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

Percentage Accurate: 99.8% → 99.9%

Time: 11.5s

Alternatives: 10

Speedup: 3.0×

• 29.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fabs x)
  (fabs
   (*
    (sqrt (/ 1.0 PI))
    (+
     2.0
     (+
      (* 0.047619047619047616 (pow x 6.0))
      (+ (* 0.2 (pow x 4.0)) (* 0.6666666666666666 (pow x 2.0)))))))))

C

double code(double x) {
	return fabs(x) * fabs((sqrt((1.0 / ((double) M_PI))) * (2.0 + ((0.047619047619047616 * pow(x, 6.0)) + ((0.2 * pow(x, 4.0)) + (0.6666666666666666 * pow(x, 2.0)))))));
}

Java

public static double code(double x) {
	return Math.abs(x) * Math.abs((Math.sqrt((1.0 / Math.PI)) * (2.0 + ((0.047619047619047616 * Math.pow(x, 6.0)) + ((0.2 * Math.pow(x, 4.0)) + (0.6666666666666666 * Math.pow(x, 2.0)))))));
}

Python

def code(x):
	return math.fabs(x) * math.fabs((math.sqrt((1.0 / math.pi)) * (2.0 + ((0.047619047619047616 * math.pow(x, 6.0)) + ((0.2 * math.pow(x, 4.0)) + (0.6666666666666666 * math.pow(x, 2.0)))))))

Julia

function code(x)
	return Float64(abs(x) * abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(2.0 + Float64(Float64(0.047619047619047616 * (x ^ 6.0)) + Float64(Float64(0.2 * (x ^ 4.0)) + Float64(0.6666666666666666 * (x ^ 2.0))))))))
end

MATLAB

function tmp = code(x)
	tmp = abs(x) * abs((sqrt((1.0 / pi)) * (2.0 + ((0.047619047619047616 * (x ^ 6.0)) + ((0.2 * (x ^ 4.0)) + (0.6666666666666666 * (x ^ 2.0)))))));
end

Wolfram

code[x_] := N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * 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[Power[x, 2.0], $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. Taylor expanded in x around 0 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 34.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2 + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (/
   (fma
    0.047619047619047616
    (pow x 6.0)
    (+ 2.0 (+ (* 0.2 (pow x 4.0)) (* 0.6666666666666666 (pow x 2.0)))))
   (sqrt PI))))

C

double code(double x) {
	return x * (fma(0.047619047619047616, pow(x, 6.0), (2.0 + ((0.2 * pow(x, 4.0)) + (0.6666666666666666 * pow(x, 2.0))))) / sqrt(((double) M_PI)));
}

Julia

function code(x)
	return Float64(x * Float64(fma(0.047619047619047616, (x ^ 6.0), Float64(2.0 + Float64(Float64(0.2 * (x ^ 4.0)) + Float64(0.6666666666666666 * (x ^ 2.0))))) / sqrt(pi)))
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

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. Taylor expanded in x around 0 99.9%

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

   1. pow1 99.9%

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

7. Applied egg-rr 29.2%

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

   1. unpow1 29.2%

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

   2. +-commutative 29.2%

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

   3. fma-undefine 29.2%

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

   4. associate-+l+ 29.2%

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

   5. fma-define 29.2%

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

   6. fma-undefine 29.2%

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

   7. +-commutative 29.2%

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

   8. *-commutative 29.2%

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

   9. fma-define 29.2%

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

9. Simplified 29.2%

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

    1. fma-undefine 29.2%

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

11. Applied egg-rr 29.2%

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

12. Final simplification 29.2%

    \[\leadsto x \cdot \frac{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2 + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)}{\sqrt{\pi}} \]
13. Add Preprocessing

Alternative 3: 34.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \mathbf{if}\;\left|x\right| \leq 0.2:\\ \;\;\;\;0.6666666666666666 \cdot \left(t\_0 \cdot {x}^{3}\right) + 2 \cdot \left(x \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (if (<= (fabs x) 0.2)
     (+ (* 0.6666666666666666 (* t_0 (pow x 3.0))) (* 2.0 (* x t_0)))
     (* t_0 (+ (* 0.2 (pow x 5.0)) (* 0.047619047619047616 (pow x 7.0)))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double tmp;
	if (fabs(x) <= 0.2) {
		tmp = (0.6666666666666666 * (t_0 * pow(x, 3.0))) + (2.0 * (x * t_0));
	} else {
		tmp = t_0 * ((0.2 * pow(x, 5.0)) + (0.047619047619047616 * pow(x, 7.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	double tmp;
	if (Math.abs(x) <= 0.2) {
		tmp = (0.6666666666666666 * (t_0 * Math.pow(x, 3.0))) + (2.0 * (x * t_0));
	} else {
		tmp = t_0 * ((0.2 * Math.pow(x, 5.0)) + (0.047619047619047616 * Math.pow(x, 7.0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / math.pi))
	tmp = 0
	if math.fabs(x) <= 0.2:
		tmp = (0.6666666666666666 * (t_0 * math.pow(x, 3.0))) + (2.0 * (x * t_0))
	else:
		tmp = t_0 * ((0.2 * math.pow(x, 5.0)) + (0.047619047619047616 * math.pow(x, 7.0)))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / pi))
	tmp = 0.0
	if (abs(x) <= 0.2)
		tmp = Float64(Float64(0.6666666666666666 * Float64(t_0 * (x ^ 3.0))) + Float64(2.0 * Float64(x * t_0)));
	else
		tmp = Float64(t_0 * Float64(Float64(0.2 * (x ^ 5.0)) + Float64(0.047619047619047616 * (x ^ 7.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 / pi));
	tmp = 0.0;
	if (abs(x) <= 0.2)
		tmp = (0.6666666666666666 * (t_0 * (x ^ 3.0))) + (2.0 * (x * t_0));
	else
		tmp = t_0 * ((0.2 * (x ^ 5.0)) + (0.047619047619047616 * (x ^ 7.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.2], N[(N[(0.6666666666666666 * N[(t$95$0 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.20000000000000001

   1. Initial program 99.8%

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

   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. Taylor expanded in x around 0 99.9%

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

      1. pow1 99.9%

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

   7. Applied egg-rr 44.8%

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

      1. unpow1 44.8%

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

      2. +-commutative 44.8%

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

      3. fma-undefine 44.8%

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

      4. associate-+l+ 44.8%

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

      5. fma-define 44.8%

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

      6. fma-undefine 44.8%

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

      7. +-commutative 44.8%

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

      8. *-commutative 44.8%

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

      9. fma-define 44.8%

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

   9. Simplified 44.8%

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

   10. Taylor expanded in x around 0 43.9%

       \[\leadsto \color{blue}{0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)} \]

   if 0.20000000000000001 < (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 100.0%

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

   5. Taylor expanded in x around 0 99.9%

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

      1. pow1 99.9%

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

   7. Applied egg-rr 0.1%

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

      1. unpow1 0.1%

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

      2. +-commutative 0.1%

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

      3. fma-undefine 0.1%

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

      4. associate-+l+ 0.1%

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

      5. fma-define 0.1%

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

      6. fma-undefine 0.1%

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

      7. +-commutative 0.1%

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

      8. *-commutative 0.1%

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

      9. fma-define 0.1%

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

   9. Simplified 0.1%

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

   10. Taylor expanded in x around inf 0.1%

       \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.2 \cdot \left({x}^{5} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   11. Step-by-step derivation

       1. +-commutative 0.1%

          \[\leadsto \color{blue}{0.2 \cdot \left({x}^{5} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)} \]

       2. associate-*r* 0.1%

          \[\leadsto \color{blue}{\left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}}} + 0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right) \]

       3. associate-*r* 0.1%

          \[\leadsto \left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}} \]

       4. distribute-rgt-out 0.1%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)} \]

   12. Simplified 0.1%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.2:\\ \;\;\;\;0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{3}\right) + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 34.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \mathbf{if}\;\left|x\right| \leq 0.2:\\ \;\;\;\;t\_0 \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (if (<= (fabs x) 0.2)
     (* t_0 (+ (* x 2.0) (* 0.6666666666666666 (pow x 3.0))))
     (* t_0 (+ (* 0.2 (pow x 5.0)) (* 0.047619047619047616 (pow x 7.0)))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double tmp;
	if (fabs(x) <= 0.2) {
		tmp = t_0 * ((x * 2.0) + (0.6666666666666666 * pow(x, 3.0)));
	} else {
		tmp = t_0 * ((0.2 * pow(x, 5.0)) + (0.047619047619047616 * pow(x, 7.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	double tmp;
	if (Math.abs(x) <= 0.2) {
		tmp = t_0 * ((x * 2.0) + (0.6666666666666666 * Math.pow(x, 3.0)));
	} else {
		tmp = t_0 * ((0.2 * Math.pow(x, 5.0)) + (0.047619047619047616 * Math.pow(x, 7.0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / math.pi))
	tmp = 0
	if math.fabs(x) <= 0.2:
		tmp = t_0 * ((x * 2.0) + (0.6666666666666666 * math.pow(x, 3.0)))
	else:
		tmp = t_0 * ((0.2 * math.pow(x, 5.0)) + (0.047619047619047616 * math.pow(x, 7.0)))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / pi))
	tmp = 0.0
	if (abs(x) <= 0.2)
		tmp = Float64(t_0 * Float64(Float64(x * 2.0) + Float64(0.6666666666666666 * (x ^ 3.0))));
	else
		tmp = Float64(t_0 * Float64(Float64(0.2 * (x ^ 5.0)) + Float64(0.047619047619047616 * (x ^ 7.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 / pi));
	tmp = 0.0;
	if (abs(x) <= 0.2)
		tmp = t_0 * ((x * 2.0) + (0.6666666666666666 * (x ^ 3.0)));
	else
		tmp = t_0 * ((0.2 * (x ^ 5.0)) + (0.047619047619047616 * (x ^ 7.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.2], N[(t$95$0 * N[(N[(x * 2.0), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.20000000000000001

   1. Initial program 99.8%

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

   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. Taylor expanded in x around 0 99.9%

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

      1. pow1 99.9%

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

   7. Applied egg-rr 44.8%

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

      1. unpow1 44.8%

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

      2. +-commutative 44.8%

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

      3. fma-undefine 44.8%

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

      4. associate-+l+ 44.8%

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

      5. fma-define 44.8%

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

      6. fma-undefine 44.8%

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

      7. +-commutative 44.8%

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

      8. *-commutative 44.8%

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

      9. fma-define 44.8%

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

   9. Simplified 44.8%

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

   10. Taylor expanded in x around 0 43.9%

       \[\leadsto \color{blue}{0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   11. Step-by-step derivation

       1. +-commutative 43.9%

          \[\leadsto \color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right) + 0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right)} \]

       2. associate-*r* 43.9%

          \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} + 0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right) \]

       3. associate-*r* 43.9%

          \[\leadsto \left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}} \]

       4. distribute-rgt-out 43.9%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)} \]

   12. Simplified 43.9%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)} \]

   if 0.20000000000000001 < (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 100.0%

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

   5. Taylor expanded in x around 0 99.9%

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

      1. pow1 99.9%

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

   7. Applied egg-rr 0.1%

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

      1. unpow1 0.1%

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

      2. +-commutative 0.1%

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

      3. fma-undefine 0.1%

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

      4. associate-+l+ 0.1%

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

      5. fma-define 0.1%

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

      6. fma-undefine 0.1%

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

      7. +-commutative 0.1%

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

      8. *-commutative 0.1%

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

      9. fma-define 0.1%

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

   9. Simplified 0.1%

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

   10. Taylor expanded in x around inf 0.1%

       \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.2 \cdot \left({x}^{5} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   11. Step-by-step derivation

       1. +-commutative 0.1%

          \[\leadsto \color{blue}{0.2 \cdot \left({x}^{5} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)} \]

       2. associate-*r* 0.1%

          \[\leadsto \color{blue}{\left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}}} + 0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right) \]

       3. associate-*r* 0.1%

          \[\leadsto \left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}} \]

       4. distribute-rgt-out 0.1%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)} \]

   12. Simplified 0.1%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.2:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.2:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{0.047619047619047616 \cdot {x}^{6}}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.2)
   (* (sqrt (/ 1.0 PI)) (+ (* x 2.0) (* 0.6666666666666666 (pow x 3.0))))
   (fabs (* x (/ (* 0.047619047619047616 (pow x 6.0)) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.2) {
		tmp = sqrt((1.0 / ((double) M_PI))) * ((x * 2.0) + (0.6666666666666666 * pow(x, 3.0)));
	} else {
		tmp = fabs((x * ((0.047619047619047616 * pow(x, 6.0)) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.2) {
		tmp = Math.sqrt((1.0 / Math.PI)) * ((x * 2.0) + (0.6666666666666666 * Math.pow(x, 3.0)));
	} else {
		tmp = Math.abs((x * ((0.047619047619047616 * Math.pow(x, 6.0)) / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.2:
		tmp = math.sqrt((1.0 / math.pi)) * ((x * 2.0) + (0.6666666666666666 * math.pow(x, 3.0)))
	else:
		tmp = math.fabs((x * ((0.047619047619047616 * math.pow(x, 6.0)) / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.2)
		tmp = Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(x * 2.0) + Float64(0.6666666666666666 * (x ^ 3.0))));
	else
		tmp = abs(Float64(x * Float64(Float64(0.047619047619047616 * (x ^ 6.0)) / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.2)
		tmp = sqrt((1.0 / pi)) * ((x * 2.0) + (0.6666666666666666 * (x ^ 3.0)));
	else
		tmp = abs((x * ((0.047619047619047616 * (x ^ 6.0)) / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.2], N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(x * 2.0), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(x * N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.20000000000000001

   1. Initial program 99.8%

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

   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. Taylor expanded in x around 0 99.9%

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

      1. pow1 99.9%

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

   7. Applied egg-rr 44.8%

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

      1. unpow1 44.8%

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

      2. +-commutative 44.8%

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

      3. fma-undefine 44.8%

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

      4. associate-+l+ 44.8%

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

      5. fma-define 44.8%

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

      6. fma-undefine 44.8%

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

      7. +-commutative 44.8%

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

      8. *-commutative 44.8%

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

      9. fma-define 44.8%

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

   9. Simplified 44.8%

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

   10. Taylor expanded in x around 0 43.9%

       \[\leadsto \color{blue}{0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   11. Step-by-step derivation

       1. +-commutative 43.9%

          \[\leadsto \color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right) + 0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right)} \]

       2. associate-*r* 43.9%

          \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} + 0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right) \]

       3. associate-*r* 43.9%

          \[\leadsto \left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}} \]

       4. distribute-rgt-out 43.9%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)} \]

   12. Simplified 43.9%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)} \]

   if 0.20000000000000001 < (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 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around inf 99.9%

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

      1. pow1 99.9%

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

      2. add-sqr-sqrt 0.0%

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

      3. fabs-sqr 0.0%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate-*r* 99.8%

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

      6. sqrt-div 99.8%

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

      7. metadata-eval 99.8%

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

      8. un-div-inv 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. unpow1 99.8%

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

       2. associate-*l* 99.9%

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

       3. associate-*r/ 99.9%

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

   12. Simplified 99.9%

       \[\leadsto \left|\color{blue}{x \cdot \frac{0.047619047619047616 \cdot {x}^{6}}{\sqrt{\pi}}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.2:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{0.047619047619047616 \cdot {x}^{6}}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 34.4% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2 + 0.6666666666666666 \cdot {x}^{2}\right)}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (/
   (fma
    0.047619047619047616
    (pow x 6.0)
    (+ 2.0 (* 0.6666666666666666 (pow x 2.0))))
   (sqrt PI))))

C

double code(double x) {
	return x * (fma(0.047619047619047616, pow(x, 6.0), (2.0 + (0.6666666666666666 * pow(x, 2.0)))) / sqrt(((double) M_PI)));
}

Julia

function code(x)
	return Float64(x * Float64(fma(0.047619047619047616, (x ^ 6.0), Float64(2.0 + Float64(0.6666666666666666 * (x ^ 2.0)))) / sqrt(pi)))
end

Wolfram

code[x_] := N[(x * N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision] + N[(2.0 + N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

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. Taylor expanded in x around 0 99.9%

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

   1. pow1 99.9%

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

7. Applied egg-rr 29.2%

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

   1. unpow1 29.2%

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

   2. +-commutative 29.2%

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

   3. fma-undefine 29.2%

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

   4. associate-+l+ 29.2%

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

   5. fma-define 29.2%

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

   6. fma-undefine 29.2%

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

   7. +-commutative 29.2%

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

   8. *-commutative 29.2%

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

   9. fma-define 29.2%

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

9. Simplified 29.2%

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

10. Taylor expanded in x around 0 28.7%

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

11. Final simplification 28.7%

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

Alternative 7: 34.4% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.2:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.2)
   (* (sqrt (/ 1.0 PI)) (+ (* x 2.0) (* 0.6666666666666666 (pow x 3.0))))
   (* 0.047619047619047616 (/ (pow x 7.0) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.2) {
		tmp = sqrt((1.0 / ((double) M_PI))) * ((x * 2.0) + (0.6666666666666666 * pow(x, 3.0)));
	} else {
		tmp = 0.047619047619047616 * (pow(x, 7.0) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.2) {
		tmp = Math.sqrt((1.0 / Math.PI)) * ((x * 2.0) + (0.6666666666666666 * Math.pow(x, 3.0)));
	} else {
		tmp = 0.047619047619047616 * (Math.pow(x, 7.0) / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.2:
		tmp = math.sqrt((1.0 / math.pi)) * ((x * 2.0) + (0.6666666666666666 * math.pow(x, 3.0)))
	else:
		tmp = 0.047619047619047616 * (math.pow(x, 7.0) / math.sqrt(math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.2)
		tmp = Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(x * 2.0) + Float64(0.6666666666666666 * (x ^ 3.0))));
	else
		tmp = Float64(0.047619047619047616 * Float64((x ^ 7.0) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.2)
		tmp = sqrt((1.0 / pi)) * ((x * 2.0) + (0.6666666666666666 * (x ^ 3.0)));
	else
		tmp = 0.047619047619047616 * ((x ^ 7.0) / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.2], N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(x * 2.0), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[x, 7.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.20000000000000001

   1. Initial program 99.8%

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

   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. Taylor expanded in x around 0 99.9%

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

      1. pow1 99.9%

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

   7. Applied egg-rr 44.8%

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

      1. unpow1 44.8%

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

      2. +-commutative 44.8%

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

      3. fma-undefine 44.8%

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

      4. associate-+l+ 44.8%

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

      5. fma-define 44.8%

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

      6. fma-undefine 44.8%

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

      7. +-commutative 44.8%

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

      8. *-commutative 44.8%

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

      9. fma-define 44.8%

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

   9. Simplified 44.8%

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

   10. Taylor expanded in x around 0 43.9%

       \[\leadsto \color{blue}{0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   11. Step-by-step derivation

       1. +-commutative 43.9%

          \[\leadsto \color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right) + 0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right)} \]

       2. associate-*r* 43.9%

          \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} + 0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right) \]

       3. associate-*r* 43.9%

          \[\leadsto \left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}} \]

       4. distribute-rgt-out 43.9%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)} \]

   12. Simplified 43.9%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)} \]

   if 0.20000000000000001 < (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 100.0%

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

   5. Taylor expanded in x around 0 99.9%

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

      1. pow1 99.9%

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

   7. Applied egg-rr 0.1%

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

      1. unpow1 0.1%

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

      2. +-commutative 0.1%

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

      3. fma-undefine 0.1%

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

      4. associate-+l+ 0.1%

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

      5. fma-define 0.1%

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

      6. fma-undefine 0.1%

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

      7. +-commutative 0.1%

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

      8. *-commutative 0.1%

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

      9. fma-define 0.1%

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

   9. Simplified 0.1%

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

   10. Taylor expanded in x around inf 0.1%

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

       1. associate-*r* 0.1%

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

       2. sqrt-div 0.1%

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

       3. metadata-eval 0.1%

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

       4. un-div-inv 0.1%

          \[\leadsto \color{blue}{\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}} \]

   12. Applied egg-rr 0.1%

       \[\leadsto \color{blue}{\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}} \]
   13. Step-by-step derivation

       1. associate-/l* 0.1%

          \[\leadsto \color{blue}{0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}} \]

   14. Simplified 0.1%

       \[\leadsto \color{blue}{0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.2:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.2% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \sqrt{\frac{{x}^{14}}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (* x (/ 2.0 (sqrt PI)))
   (* 0.047619047619047616 (sqrt (/ (pow x 14.0) PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = 0.047619047619047616 * sqrt((pow(x, 14.0) / ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = 0.047619047619047616 * Math.sqrt((Math.pow(x, 14.0) / Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = 0.047619047619047616 * math.sqrt((math.pow(x, 14.0) / math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(0.047619047619047616 * sqrt(Float64((x ^ 14.0) / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = 0.047619047619047616 * sqrt(((x ^ 14.0) / pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[Sqrt[N[(N[Power[x, 14.0], $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   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. Taylor expanded in x around 0 99.9%

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

      1. pow1 99.9%

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

   7. Applied egg-rr 29.2%

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

      1. unpow1 29.2%

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

      2. +-commutative 29.2%

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

      3. fma-undefine 29.2%

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

      4. associate-+l+ 29.2%

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

      5. fma-define 29.2%

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

      6. fma-undefine 29.2%

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

      7. +-commutative 29.2%

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

      8. *-commutative 29.2%

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

      9. fma-define 29.2%

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

   9. Simplified 29.2%

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

   10. Taylor expanded in x around 0 28.4%

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

       1. associate-*r* 28.4%

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

   12. Simplified 28.4%

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

       1. sqrt-div 28.4%

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

       2. metadata-eval 28.4%

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

       3. un-div-inv 28.2%

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

       4. *-commutative 28.2%

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

   14. Applied egg-rr 28.2%

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

       1. associate-/l* 28.4%

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

   16. Simplified 28.4%

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

   if 1.8500000000000001 < 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. Taylor expanded in x around 0 99.9%

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

      1. pow1 99.9%

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

   7. Applied egg-rr 29.2%

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

      1. unpow1 29.2%

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

      2. +-commutative 29.2%

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

      3. fma-undefine 29.2%

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

      4. associate-+l+ 29.2%

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

      5. fma-define 29.2%

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

      6. fma-undefine 29.2%

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

      7. +-commutative 29.2%

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

      8. *-commutative 29.2%

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

      9. fma-define 29.2%

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

   9. Simplified 29.2%

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

   10. Taylor expanded in x around inf 3.7%

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

       1. add-sqr-sqrt 3.6%

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

       2. sqrt-unprod 37.5%

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

       3. *-commutative 37.5%

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

       4. *-commutative 37.5%

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

       5. swap-sqr 37.5%

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

       6. add-sqr-sqrt 37.5%

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

       7. pow-prod-up 37.5%

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

       8. metadata-eval 37.5%

          \[\leadsto 0.047619047619047616 \cdot \sqrt{\frac{1}{\pi} \cdot {x}^{\color{blue}{14}}} \]

   12. Applied egg-rr 37.5%

       \[\leadsto 0.047619047619047616 \cdot \color{blue}{\sqrt{\frac{1}{\pi} \cdot {x}^{14}}} \]
   13. Step-by-step derivation

       1. associate-*l/ 37.5%

          \[\leadsto 0.047619047619047616 \cdot \sqrt{\color{blue}{\frac{1 \cdot {x}^{14}}{\pi}}} \]

       2. *-lft-identity 37.5%

          \[\leadsto 0.047619047619047616 \cdot \sqrt{\frac{\color{blue}{{x}^{14}}}{\pi}} \]

   14. Simplified 37.5%

       \[\leadsto 0.047619047619047616 \cdot \color{blue}{\sqrt{\frac{{x}^{14}}{\pi}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \sqrt{\frac{{x}^{14}}{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.2% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (* x (/ 2.0 (sqrt PI)))
   (* 0.047619047619047616 (/ (pow x 7.0) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = 0.047619047619047616 * (pow(x, 7.0) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = 0.047619047619047616 * (Math.pow(x, 7.0) / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = 0.047619047619047616 * (math.pow(x, 7.0) / math.sqrt(math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(0.047619047619047616 * Float64((x ^ 7.0) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = 0.047619047619047616 * ((x ^ 7.0) / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[x, 7.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   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. Taylor expanded in x around 0 99.9%

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

      1. pow1 99.9%

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

   7. Applied egg-rr 29.2%

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

      1. unpow1 29.2%

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

      2. +-commutative 29.2%

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

      3. fma-undefine 29.2%

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

      4. associate-+l+ 29.2%

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

      5. fma-define 29.2%

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

      6. fma-undefine 29.2%

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

      7. +-commutative 29.2%

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

      8. *-commutative 29.2%

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

      9. fma-define 29.2%

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

   9. Simplified 29.2%

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

   10. Taylor expanded in x around 0 28.4%

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

       1. associate-*r* 28.4%

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

   12. Simplified 28.4%

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

       1. sqrt-div 28.4%

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

       2. metadata-eval 28.4%

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

       3. un-div-inv 28.2%

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

       4. *-commutative 28.2%

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

   14. Applied egg-rr 28.2%

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

       1. associate-/l* 28.4%

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

   16. Simplified 28.4%

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

   if 1.8500000000000001 < 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. Taylor expanded in x around 0 99.9%

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

      1. pow1 99.9%

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

   7. Applied egg-rr 29.2%

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

      1. unpow1 29.2%

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

      2. +-commutative 29.2%

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

      3. fma-undefine 29.2%

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

      4. associate-+l+ 29.2%

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

      5. fma-define 29.2%

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

      6. fma-undefine 29.2%

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

      7. +-commutative 29.2%

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

      8. *-commutative 29.2%

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

      9. fma-define 29.2%

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

   9. Simplified 29.2%

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

   10. Taylor expanded in x around inf 3.7%

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

       1. associate-*r* 3.7%

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

       2. sqrt-div 3.7%

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

       3. metadata-eval 3.7%

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

       4. un-div-inv 3.7%

          \[\leadsto \color{blue}{\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}} \]

   12. Applied egg-rr 3.7%

       \[\leadsto \color{blue}{\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}} \]
   13. Step-by-step derivation

       1. associate-/l* 3.7%

          \[\leadsto \color{blue}{0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}} \]

   14. Simplified 3.7%

       \[\leadsto \color{blue}{0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 34.2% accurate, 17.6× speedup

Math

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

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(x * Float64(2.0 / sqrt(pi)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

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. Taylor expanded in x around 0 99.9%

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

   1. pow1 99.9%

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

7. Applied egg-rr 29.2%

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

   1. unpow1 29.2%

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

   2. +-commutative 29.2%

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

   3. fma-undefine 29.2%

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

   4. associate-+l+ 29.2%

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

   5. fma-define 29.2%

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

   6. fma-undefine 29.2%

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

   7. +-commutative 29.2%

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

   8. *-commutative 29.2%

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

   9. fma-define 29.2%

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

9. Simplified 29.2%

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

10. Taylor expanded in x around 0 28.4%

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

    1. associate-*r* 28.4%

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

12. Simplified 28.4%

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

    1. sqrt-div 28.4%

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

    2. metadata-eval 28.4%

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

    3. un-div-inv 28.2%

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

    4. *-commutative 28.2%

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

14. Applied egg-rr 28.2%

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

    1. associate-/l* 28.4%

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

16. Simplified 28.4%

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

17. Final simplification 28.4%

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

Reproduce

herbie shell --seed 2024039 
(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)))))))