}} \]

    3. fma-define 98.8%

       \[\leadsto \left|x \cdot \color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}\right| \cdot \frac{1}{\sqrt{\pi}} \]

    4. pow1/2 98.8%

       \[\leadsto \left|x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right| \cdot \frac{1}{\color{blue}{{\pi}^{0.5}}} \]

    5. pow-flip 98.8%

       \[\leadsto \left|x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right| \cdot \color{blue}{{\pi}^{\left(-0.5\right)}} \]

    6. metadata-eval 98.8%

       \[\leadsto \left|x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right| \cdot {\pi}^{\color{blue}{-0.5}} \]

11. Applied egg-rr 98.8%

    \[\leadsto \color{blue}{\left|x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right| \cdot {\pi}^{-0.5}} \]
12. Step-by-step derivation

    1. *-commutative 98.8%

       \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left|x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right|} \]

    2. rem-square-sqrt 37.7%

       \[\leadsto {\pi}^{-0.5} \cdot \left|\color{blue}{\sqrt{x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)} \cdot \sqrt{x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}}\right| \]

    3. fabs-sqr 37.7%

       \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(\sqrt{x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)} \cdot \sqrt{x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}\right)} \]

    4. rem-square-sqrt 39.1%

       \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right)} \]

    5. fma-undefine 39.1%

       \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \color{blue}{\left(0.047619047619047616 \cdot {x}^{6} + 2\right)}\right) \]

    6. +-commutative 39.1%

       \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \color{blue}{\left(2 + 0.047619047619047616 \cdot {x}^{6}\right)}\right) \]

    7. distribute-lft-in 39.1%

       \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(x \cdot 2 + x \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)} \]

    8. fma-define 39.1%

       \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\mathsf{fma}\left(x, 2, x \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)} \]

    9. *-commutative 39.1%

       \[\leadsto {\pi}^{-0.5} \cdot \mathsf{fma}\left(x, 2, \color{blue}{\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot x}\right) \]

    10. associate-*l* 39.1%

        \[\leadsto {\pi}^{-0.5} \cdot \mathsf{fma}\left(x, 2, \color{blue}{0.047619047619047616 \cdot \left({x}^{6} \cdot x\right)}\right) \]

    11. pow-plus 39.1%

        \[\leadsto {\pi}^{-0.5} \cdot \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot \color{blue}{{x}^{\left(6 + 1\right)}}\right) \]

    12. metadata-eval 39.1%

        \[\leadsto {\pi}^{-0.5} \cdot \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {x}^{\color{blue}{7}}\right) \]

13. Simplified 39.1%

    \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {x}^{7}\right)} \]

14. Taylor expanded in x around 0 39.1%

    \[\leadsto \color{blue}{x \cdot \left(0.047619047619047616 \cdot \left({x}^{6} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \sqrt{\frac{1}{\pi}}\right)} \]
15. Step-by-step derivation

    1. associate-*r* 39.1%

       \[\leadsto x \cdot \left(\color{blue}{\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot \sqrt{\frac{1}{\pi}}} + 2 \cdot \sqrt{\frac{1}{\pi}}\right) \]

    2. distribute-rgt-in 39.1%

       \[\leadsto x \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{6} + 2\right)\right)} \]

    3. fma-undefine 39.1%

       \[\leadsto x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}\right) \]

    4. rem-exp-log 39.1%

       \[\leadsto x \cdot \left(\sqrt{\frac{1}{\color{blue}{e^{\log \pi}}}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right) \]

    5. exp-neg 39.1%

       \[\leadsto x \cdot \left(\sqrt{\color{blue}{e^{-\log \pi}}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right) \]

    6. unpow1/2 39.1%

       \[\leadsto x \cdot \left(\color{blue}{{\left(e^{-\log \pi}\right)}^{0.5}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right) \]

    7. exp-prod 39.1%

       \[\leadsto x \cdot \left(\color{blue}{e^{\left(-\log \pi\right) \cdot 0.5}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right) \]

    8. distribute-lft-neg-out 39.1%

       \[\leadsto x \cdot \left(e^{\color{blue}{-\log \pi \cdot 0.5}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right) \]

    9. distribute-rgt-neg-in 39.1%

       \[\leadsto x \cdot \left(e^{\color{blue}{\log \pi \cdot \left(-0.5\right)}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right) \]

    10. metadata-eval 39.1%

        \[\leadsto x \cdot \left(e^{\log \pi \cdot \color{blue}{-0.5}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right) \]

    11. exp-to-pow 39.1%

        \[\leadsto x \cdot \left(\color{blue}{{\pi}^{-0.5}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right) \]

16. Simplified 39.1%

    \[\leadsto \color{blue}{x \cdot \left({\pi}^{-0.5} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right)} \]
17. Add Preprocessing

Alternative 4: 35.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;x \cdot \left({\pi}^{-0.5} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.9)
   (* x (* (pow PI -0.5) 2.0))
   (* 0.047619047619047616 (* (pow PI -0.5) (pow x 7.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = x * (pow(((double) M_PI), -0.5) * 2.0);
	} else {
		tmp = 0.047619047619047616 * (pow(((double) M_PI), -0.5) * pow(x, 7.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = x * (Math.pow(Math.PI, -0.5) * 2.0);
	} else {
		tmp = 0.047619047619047616 * (Math.pow(Math.PI, -0.5) * Math.pow(x, 7.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.9:
		tmp = x * (math.pow(math.pi, -0.5) * 2.0)
	else:
		tmp = 0.047619047619047616 * (math.pow(math.pi, -0.5) * math.pow(x, 7.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.9)
		tmp = Float64(x * Float64((pi ^ -0.5) * 2.0));
	else
		tmp = Float64(0.047619047619047616 * Float64((pi ^ -0.5) * (x ^ 7.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.9)
		tmp = x * ((pi ^ -0.5) * 2.0);
	else
		tmp = 0.047619047619047616 * ((pi ^ -0.5) * (x ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.9], N[(x * N[(N[Power[Pi, -0.5], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 99.9%

         \[\leadsto \color{blue}{{\left(\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)}^{1}} \]

      2. mul-fabs 99.9%

         \[\leadsto {\color{blue}{\left(\left|x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)}}^{1} \]

      3. +-commutative 99.9%

         \[\leadsto {\left(\left|x \cdot \frac{\color{blue}{\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}}{\sqrt{\pi}}\right|\right)}^{1} \]

      4. pow2 99.9%

         \[\leadsto {\left(\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, \color{blue}{{x}^{2}}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|\right)}^{1} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{{\left(\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 99.9%

         \[\leadsto \color{blue}{\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|} \]

      2. associate-*r/ 99.4%

         \[\leadsto \left|\color{blue}{\frac{x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)}{\sqrt{\pi}}}\right| \]

      3. fabs-div 99.4%

         \[\leadsto \color{blue}{\frac{\left|x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|}{\left|\sqrt{\pi}\right|}} \]

   8. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\left|x \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|}{\sqrt{\pi}}} \]

   9. Taylor expanded in x around inf 98.4%

      \[\leadsto \frac{\left|x \cdot \left(2 + \color{blue}{0.047619047619047616 \cdot {x}^{6}}\right)\right|}{\sqrt{\pi}} \]

   10. Taylor expanded in x around 0 69.6%

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

       1. *-commutative 69.6%

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

   12. Simplified 69.6%

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

   13. Taylor expanded in x around 0 70.1%

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

       1. *-commutative 70.1%

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

       2. rem-square-sqrt 37.7%

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

       3. fabs-sqr 37.7%

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

       4. rem-square-sqrt 39.2%

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

       5. associate-*l* 39.2%

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

       6. *-commutative 39.2%

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

       7. associate-*r* 39.2%

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

       8. rem-exp-log 39.2%

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

       9. exp-neg 39.2%

          \[\leadsto x \cdot \left(\sqrt{\color{blue}{e^{-\log \pi}}} \cdot 2\right) \]

       10. unpow1/2 39.2%

           \[\leadsto x \cdot \left(\color{blue}{{\left(e^{-\log \pi}\right)}^{0.5}} \cdot 2\right) \]

       11. exp-prod 39.2%

           \[\leadsto x \cdot \left(\color{blue}{e^{\left(-\log \pi\right) \cdot 0.5}} \cdot 2\right) \]

       12. distribute-lft-neg-out 39.2%

           \[\leadsto x \cdot \left(e^{\color{blue}{-\log \pi \cdot 0.5}} \cdot 2\right) \]

       13. distribute-rgt-neg-in 39.2%

           \[\leadsto x \cdot \left(e^{\color{blue}{\log \pi \cdot \left(-0.5\right)}} \cdot 2\right) \]

       14. metadata-eval 39.2%

           \[\leadsto x \cdot \left(e^{\log \pi \cdot \color{blue}{-0.5}} \cdot 2\right) \]

       15. exp-to-pow 39.2%

           \[\leadsto x \cdot \left(\color{blue}{{\pi}^{-0.5}} \cdot 2\right) \]

       16. *-commutative 39.2%

           \[\leadsto x \cdot \color{blue}{\left(2 \cdot {\pi}^{-0.5}\right)} \]

   15. Simplified 39.2%

       \[\leadsto \color{blue}{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)} \]

   if 1.8999999999999999 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. pow1 99.9%

         \[\leadsto \color{blue}{{\left(\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)}^{1}} \]

      2. mul-fabs 99.9%

         \[\leadsto {\color{blue}{\left(\left|x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)}}^{1} \]

      3. +-commutative 99.9%

         \[\leadsto {\left(\left|x \cdot \frac{\color{blue}{\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}}{\sqrt{\pi}}\right|\right)}^{1} \]

      4. pow2 99.9%

         \[\leadsto {\left(\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, \color{blue}{{x}^{2}}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|\right)}^{1} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{{\left(\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|\right)}^{1}} \]
   7. Step-by-step derivation

      1. unpow1 99.9%

         \[\leadsto \color{blue}{\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|} \]

      2. associate-*r/ 99.4%

         \[\leadsto \left|\color{blue}{\frac{x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)}{\sqrt{\pi}}}\right| \]

      3. fabs-div 99.4%

         \[\leadsto \color{blue}{\frac{\left|x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|}{\left|\sqrt{\pi}\right|}} \]

   8. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\left|x \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|}{\sqrt{\pi}}} \]

   9. Taylor expanded in x around inf 98.4%

      \[\leadsto \frac{\left|x \cdot \left(2 + \color{blue}{0.047619047619047616 \cdot {x}^{6}}\right)\right|}{\sqrt{\pi}} \]
   10. Step-by-step derivation

       1. div-inv 98.8%

          \[\leadsto \color{blue}{\left|x \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \cdot \frac{1}{\sqrt{\pi}}} \]

       2. +-commutative 98.8%

          \[\leadsto \left|x \cdot \color{blue}{\left(0.047619047619047616 \cdot {x}^{6} + 2\right)}\right| \cdot \frac{1}{\sqrt{\pi}} \]

       3. fma-define 98.8%

          \[\leadsto \left|x \cdot \color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}\right| \cdot \frac{1}{\sqrt{\pi}} \]

       4. pow1/2 98.8%

          \[\leadsto \left|x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right| \cdot \frac{1}{\color{blue}{{\pi}^{0.5}}} \]

       5. pow-flip 98.8%

          \[\leadsto \left|x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right| \cdot \color{blue}{{\pi}^{\left(-0.5\right)}} \]

       6. metadata-eval 98.8%

          \[\leadsto \left|x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right| \cdot {\pi}^{\color{blue}{-0.5}} \]

   11. Applied egg-rr 98.8%

       \[\leadsto \color{blue}{\left|x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right| \cdot {\pi}^{-0.5}} \]
   12. Step-by-step derivation

       1. *-commutative 98.8%

          \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left|x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right|} \]

       2. rem-square-sqrt 37.7%

          \[\leadsto {\pi}^{-0.5} \cdot \left|\color{blue}{\sqrt{x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)} \cdot \sqrt{x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}}\right| \]

       3. fabs-sqr 37.7%

          \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(\sqrt{x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)} \cdot \sqrt{x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}\right)} \]

       4. rem-square-sqrt 39.1%

          \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right)} \]

       5. fma-undefine 39.1%

          \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \color{blue}{\left(0.047619047619047616 \cdot {x}^{6} + 2\right)}\right) \]

       6. +-commutative 39.1%

          \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \color{blue}{\left(2 + 0.047619047619047616 \cdot {x}^{6}\right)}\right) \]

       7. distribute-lft-in 39.1%

          \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(x \cdot 2 + x \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)} \]

       8. fma-define 39.1%

          \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\mathsf{fma}\left(x, 2, x \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)} \]

       9. *-commutative 39.1%

          \[\leadsto {\pi}^{-0.5} \cdot \mathsf{fma}\left(x, 2, \color{blue}{\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot x}\right) \]

       10. associate-*l* 39.1%

           \[\leadsto {\pi}^{-0.5} \cdot \mathsf{fma}\left(x, 2, \color{blue}{0.047619047619047616 \cdot \left({x}^{6} \cdot x\right)}\right) \]

       11. pow-plus 39.1%

           \[\leadsto {\pi}^{-0.5} \cdot \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot \color{blue}{{x}^{\left(6 + 1\right)}}\right) \]

       12. metadata-eval 39.1%

           \[\leadsto {\pi}^{-0.5} \cdot \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {x}^{\color{blue}{7}}\right) \]

   13. Simplified 39.1%

       \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {x}^{7}\right)} \]

   14. Taylor expanded in x around inf 3.8%

       \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   15. Step-by-step derivation

       1. rem-exp-log 3.8%

          \[\leadsto 0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\color{blue}{e^{\log \pi}}}}\right) \]

       2. exp-neg 3.8%

          \[\leadsto 0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\color{blue}{e^{-\log \pi}}}\right) \]

       3. unpow1/2 3.8%

          \[\leadsto 0.047619047619047616 \cdot \left({x}^{7} \cdot \color{blue}{{\left(e^{-\log \pi}\right)}^{0.5}}\right) \]

       4. exp-prod 3.8%

          \[\leadsto 0.047619047619047616 \cdot \left({x}^{7} \cdot \color{blue}{e^{\left(-\log \pi\right) \cdot 0.5}}\right) \]

       5. distribute-lft-neg-out 3.8%

          \[\leadsto 0.047619047619047616 \cdot \left({x}^{7} \cdot e^{\color{blue}{-\log \pi \cdot 0.5}}\right) \]

       6. distribute-rgt-neg-in 3.8%

          \[\leadsto 0.047619047619047616 \cdot \left({x}^{7} \cdot e^{\color{blue}{\log \pi \cdot \left(-0.5\right)}}\right) \]

       7. metadata-eval 3.8%

          \[\leadsto 0.047619047619047616 \cdot \left({x}^{7} \cdot e^{\log \pi \cdot \color{blue}{-0.5}}\right) \]

       8. exp-to-pow 3.8%

          \[\leadsto 0.047619047619047616 \cdot \left({x}^{7} \cdot \color{blue}{{\pi}^{-0.5}}\right) \]

   16. Simplified 3.8%

       \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;x \cdot \left({\pi}^{-0.5} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 35.7% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ {\pi}^{-0.5} \cdot \left(x \cdot 2 + 0.047619047619047616 \cdot {x}^{7}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (pow PI -0.5) (+ (* x 2.0) (* 0.047619047619047616 (pow x 7.0)))))

C

double code(double x) {
	return pow(((double) M_PI), -0.5) * ((x * 2.0) + (0.047619047619047616 * pow(x, 7.0)));
}

Java

public static double code(double x) {
	return Math.pow(Math.PI, -0.5) * ((x * 2.0) + (0.047619047619047616 * Math.pow(x, 7.0)));
}

Python

def code(x):
	return math.pow(math.pi, -0.5) * ((x * 2.0) + (0.047619047619047616 * math.pow(x, 7.0)))

Julia

function code(x)
	return Float64((pi ^ -0.5) * Float64(Float64(x * 2.0) + Float64(0.047619047619047616 * (x ^ 7.0))))
end

MATLAB

function tmp = code(x)
	tmp = (pi ^ -0.5) * ((x * 2.0) + (0.047619047619047616 * (x ^ 7.0)));
end

Wolfram

code[x_] := N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(N[(x * 2.0), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. pow1 99.9%

      \[\leadsto \color{blue}{{\left(\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)}^{1}} \]

   2. mul-fabs 99.9%

      \[\leadsto {\color{blue}{\left(\left|x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)}}^{1} \]

   3. +-commutative 99.9%

      \[\leadsto {\left(\left|x \cdot \frac{\color{blue}{\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}}{\sqrt{\pi}}\right|\right)}^{1} \]

   4. pow2 99.9%

      \[\leadsto {\left(\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, \color{blue}{{x}^{2}}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|\right)}^{1} \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{{\left(\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|\right)}^{1}} \]
7. Step-by-step derivation

   1. unpow1 99.9%

      \[\leadsto \color{blue}{\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|} \]

   2. associate-*r/ 99.4%

      \[\leadsto \left|\color{blue}{\frac{x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)}{\sqrt{\pi}}}\right| \]

   3. fabs-div 99.4%

      \[\leadsto \color{blue}{\frac{\left|x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|}{\left|\sqrt{\pi}\right|}} \]

8. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{\left|x \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|}{\sqrt{\pi}}} \]

9. Taylor expanded in x around inf 98.4%

   \[\leadsto \frac{\left|x \cdot \left(2 + \color{blue}{0.047619047619047616 \cdot {x}^{6}}\right)\right|}{\sqrt{\pi}} \]
10. Step-by-step derivation

    1. div-inv 98.8%

       \[\leadsto \color{blue}{\left|x \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \cdot \frac{1}{\sqrt{\pi}}} \]

    2. +-commutative 98.8%

       \[\leadsto \left|x \cdot \color{blue}{\left(0.047619047619047616 \cdot {x}^{6} + 2\right)}\right| \cdot \frac{1}{\sqrt{\pi}} \]

    3. fma-define 98.8%

       \[\leadsto \left|x \cdot \color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}\right| \cdot \frac{1}{\sqrt{\pi}} \]

    4. pow1/2 98.8%

       \[\leadsto \left|x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right| \cdot \frac{1}{\color{blue}{{\pi}^{0.5}}} \]

    5. pow-flip 98.8%

       \[\leadsto \left|x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right| \cdot \color{blue}{{\pi}^{\left(-0.5\right)}} \]

    6. metadata-eval 98.8%

       \[\leadsto \left|x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right| \cdot {\pi}^{\color{blue}{-0.5}} \]

11. Applied egg-rr 98.8%

    \[\leadsto \color{blue}{\left|x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right| \cdot {\pi}^{-0.5}} \]
12. Step-by-step derivation

    1. *-commutative 98.8%

       \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left|x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right|} \]

    2. rem-square-sqrt 37.7%

       \[\leadsto {\pi}^{-0.5} \cdot \left|\color{blue}{\sqrt{x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)} \cdot \sqrt{x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}}\right| \]

    3. fabs-sqr 37.7%

       \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(\sqrt{x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)} \cdot \sqrt{x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}\right)} \]

    4. rem-square-sqrt 39.1%

       \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)\right)} \]

    5. fma-undefine 39.1%

       \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \color{blue}{\left(0.047619047619047616 \cdot {x}^{6} + 2\right)}\right) \]

    6. +-commutative 39.1%

       \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \color{blue}{\left(2 + 0.047619047619047616 \cdot {x}^{6}\right)}\right) \]

    7. distribute-lft-in 39.1%

       \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(x \cdot 2 + x \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)} \]

    8. fma-define 39.1%

       \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\mathsf{fma}\left(x, 2, x \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)\right)} \]

    9. *-commutative 39.1%

       \[\leadsto {\pi}^{-0.5} \cdot \mathsf{fma}\left(x, 2, \color{blue}{\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot x}\right) \]

    10. associate-*l* 39.1%

        \[\leadsto {\pi}^{-0.5} \cdot \mathsf{fma}\left(x, 2, \color{blue}{0.047619047619047616 \cdot \left({x}^{6} \cdot x\right)}\right) \]

    11. pow-plus 39.1%

        \[\leadsto {\pi}^{-0.5} \cdot \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot \color{blue}{{x}^{\left(6 + 1\right)}}\right) \]

    12. metadata-eval 39.1%

        \[\leadsto {\pi}^{-0.5} \cdot \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {x}^{\color{blue}{7}}\right) \]

13. Simplified 39.1%

    \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {x}^{7}\right)} \]
14. Step-by-step derivation

    1. fma-undefine 39.1%

       \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(x \cdot 2 + 0.047619047619047616 \cdot {x}^{7}\right)} \]

15. Applied egg-rr 39.1%

    \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(x \cdot 2 + 0.047619047619047616 \cdot {x}^{7}\right)} \]
16. Add Preprocessing

Alternative 6: 35.8% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ x \cdot \left({\pi}^{-0.5} \cdot 2\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (* (pow PI -0.5) 2.0)))

C

double code(double x) {
	return x * (pow(((double) M_PI), -0.5) * 2.0);
}

Java

public static double code(double x) {
	return x * (Math.pow(Math.PI, -0.5) * 2.0);
}

Python

def code(x):
	return x * (math.pow(math.pi, -0.5) * 2.0)

Julia

function code(x)
	return Float64(x * Float64((pi ^ -0.5) * 2.0))
end

MATLAB

function tmp = code(x)
	tmp = x * ((pi ^ -0.5) * 2.0);
end

Wolfram

code[x_] := N[(x * N[(N[Power[Pi, -0.5], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. pow1 99.9%

      \[\leadsto \color{blue}{{\left(\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)}^{1}} \]

   2. mul-fabs 99.9%

      \[\leadsto {\color{blue}{\left(\left|x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)}}^{1} \]

   3. +-commutative 99.9%

      \[\leadsto {\left(\left|x \cdot \frac{\color{blue}{\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}}{\sqrt{\pi}}\right|\right)}^{1} \]

   4. pow2 99.9%

      \[\leadsto {\left(\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, \color{blue}{{x}^{2}}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|\right)}^{1} \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{{\left(\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|\right)}^{1}} \]
7. Step-by-step derivation

   1. unpow1 99.9%

      \[\leadsto \color{blue}{\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|} \]

   2. associate-*r/ 99.4%

      \[\leadsto \left|\color{blue}{\frac{x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)}{\sqrt{\pi}}}\right| \]

   3. fabs-div 99.4%

      \[\leadsto \color{blue}{\frac{\left|x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|}{\left|\sqrt{\pi}\right|}} \]

8. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{\left|x \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|}{\sqrt{\pi}}} \]

9. Taylor expanded in x around inf 98.4%

   \[\leadsto \frac{\left|x \cdot \left(2 + \color{blue}{0.047619047619047616 \cdot {x}^{6}}\right)\right|}{\sqrt{\pi}} \]

10. Taylor expanded in x around 0 69.6%

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

    1. *-commutative 69.6%

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

12. Simplified 69.6%

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

13. Taylor expanded in x around 0 70.1%

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

    1. *-commutative 70.1%

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

    2. rem-square-sqrt 37.7%

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

    3. fabs-sqr 37.7%

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

    4. rem-square-sqrt 39.2%

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

    5. associate-*l* 39.2%

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

    6. *-commutative 39.2%

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

    7. associate-*r* 39.2%

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

    8. rem-exp-log 39.2%

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

    9. exp-neg 39.2%

       \[\leadsto x \cdot \left(\sqrt{\color{blue}{e^{-\log \pi}}} \cdot 2\right) \]

    10. unpow1/2 39.2%

        \[\leadsto x \cdot \left(\color{blue}{{\left(e^{-\log \pi}\right)}^{0.5}} \cdot 2\right) \]

    11. exp-prod 39.2%

        \[\leadsto x \cdot \left(\color{blue}{e^{\left(-\log \pi\right) \cdot 0.5}} \cdot 2\right) \]

    12. distribute-lft-neg-out 39.2%

        \[\leadsto x \cdot \left(e^{\color{blue}{-\log \pi \cdot 0.5}} \cdot 2\right) \]

    13. distribute-rgt-neg-in 39.2%

        \[\leadsto x \cdot \left(e^{\color{blue}{\log \pi \cdot \left(-0.5\right)}} \cdot 2\right) \]

    14. metadata-eval 39.2%

        \[\leadsto x \cdot \left(e^{\log \pi \cdot \color{blue}{-0.5}} \cdot 2\right) \]

    15. exp-to-pow 39.2%

        \[\leadsto x \cdot \left(\color{blue}{{\pi}^{-0.5}} \cdot 2\right) \]

    16. *-commutative 39.2%

        \[\leadsto x \cdot \color{blue}{\left(2 \cdot {\pi}^{-0.5}\right)} \]

15. Simplified 39.2%

    \[\leadsto \color{blue}{x \cdot \left(2 \cdot {\pi}^{-0.5}\right)} \]

16. Final simplification 39.2%

    \[\leadsto x \cdot \left({\pi}^{-0.5} \cdot 2\right) \]
17. Add Preprocessing

Alternative 7: 35.5% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. pow1 99.9%

      \[\leadsto \color{blue}{{\left(\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)}^{1}} \]

   2. mul-fabs 99.9%

      \[\leadsto {\color{blue}{\left(\left|x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)}}^{1} \]

   3. +-commutative 99.9%

      \[\leadsto {\left(\left|x \cdot \frac{\color{blue}{\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}}{\sqrt{\pi}}\right|\right)}^{1} \]

   4. pow2 99.9%

      \[\leadsto {\left(\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, \color{blue}{{x}^{2}}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|\right)}^{1} \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{{\left(\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|\right)}^{1}} \]
7. Step-by-step derivation

   1. unpow1 99.9%

      \[\leadsto \color{blue}{\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|} \]

   2. associate-*r/ 99.4%

      \[\leadsto \left|\color{blue}{\frac{x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)}{\sqrt{\pi}}}\right| \]

   3. fabs-div 99.4%

      \[\leadsto \color{blue}{\frac{\left|x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|}{\left|\sqrt{\pi}\right|}} \]

8. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{\left|x \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.2, {x}^{4}, 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|}{\sqrt{\pi}}} \]

9. Taylor expanded in x around inf 98.4%

   \[\leadsto \frac{\left|x \cdot \left(2 + \color{blue}{0.047619047619047616 \cdot {x}^{6}}\right)\right|}{\sqrt{\pi}} \]

10. Taylor expanded in x around 0 69.6%

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

    1. *-commutative 69.6%

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

12. Simplified 69.6%

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

13. Taylor expanded in x around 0 69.6%

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

    1. *-commutative 69.6%

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

    2. rem-square-sqrt 37.6%

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

    3. fabs-sqr 37.6%

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

    4. rem-square-sqrt 39.0%

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

    5. *-commutative 39.0%

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

15. Simplified 39.0%

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

16. Final simplification 39.0%

    \[\leadsto \frac{x \cdot 2}{\sqrt{\pi}} \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024170 
(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)))))))