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

Percentage Accurate: 99.8% → 99.8%

Time: 12.1s

Alternatives: 10

Speedup: 3.0×

• 27.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot \left(x \cdot x\right)\\ t_1 := \left|x\right| \cdot \left(\left|x\right| \cdot t\_0\right)\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + 0.6666666666666666 \cdot t\_0\right) + 0.2 \cdot t\_1\right) + 0.047619047619047616 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot t\_1\right)\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (* x x))) (t_1 (* (fabs x) (* (fabs x) t_0))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* 0.6666666666666666 t_0)) (* 0.2 t_1))
      (* 0.047619047619047616 (* (fabs x) (* (fabs x) t_1))))))))

C

double code(double x) {
	double t_0 = fabs(x) * (x * x);
	double t_1 = fabs(x) * (fabs(x) * t_0);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (fabs(x) * (fabs(x) * t_1))))));
}

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * (x * x);
	double t_1 = Math.abs(x) * (Math.abs(x) * t_0);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (Math.abs(x) * (Math.abs(x) * t_1))))));
}

Python

def code(x):
	t_0 = math.fabs(x) * (x * x)
	t_1 = math.fabs(x) * (math.fabs(x) * t_0)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (math.fabs(x) * (math.fabs(x) * t_1))))))

Julia

function code(x)
	t_0 = Float64(abs(x) * Float64(x * x))
	t_1 = Float64(abs(x) * Float64(abs(x) * t_0))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(0.6666666666666666 * t_0)) + Float64(0.2 * t_1)) + Float64(0.047619047619047616 * Float64(abs(x) * Float64(abs(x) * t_1))))))
end

MATLAB

function tmp = code(x)
	t_0 = abs(x) * (x * x);
	t_1 = abs(x) * (abs(x) * t_0);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (abs(x) * (abs(x) * t_1))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$0), $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[(0.6666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.2 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 3.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fabs x)
  (fabs
   (*
    (pow PI -0.5)
    (+
     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((pow(((double) M_PI), -0.5) * (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.pow(Math.PI, -0.5) * (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.pow(math.pi, -0.5) * (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((pi ^ -0.5) * 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(((pi ^ -0.5) * (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[Power[Pi, -0.5], $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. Step-by-step derivation

   1. *-un-lft-identity 99.9%

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

   2. inv-pow 99.9%

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

   3. sqrt-pow1 99.9%

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

   4. metadata-eval 99.9%

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

7. Applied egg-rr 99.9%

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

   1. *-lft-identity 99.9%

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

9. Simplified 99.9%

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

Alternative 3: 35.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot \left(2 + {x}^{2} \cdot \left(0.6666666666666666 + 0.2 \cdot {x}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.1)
   (*
    (pow PI -0.5)
    (* x (+ 2.0 (* (pow x 2.0) (+ 0.6666666666666666 (* 0.2 (pow x 2.0)))))))
   (*
    (pow x 7.0)
    (* (sqrt (/ 1.0 PI)) (+ 0.047619047619047616 (/ 0.2 (pow x 2.0)))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.1) {
		tmp = pow(((double) M_PI), -0.5) * (x * (2.0 + (pow(x, 2.0) * (0.6666666666666666 + (0.2 * pow(x, 2.0))))));
	} else {
		tmp = pow(x, 7.0) * (sqrt((1.0 / ((double) M_PI))) * (0.047619047619047616 + (0.2 / pow(x, 2.0))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.1) {
		tmp = Math.pow(Math.PI, -0.5) * (x * (2.0 + (Math.pow(x, 2.0) * (0.6666666666666666 + (0.2 * Math.pow(x, 2.0))))));
	} else {
		tmp = Math.pow(x, 7.0) * (Math.sqrt((1.0 / Math.PI)) * (0.047619047619047616 + (0.2 / Math.pow(x, 2.0))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.1:
		tmp = math.pow(math.pi, -0.5) * (x * (2.0 + (math.pow(x, 2.0) * (0.6666666666666666 + (0.2 * math.pow(x, 2.0))))))
	else:
		tmp = math.pow(x, 7.0) * (math.sqrt((1.0 / math.pi)) * (0.047619047619047616 + (0.2 / math.pow(x, 2.0))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.1)
		tmp = Float64((pi ^ -0.5) * Float64(x * Float64(2.0 + Float64((x ^ 2.0) * Float64(0.6666666666666666 + Float64(0.2 * (x ^ 2.0)))))));
	else
		tmp = Float64((x ^ 7.0) * Float64(sqrt(Float64(1.0 / pi)) * Float64(0.047619047619047616 + Float64(0.2 / (x ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.1)
		tmp = (pi ^ -0.5) * (x * (2.0 + ((x ^ 2.0) * (0.6666666666666666 + (0.2 * (x ^ 2.0))))));
	else
		tmp = (x ^ 7.0) * (sqrt((1.0 / pi)) * (0.047619047619047616 + (0.2 / (x ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.1], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * N[(2.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.6666666666666666 + N[(0.2 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 7.0], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(0.047619047619047616 + N[(0.2 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.10000000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.8%

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

   5. 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. *-un-lft-identity 99.9%

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

      2. inv-pow 99.9%

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

      3. sqrt-pow1 99.9%

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

      4. metadata-eval 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. *-lft-identity 99.9%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in x around 0 99.9%

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

   12. Simplified 47.8%

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

   13. Taylor expanded in x around 0 47.8%

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

       1. *-commutative 47.8%

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

   15. Simplified 47.8%

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

   if 0.10000000000000001 < (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%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. inv-pow 99.8%

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

      3. sqrt-pow1 99.8%

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

      4. metadata-eval 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. *-lft-identity 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in x around 0 99.8%

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

   12. Simplified 0.1%

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

   13. Taylor expanded in x around inf 0.1%

       \[\leadsto \color{blue}{{x}^{7} \cdot \left(0.047619047619047616 \cdot \sqrt{\frac{1}{\pi}} + 0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]
   14. Step-by-step derivation

       1. associate-*r* 0.1%

          \[\leadsto {x}^{7} \cdot \left(0.047619047619047616 \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \sqrt{\frac{1}{\pi}}}\right) \]

       2. distribute-rgt-out 0.1%

          \[\leadsto {x}^{7} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + 0.2 \cdot \frac{1}{{x}^{2}}\right)\right)} \]

       3. associate-*r/ 0.1%

          \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \color{blue}{\frac{0.2 \cdot 1}{{x}^{2}}}\right)\right) \]

       4. metadata-eval 0.1%

          \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \frac{\color{blue}{0.2}}{{x}^{2}}\right)\right) \]

   15. Simplified 0.1%

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

4. Final simplification 32.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot \left(2 + {x}^{2} \cdot \left(0.6666666666666666 + 0.2 \cdot {x}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 35.6% accurate, 4.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (pow PI -0.5)
  (*
   x
   (+
    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 pow(((double) M_PI), -0.5) * (x * (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.pow(Math.PI, -0.5) * (x * (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.pow(math.pi, -0.5) * (x * (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((pi ^ -0.5) * Float64(x * 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 = (pi ^ -0.5) * (x * (2.0 + ((0.047619047619047616 * (x ^ 6.0)) + ((0.2 * (x ^ 4.0)) + (0.6666666666666666 * (x ^ 2.0))))));
end

Wolfram

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

   1. *-un-lft-identity 99.9%

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

   2. inv-pow 99.9%

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

   3. sqrt-pow1 99.9%

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

   4. metadata-eval 99.9%

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

7. Applied egg-rr 99.9%

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

   1. *-lft-identity 99.9%

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

9. Simplified 99.9%

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

10. Taylor expanded in x around 0 99.9%

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

12. Simplified 32.1%

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

    1. fma-undefine 32.1%

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

    2. fma-undefine 32.1%

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

    3. associate-+r+ 32.1%

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

14. Applied egg-rr 32.1%

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

15. Final simplification 32.1%

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

Alternative 5: 35.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;x \cdot \left(t\_0 \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot \left(t\_0 \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (if (<= (fabs x) 0.1)
     (* x (* t_0 (+ 2.0 (* 0.6666666666666666 (pow x 2.0)))))
     (* (pow x 7.0) (* t_0 (+ 0.047619047619047616 (/ 0.2 (pow x 2.0))))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double tmp;
	if (fabs(x) <= 0.1) {
		tmp = x * (t_0 * (2.0 + (0.6666666666666666 * pow(x, 2.0))));
	} else {
		tmp = pow(x, 7.0) * (t_0 * (0.047619047619047616 + (0.2 / pow(x, 2.0))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	double tmp;
	if (Math.abs(x) <= 0.1) {
		tmp = x * (t_0 * (2.0 + (0.6666666666666666 * Math.pow(x, 2.0))));
	} else {
		tmp = Math.pow(x, 7.0) * (t_0 * (0.047619047619047616 + (0.2 / Math.pow(x, 2.0))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / math.pi))
	tmp = 0
	if math.fabs(x) <= 0.1:
		tmp = x * (t_0 * (2.0 + (0.6666666666666666 * math.pow(x, 2.0))))
	else:
		tmp = math.pow(x, 7.0) * (t_0 * (0.047619047619047616 + (0.2 / math.pow(x, 2.0))))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / pi))
	tmp = 0.0
	if (abs(x) <= 0.1)
		tmp = Float64(x * Float64(t_0 * Float64(2.0 + Float64(0.6666666666666666 * (x ^ 2.0)))));
	else
		tmp = Float64((x ^ 7.0) * Float64(t_0 * Float64(0.047619047619047616 + Float64(0.2 / (x ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 / pi));
	tmp = 0.0;
	if (abs(x) <= 0.1)
		tmp = x * (t_0 * (2.0 + (0.6666666666666666 * (x ^ 2.0))));
	else
		tmp = (x ^ 7.0) * (t_0 * (0.047619047619047616 + (0.2 / (x ^ 2.0))));
	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], 0.1], N[(x * N[(t$95$0 * N[(2.0 + N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 7.0], $MachinePrecision] * N[(t$95$0 * N[(0.047619047619047616 + N[(0.2 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.10000000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.8%

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

   5. 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. *-un-lft-identity 99.9%

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

      2. inv-pow 99.9%

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

      3. sqrt-pow1 99.9%

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

      4. metadata-eval 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. *-lft-identity 99.9%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in x around 0 99.9%

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

   12. Simplified 47.8%

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

   13. Taylor expanded in x around 0 47.8%

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

       1. +-commutative 47.8%

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

       2. associate-*r* 47.8%

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

       3. distribute-rgt-out 47.8%

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

   15. Simplified 47.8%

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

   if 0.10000000000000001 < (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%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. inv-pow 99.8%

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

      3. sqrt-pow1 99.8%

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

      4. metadata-eval 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. *-lft-identity 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in x around 0 99.8%

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

   12. Simplified 0.1%

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

   13. Taylor expanded in x around inf 0.1%

       \[\leadsto \color{blue}{{x}^{7} \cdot \left(0.047619047619047616 \cdot \sqrt{\frac{1}{\pi}} + 0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]
   14. Step-by-step derivation

       1. associate-*r* 0.1%

          \[\leadsto {x}^{7} \cdot \left(0.047619047619047616 \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \sqrt{\frac{1}{\pi}}}\right) \]

       2. distribute-rgt-out 0.1%

          \[\leadsto {x}^{7} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + 0.2 \cdot \frac{1}{{x}^{2}}\right)\right)} \]

       3. associate-*r/ 0.1%

          \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \color{blue}{\frac{0.2 \cdot 1}{{x}^{2}}}\right)\right) \]

       4. metadata-eval 0.1%

          \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \frac{\color{blue}{0.2}}{{x}^{2}}\right)\right) \]

   15. Simplified 0.1%

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

Alternative 6: 35.4% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.1)
   (* x (* (sqrt (/ 1.0 PI)) (+ 2.0 (* 0.6666666666666666 (pow x 2.0)))))
   (* (pow x 7.0) (/ 0.047619047619047616 (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.1) {
		tmp = x * (sqrt((1.0 / ((double) M_PI))) * (2.0 + (0.6666666666666666 * pow(x, 2.0))));
	} else {
		tmp = pow(x, 7.0) * (0.047619047619047616 / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.1) {
		tmp = x * (Math.sqrt((1.0 / Math.PI)) * (2.0 + (0.6666666666666666 * Math.pow(x, 2.0))));
	} else {
		tmp = Math.pow(x, 7.0) * (0.047619047619047616 / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.1:
		tmp = x * (math.sqrt((1.0 / math.pi)) * (2.0 + (0.6666666666666666 * math.pow(x, 2.0))))
	else:
		tmp = math.pow(x, 7.0) * (0.047619047619047616 / math.sqrt(math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.1)
		tmp = Float64(x * Float64(sqrt(Float64(1.0 / pi)) * Float64(2.0 + Float64(0.6666666666666666 * (x ^ 2.0)))));
	else
		tmp = Float64((x ^ 7.0) * Float64(0.047619047619047616 / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.1)
		tmp = x * (sqrt((1.0 / pi)) * (2.0 + (0.6666666666666666 * (x ^ 2.0))));
	else
		tmp = (x ^ 7.0) * (0.047619047619047616 / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.1], N[(x * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(2.0 + N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 7.0], $MachinePrecision] * N[(0.047619047619047616 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.10000000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.8%

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

   5. 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. *-un-lft-identity 99.9%

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

      2. inv-pow 99.9%

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

      3. sqrt-pow1 99.9%

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

      4. metadata-eval 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. *-lft-identity 99.9%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in x around 0 99.9%

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

   12. Simplified 47.8%

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

   13. Taylor expanded in x around 0 47.8%

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

       1. +-commutative 47.8%

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

       2. associate-*r* 47.8%

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

       3. distribute-rgt-out 47.8%

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

   15. Simplified 47.8%

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

   if 0.10000000000000001 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 99.5%

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

      1. *-commutative 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-*l* 99.5%

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

      4. *-commutative 99.5%

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

      5. *-commutative 99.5%

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

   7. Simplified 99.5%

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

      1. add-cube-cbrt 99.4%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \color{blue}{\left(\left(\sqrt[3]{\left|x\right| \cdot {x}^{6}} \cdot \sqrt[3]{\left|x\right| \cdot {x}^{6}}\right) \cdot \sqrt[3]{\left|x\right| \cdot {x}^{6}}\right)}\right)\right| \]

      2. pow3 99.4%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \color{blue}{{\left(\sqrt[3]{\left|x\right| \cdot {x}^{6}}\right)}^{3}}\right)\right| \]

      3. *-commutative 99.4%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\sqrt[3]{\color{blue}{{x}^{6} \cdot \left|x\right|}}\right)}^{3}\right)\right| \]

      4. cbrt-prod 99.3%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\color{blue}{\left(\sqrt[3]{{x}^{6}} \cdot \sqrt[3]{\left|x\right|}\right)}}^{3}\right)\right| \]

      5. metadata-eval 99.3%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\sqrt[3]{{x}^{\color{blue}{\left(3 + 3\right)}}} \cdot \sqrt[3]{\left|x\right|}\right)}^{3}\right)\right| \]

      6. pow-prod-up 99.3%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\sqrt[3]{\color{blue}{{x}^{3} \cdot {x}^{3}}} \cdot \sqrt[3]{\left|x\right|}\right)}^{3}\right)\right| \]

      7. pow-prod-down 99.3%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\sqrt[3]{\color{blue}{{\left(x \cdot x\right)}^{3}}} \cdot \sqrt[3]{\left|x\right|}\right)}^{3}\right)\right| \]

      8. pow3 99.3%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\sqrt[3]{\color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)}} \cdot \sqrt[3]{\left|x\right|}\right)}^{3}\right)\right| \]

      9. add-cbrt-cube 99.4%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\color{blue}{\left(x \cdot x\right)} \cdot \sqrt[3]{\left|x\right|}\right)}^{3}\right)\right| \]

      10. pow2 99.3%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\color{blue}{{x}^{2}} \cdot \sqrt[3]{\left|x\right|}\right)}^{3}\right)\right| \]

   9. Applied egg-rr 99.3%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \color{blue}{{\left({x}^{2} \cdot \sqrt[3]{\left|x\right|}\right)}^{3}}\right)\right| \]
   10. Step-by-step derivation

       1. cube-prod 99.4%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \color{blue}{\left({\left({x}^{2}\right)}^{3} \cdot {\left(\sqrt[3]{\left|x\right|}\right)}^{3}\right)}\right)\right| \]

       2. cube-mult 99.4%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right)} \cdot {\left(\sqrt[3]{\left|x\right|}\right)}^{3}\right)\right)\right| \]

       3. unpow2 99.4%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\left(\color{blue}{\left(x \cdot x\right)} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot {\left(\sqrt[3]{\left|x\right|}\right)}^{3}\right)\right)\right| \]

       4. sqr-abs 99.4%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\left(\color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot {\left(\sqrt[3]{\left|x\right|}\right)}^{3}\right)\right)\right| \]

       5. swap-sqr 99.3%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{\left(\left(\left|x\right| \cdot {x}^{2}\right) \cdot \left(\left|x\right| \cdot {x}^{2}\right)\right)} \cdot {\left(\sqrt[3]{\left|x\right|}\right)}^{3}\right)\right)\right| \]

       6. rem-cube-cbrt 99.3%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\left(\color{blue}{{\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{3}} \cdot \left(\left|x\right| \cdot {x}^{2}\right)\right) \cdot {\left(\sqrt[3]{\left|x\right|}\right)}^{3}\right)\right)\right| \]

       7. rem-cube-cbrt 99.2%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\left({\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{3} \cdot \color{blue}{{\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{3}}\right) \cdot {\left(\sqrt[3]{\left|x\right|}\right)}^{3}\right)\right)\right| \]

       8. pow-sqr 99.2%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{{\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{\left(2 \cdot 3\right)}} \cdot {\left(\sqrt[3]{\left|x\right|}\right)}^{3}\right)\right)\right| \]

       9. metadata-eval 99.2%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left({\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{\color{blue}{6}} \cdot {\left(\sqrt[3]{\left|x\right|}\right)}^{3}\right)\right)\right| \]

       10. rem-cube-cbrt 99.2%

           \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left({\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{6} \cdot \color{blue}{\left|x\right|}\right)\right)\right| \]

       11. rem-cbrt-cube 99.2%

           \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left({\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{6} \cdot \color{blue}{\sqrt[3]{{\left(\left|x\right|\right)}^{3}}}\right)\right)\right| \]

       12. cube-mult 99.2%

           \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left({\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{6} \cdot \sqrt[3]{\color{blue}{\left|x\right| \cdot \left(\left|x\right| \cdot \left|x\right|\right)}}\right)\right)\right| \]

       13. sqr-abs 99.2%

           \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left({\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{6} \cdot \sqrt[3]{\left|x\right| \cdot \color{blue}{\left(x \cdot x\right)}}\right)\right)\right| \]

       14. unpow2 99.2%

           \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left({\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{6} \cdot \sqrt[3]{\left|x\right| \cdot \color{blue}{{x}^{2}}}\right)\right)\right| \]

       15. pow-plus 99.2%

           \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \color{blue}{{\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{\left(6 + 1\right)}}\right)\right| \]

       16. unpow2 99.2%

           \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\sqrt[3]{\left|x\right| \cdot \color{blue}{\left(x \cdot x\right)}}\right)}^{\left(6 + 1\right)}\right)\right| \]

       17. sqr-abs 99.2%

           \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\sqrt[3]{\left|x\right| \cdot \color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)}}\right)}^{\left(6 + 1\right)}\right)\right| \]

       18. cube-mult 99.2%

           \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\sqrt[3]{\color{blue}{{\left(\left|x\right|\right)}^{3}}}\right)}^{\left(6 + 1\right)}\right)\right| \]

       19. rem-cbrt-cube 99.5%

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

       20. metadata-eval 99.5%

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

   11. Simplified 99.5%

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

       1. add-sqr-sqrt 99.5%

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

       2. fabs-sqr 99.5%

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

       3. add-sqr-sqrt 99.5%

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

       4. inv-pow 99.5%

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

       5. sqrt-pow1 99.5%

          \[\leadsto \color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right) \]

       6. metadata-eval 99.5%

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

       7. *-commutative 99.5%

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

       8. *-un-lft-identity 99.5%

          \[\leadsto \color{blue}{1 \cdot \left(\left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right) \cdot {\pi}^{-0.5}\right)} \]

       9. *-commutative 99.5%

          \[\leadsto 1 \cdot \left(\color{blue}{\left({\left(\left|x\right|\right)}^{7} \cdot 0.047619047619047616\right)} \cdot {\pi}^{-0.5}\right) \]

       10. associate-*l* 99.5%

           \[\leadsto 1 \cdot \color{blue}{\left({\left(\left|x\right|\right)}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)} \]

       11. add-sqr-sqrt 0.0%

           \[\leadsto 1 \cdot \left({\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right) \]

       12. fabs-sqr 0.0%

           \[\leadsto 1 \cdot \left({\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right) \]

       13. add-sqr-sqrt 0.1%

           \[\leadsto 1 \cdot \left({\color{blue}{x}}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right) \]

       14. *-commutative 0.1%

           \[\leadsto 1 \cdot \left({x}^{7} \cdot \color{blue}{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right)}\right) \]

   13. Applied egg-rr 0.1%

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

       1. *-lft-identity 0.1%

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

   15. Simplified 0.1%

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

Alternative 7: 35.4% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.1)
   (* (pow PI -0.5) (* x (+ 2.0 (* 0.6666666666666666 (pow x 2.0)))))
   (* (pow x 7.0) (/ 0.047619047619047616 (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.1) {
		tmp = pow(((double) M_PI), -0.5) * (x * (2.0 + (0.6666666666666666 * pow(x, 2.0))));
	} else {
		tmp = pow(x, 7.0) * (0.047619047619047616 / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.1) {
		tmp = Math.pow(Math.PI, -0.5) * (x * (2.0 + (0.6666666666666666 * Math.pow(x, 2.0))));
	} else {
		tmp = Math.pow(x, 7.0) * (0.047619047619047616 / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.1:
		tmp = math.pow(math.pi, -0.5) * (x * (2.0 + (0.6666666666666666 * math.pow(x, 2.0))))
	else:
		tmp = math.pow(x, 7.0) * (0.047619047619047616 / math.sqrt(math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.1)
		tmp = Float64((pi ^ -0.5) * Float64(x * Float64(2.0 + Float64(0.6666666666666666 * (x ^ 2.0)))));
	else
		tmp = Float64((x ^ 7.0) * Float64(0.047619047619047616 / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.1)
		tmp = (pi ^ -0.5) * (x * (2.0 + (0.6666666666666666 * (x ^ 2.0))));
	else
		tmp = (x ^ 7.0) * (0.047619047619047616 / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.1], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * N[(2.0 + N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 7.0], $MachinePrecision] * N[(0.047619047619047616 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.10000000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.8%

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

   5. 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. *-un-lft-identity 99.9%

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

      2. inv-pow 99.9%

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

      3. sqrt-pow1 99.9%

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

      4. metadata-eval 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. *-lft-identity 99.9%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in x around 0 99.9%

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

   12. Simplified 47.8%

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

   13. Taylor expanded in x around 0 47.8%

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

   if 0.10000000000000001 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 99.5%

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

      1. *-commutative 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-*l* 99.5%

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

      4. *-commutative 99.5%

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

      5. *-commutative 99.5%

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

   7. Simplified 99.5%

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

      1. add-cube-cbrt 99.4%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \color{blue}{\left(\left(\sqrt[3]{\left|x\right| \cdot {x}^{6}} \cdot \sqrt[3]{\left|x\right| \cdot {x}^{6}}\right) \cdot \sqrt[3]{\left|x\right| \cdot {x}^{6}}\right)}\right)\right| \]

      2. pow3 99.4%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \color{blue}{{\left(\sqrt[3]{\left|x\right| \cdot {x}^{6}}\right)}^{3}}\right)\right| \]

      3. *-commutative 99.4%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\sqrt[3]{\color{blue}{{x}^{6} \cdot \left|x\right|}}\right)}^{3}\right)\right| \]

      4. cbrt-prod 99.3%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\color{blue}{\left(\sqrt[3]{{x}^{6}} \cdot \sqrt[3]{\left|x\right|}\right)}}^{3}\right)\right| \]

      5. metadata-eval 99.3%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\sqrt[3]{{x}^{\color{blue}{\left(3 + 3\right)}}} \cdot \sqrt[3]{\left|x\right|}\right)}^{3}\right)\right| \]

      6. pow-prod-up 99.3%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\sqrt[3]{\color{blue}{{x}^{3} \cdot {x}^{3}}} \cdot \sqrt[3]{\left|x\right|}\right)}^{3}\right)\right| \]

      7. pow-prod-down 99.3%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\sqrt[3]{\color{blue}{{\left(x \cdot x\right)}^{3}}} \cdot \sqrt[3]{\left|x\right|}\right)}^{3}\right)\right| \]

      8. pow3 99.3%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\sqrt[3]{\color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)}} \cdot \sqrt[3]{\left|x\right|}\right)}^{3}\right)\right| \]

      9. add-cbrt-cube 99.4%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\color{blue}{\left(x \cdot x\right)} \cdot \sqrt[3]{\left|x\right|}\right)}^{3}\right)\right| \]

      10. pow2 99.3%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\color{blue}{{x}^{2}} \cdot \sqrt[3]{\left|x\right|}\right)}^{3}\right)\right| \]

   9. Applied egg-rr 99.3%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \color{blue}{{\left({x}^{2} \cdot \sqrt[3]{\left|x\right|}\right)}^{3}}\right)\right| \]
   10. Step-by-step derivation

       1. cube-prod 99.4%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \color{blue}{\left({\left({x}^{2}\right)}^{3} \cdot {\left(\sqrt[3]{\left|x\right|}\right)}^{3}\right)}\right)\right| \]

       2. cube-mult 99.4%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right)} \cdot {\left(\sqrt[3]{\left|x\right|}\right)}^{3}\right)\right)\right| \]

       3. unpow2 99.4%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\left(\color{blue}{\left(x \cdot x\right)} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot {\left(\sqrt[3]{\left|x\right|}\right)}^{3}\right)\right)\right| \]

       4. sqr-abs 99.4%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\left(\color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot {\left(\sqrt[3]{\left|x\right|}\right)}^{3}\right)\right)\right| \]

       5. swap-sqr 99.3%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{\left(\left(\left|x\right| \cdot {x}^{2}\right) \cdot \left(\left|x\right| \cdot {x}^{2}\right)\right)} \cdot {\left(\sqrt[3]{\left|x\right|}\right)}^{3}\right)\right)\right| \]

       6. rem-cube-cbrt 99.3%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\left(\color{blue}{{\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{3}} \cdot \left(\left|x\right| \cdot {x}^{2}\right)\right) \cdot {\left(\sqrt[3]{\left|x\right|}\right)}^{3}\right)\right)\right| \]

       7. rem-cube-cbrt 99.2%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\left({\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{3} \cdot \color{blue}{{\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{3}}\right) \cdot {\left(\sqrt[3]{\left|x\right|}\right)}^{3}\right)\right)\right| \]

       8. pow-sqr 99.2%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{{\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{\left(2 \cdot 3\right)}} \cdot {\left(\sqrt[3]{\left|x\right|}\right)}^{3}\right)\right)\right| \]

       9. metadata-eval 99.2%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left({\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{\color{blue}{6}} \cdot {\left(\sqrt[3]{\left|x\right|}\right)}^{3}\right)\right)\right| \]

       10. rem-cube-cbrt 99.2%

           \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left({\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{6} \cdot \color{blue}{\left|x\right|}\right)\right)\right| \]

       11. rem-cbrt-cube 99.2%

           \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left({\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{6} \cdot \color{blue}{\sqrt[3]{{\left(\left|x\right|\right)}^{3}}}\right)\right)\right| \]

       12. cube-mult 99.2%

           \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left({\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{6} \cdot \sqrt[3]{\color{blue}{\left|x\right| \cdot \left(\left|x\right| \cdot \left|x\right|\right)}}\right)\right)\right| \]

       13. sqr-abs 99.2%

           \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left({\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{6} \cdot \sqrt[3]{\left|x\right| \cdot \color{blue}{\left(x \cdot x\right)}}\right)\right)\right| \]

       14. unpow2 99.2%

           \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left({\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{6} \cdot \sqrt[3]{\left|x\right| \cdot \color{blue}{{x}^{2}}}\right)\right)\right| \]

       15. pow-plus 99.2%

           \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \color{blue}{{\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{\left(6 + 1\right)}}\right)\right| \]

       16. unpow2 99.2%

           \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\sqrt[3]{\left|x\right| \cdot \color{blue}{\left(x \cdot x\right)}}\right)}^{\left(6 + 1\right)}\right)\right| \]

       17. sqr-abs 99.2%

           \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\sqrt[3]{\left|x\right| \cdot \color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)}}\right)}^{\left(6 + 1\right)}\right)\right| \]

       18. cube-mult 99.2%

           \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\sqrt[3]{\color{blue}{{\left(\left|x\right|\right)}^{3}}}\right)}^{\left(6 + 1\right)}\right)\right| \]

       19. rem-cbrt-cube 99.5%

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

       20. metadata-eval 99.5%

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

   11. Simplified 99.5%

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

       1. add-sqr-sqrt 99.5%

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

       2. fabs-sqr 99.5%

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

       3. add-sqr-sqrt 99.5%

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

       4. inv-pow 99.5%

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

       5. sqrt-pow1 99.5%

          \[\leadsto \color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right) \]

       6. metadata-eval 99.5%

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

       7. *-commutative 99.5%

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

       8. *-un-lft-identity 99.5%

          \[\leadsto \color{blue}{1 \cdot \left(\left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right) \cdot {\pi}^{-0.5}\right)} \]

       9. *-commutative 99.5%

          \[\leadsto 1 \cdot \left(\color{blue}{\left({\left(\left|x\right|\right)}^{7} \cdot 0.047619047619047616\right)} \cdot {\pi}^{-0.5}\right) \]

       10. associate-*l* 99.5%

           \[\leadsto 1 \cdot \color{blue}{\left({\left(\left|x\right|\right)}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)} \]

       11. add-sqr-sqrt 0.0%

           \[\leadsto 1 \cdot \left({\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right) \]

       12. fabs-sqr 0.0%

           \[\leadsto 1 \cdot \left({\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right) \]

       13. add-sqr-sqrt 0.1%

           \[\leadsto 1 \cdot \left({\color{blue}{x}}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right) \]

       14. *-commutative 0.1%

           \[\leadsto 1 \cdot \left({x}^{7} \cdot \color{blue}{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right)}\right) \]

   13. Applied egg-rr 0.1%

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

       1. *-lft-identity 0.1%

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

   15. Simplified 0.1%

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

Alternative 8: 35.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.1)
   (* 2.0 (* x (pow PI -0.5)))
   (* (pow x 7.0) (/ 0.047619047619047616 (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.1) {
		tmp = 2.0 * (x * pow(((double) M_PI), -0.5));
	} else {
		tmp = pow(x, 7.0) * (0.047619047619047616 / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.1) {
		tmp = 2.0 * (x * Math.pow(Math.PI, -0.5));
	} else {
		tmp = Math.pow(x, 7.0) * (0.047619047619047616 / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.1:
		tmp = 2.0 * (x * math.pow(math.pi, -0.5))
	else:
		tmp = math.pow(x, 7.0) * (0.047619047619047616 / math.sqrt(math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.1)
		tmp = Float64(2.0 * Float64(x * (pi ^ -0.5)));
	else
		tmp = Float64((x ^ 7.0) * Float64(0.047619047619047616 / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.1)
		tmp = 2.0 * (x * (pi ^ -0.5));
	else
		tmp = (x ^ 7.0) * (0.047619047619047616 / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.1], N[(2.0 * N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 7.0], $MachinePrecision] * N[(0.047619047619047616 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.10000000000000001

   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%

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

   5. Taylor expanded in x around 0 99.0%

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

      1. associate-*r* 99.0%

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

   7. Simplified 99.0%

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

      1. associate-*r* 99.0%

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

      2. add-exp-log 90.5%

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

      3. inv-pow 90.5%

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

      4. sqrt-pow1 90.5%

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

      5. metadata-eval 90.5%

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

   9. Applied egg-rr 90.5%

      \[\leadsto \left|\color{blue}{e^{\log \left(2 \cdot \left({\pi}^{-0.5} \cdot \left|x\right|\right)\right)}}\right| \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 90.5%

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

       2. fabs-sqr 90.5%

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

       3. add-sqr-sqrt 90.5%

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

       4. rem-exp-log 99.0%

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

       5. *-commutative 99.0%

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

       6. add-sqr-sqrt 45.2%

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

       7. fabs-sqr 45.2%

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

       8. add-sqr-sqrt 47.5%

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

   11. Applied egg-rr 47.5%

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

   if 0.10000000000000001 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 99.5%

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

      1. *-commutative 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-*l* 99.5%

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

      4. *-commutative 99.5%

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

      5. *-commutative 99.5%

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

   7. Simplified 99.5%

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

      1. add-cube-cbrt 99.4%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \color{blue}{\left(\left(\sqrt[3]{\left|x\right| \cdot {x}^{6}} \cdot \sqrt[3]{\left|x\right| \cdot {x}^{6}}\right) \cdot \sqrt[3]{\left|x\right| \cdot {x}^{6}}\right)}\right)\right| \]

      2. pow3 99.4%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \color{blue}{{\left(\sqrt[3]{\left|x\right| \cdot {x}^{6}}\right)}^{3}}\right)\right| \]

      3. *-commutative 99.4%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\sqrt[3]{\color{blue}{{x}^{6} \cdot \left|x\right|}}\right)}^{3}\right)\right| \]

      4. cbrt-prod 99.3%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\color{blue}{\left(\sqrt[3]{{x}^{6}} \cdot \sqrt[3]{\left|x\right|}\right)}}^{3}\right)\right| \]

      5. metadata-eval 99.3%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\sqrt[3]{{x}^{\color{blue}{\left(3 + 3\right)}}} \cdot \sqrt[3]{\left|x\right|}\right)}^{3}\right)\right| \]

      6. pow-prod-up 99.3%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\sqrt[3]{\color{blue}{{x}^{3} \cdot {x}^{3}}} \cdot \sqrt[3]{\left|x\right|}\right)}^{3}\right)\right| \]

      7. pow-prod-down 99.3%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\sqrt[3]{\color{blue}{{\left(x \cdot x\right)}^{3}}} \cdot \sqrt[3]{\left|x\right|}\right)}^{3}\right)\right| \]

      8. pow3 99.3%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\sqrt[3]{\color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)}} \cdot \sqrt[3]{\left|x\right|}\right)}^{3}\right)\right| \]

      9. add-cbrt-cube 99.4%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\color{blue}{\left(x \cdot x\right)} \cdot \sqrt[3]{\left|x\right|}\right)}^{3}\right)\right| \]

      10. pow2 99.3%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\color{blue}{{x}^{2}} \cdot \sqrt[3]{\left|x\right|}\right)}^{3}\right)\right| \]

   9. Applied egg-rr 99.3%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \color{blue}{{\left({x}^{2} \cdot \sqrt[3]{\left|x\right|}\right)}^{3}}\right)\right| \]
   10. Step-by-step derivation

       1. cube-prod 99.4%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \color{blue}{\left({\left({x}^{2}\right)}^{3} \cdot {\left(\sqrt[3]{\left|x\right|}\right)}^{3}\right)}\right)\right| \]

       2. cube-mult 99.4%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right)} \cdot {\left(\sqrt[3]{\left|x\right|}\right)}^{3}\right)\right)\right| \]

       3. unpow2 99.4%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\left(\color{blue}{\left(x \cdot x\right)} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot {\left(\sqrt[3]{\left|x\right|}\right)}^{3}\right)\right)\right| \]

       4. sqr-abs 99.4%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\left(\color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot {\left(\sqrt[3]{\left|x\right|}\right)}^{3}\right)\right)\right| \]

       5. swap-sqr 99.3%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{\left(\left(\left|x\right| \cdot {x}^{2}\right) \cdot \left(\left|x\right| \cdot {x}^{2}\right)\right)} \cdot {\left(\sqrt[3]{\left|x\right|}\right)}^{3}\right)\right)\right| \]

       6. rem-cube-cbrt 99.3%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\left(\color{blue}{{\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{3}} \cdot \left(\left|x\right| \cdot {x}^{2}\right)\right) \cdot {\left(\sqrt[3]{\left|x\right|}\right)}^{3}\right)\right)\right| \]

       7. rem-cube-cbrt 99.2%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\left({\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{3} \cdot \color{blue}{{\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{3}}\right) \cdot {\left(\sqrt[3]{\left|x\right|}\right)}^{3}\right)\right)\right| \]

       8. pow-sqr 99.2%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{{\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{\left(2 \cdot 3\right)}} \cdot {\left(\sqrt[3]{\left|x\right|}\right)}^{3}\right)\right)\right| \]

       9. metadata-eval 99.2%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left({\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{\color{blue}{6}} \cdot {\left(\sqrt[3]{\left|x\right|}\right)}^{3}\right)\right)\right| \]

       10. rem-cube-cbrt 99.2%

           \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left({\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{6} \cdot \color{blue}{\left|x\right|}\right)\right)\right| \]

       11. rem-cbrt-cube 99.2%

           \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left({\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{6} \cdot \color{blue}{\sqrt[3]{{\left(\left|x\right|\right)}^{3}}}\right)\right)\right| \]

       12. cube-mult 99.2%

           \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left({\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{6} \cdot \sqrt[3]{\color{blue}{\left|x\right| \cdot \left(\left|x\right| \cdot \left|x\right|\right)}}\right)\right)\right| \]

       13. sqr-abs 99.2%

           \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left({\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{6} \cdot \sqrt[3]{\left|x\right| \cdot \color{blue}{\left(x \cdot x\right)}}\right)\right)\right| \]

       14. unpow2 99.2%

           \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \left({\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{6} \cdot \sqrt[3]{\left|x\right| \cdot \color{blue}{{x}^{2}}}\right)\right)\right| \]

       15. pow-plus 99.2%

           \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot \color{blue}{{\left(\sqrt[3]{\left|x\right| \cdot {x}^{2}}\right)}^{\left(6 + 1\right)}}\right)\right| \]

       16. unpow2 99.2%

           \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\sqrt[3]{\left|x\right| \cdot \color{blue}{\left(x \cdot x\right)}}\right)}^{\left(6 + 1\right)}\right)\right| \]

       17. sqr-abs 99.2%

           \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\sqrt[3]{\left|x\right| \cdot \color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)}}\right)}^{\left(6 + 1\right)}\right)\right| \]

       18. cube-mult 99.2%

           \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\sqrt[3]{\color{blue}{{\left(\left|x\right|\right)}^{3}}}\right)}^{\left(6 + 1\right)}\right)\right| \]

       19. rem-cbrt-cube 99.5%

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

       20. metadata-eval 99.5%

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

   11. Simplified 99.5%

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

       1. add-sqr-sqrt 99.5%

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

       2. fabs-sqr 99.5%

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

       3. add-sqr-sqrt 99.5%

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

       4. inv-pow 99.5%

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

       5. sqrt-pow1 99.5%

          \[\leadsto \color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right) \]

       6. metadata-eval 99.5%

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

       7. *-commutative 99.5%

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

       8. *-un-lft-identity 99.5%

          \[\leadsto \color{blue}{1 \cdot \left(\left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right) \cdot {\pi}^{-0.5}\right)} \]

       9. *-commutative 99.5%

          \[\leadsto 1 \cdot \left(\color{blue}{\left({\left(\left|x\right|\right)}^{7} \cdot 0.047619047619047616\right)} \cdot {\pi}^{-0.5}\right) \]

       10. associate-*l* 99.5%

           \[\leadsto 1 \cdot \color{blue}{\left({\left(\left|x\right|\right)}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)} \]

       11. add-sqr-sqrt 0.0%

           \[\leadsto 1 \cdot \left({\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right) \]

       12. fabs-sqr 0.0%

           \[\leadsto 1 \cdot \left({\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right) \]

       13. add-sqr-sqrt 0.1%

           \[\leadsto 1 \cdot \left({\color{blue}{x}}^{7} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right) \]

       14. *-commutative 0.1%

           \[\leadsto 1 \cdot \left({x}^{7} \cdot \color{blue}{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right)}\right) \]

   13. Applied egg-rr 0.1%

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

       1. *-lft-identity 0.1%

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

   15. Simplified 0.1%

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

4. Final simplification 31.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.7% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-10}:\\ \;\;\;\;2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{\frac{{x}^{2}}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 1e-10)
   (* 2.0 (* x (pow PI -0.5)))
   (* 2.0 (sqrt (/ (pow x 2.0) PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 1e-10) {
		tmp = 2.0 * (x * pow(((double) M_PI), -0.5));
	} else {
		tmp = 2.0 * sqrt((pow(x, 2.0) / ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 1e-10) {
		tmp = 2.0 * (x * Math.pow(Math.PI, -0.5));
	} else {
		tmp = 2.0 * Math.sqrt((Math.pow(x, 2.0) / Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 1e-10:
		tmp = 2.0 * (x * math.pow(math.pi, -0.5))
	else:
		tmp = 2.0 * math.sqrt((math.pow(x, 2.0) / math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 1e-10)
		tmp = Float64(2.0 * Float64(x * (pi ^ -0.5)));
	else
		tmp = Float64(2.0 * sqrt(Float64((x ^ 2.0) / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 1e-10)
		tmp = 2.0 * (x * (pi ^ -0.5));
	else
		tmp = 2.0 * sqrt(((x ^ 2.0) / pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 1e-10], N[(2.0 * N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(N[Power[x, 2.0], $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000004e-10

   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%

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

   5. Taylor expanded in x around 0 99.9%

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

      1. associate-*r* 99.9%

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

   7. Simplified 99.9%

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

      1. associate-*r* 99.9%

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

      2. add-exp-log 91.0%

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

      3. inv-pow 91.0%

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

      4. sqrt-pow1 91.0%

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

      5. metadata-eval 91.0%

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

   9. Applied egg-rr 91.0%

      \[\leadsto \left|\color{blue}{e^{\log \left(2 \cdot \left({\pi}^{-0.5} \cdot \left|x\right|\right)\right)}}\right| \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 91.0%

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

       2. fabs-sqr 91.0%

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

       3. add-sqr-sqrt 91.0%

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

       4. rem-exp-log 99.9%

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

       5. *-commutative 99.9%

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

       6. add-sqr-sqrt 45.1%

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

       7. fabs-sqr 45.1%

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

       8. add-sqr-sqrt 47.4%

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

   11. Applied egg-rr 47.4%

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

   if 1.00000000000000004e-10 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 11.1%

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

      1. associate-*r* 11.1%

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

   7. Simplified 11.1%

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

      1. associate-*r* 11.1%

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

      2. add-exp-log 11.0%

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

      3. inv-pow 11.0%

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

      4. sqrt-pow1 11.0%

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

      5. metadata-eval 11.0%

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

   9. Applied egg-rr 11.0%

      \[\leadsto \left|\color{blue}{e^{\log \left(2 \cdot \left({\pi}^{-0.5} \cdot \left|x\right|\right)\right)}}\right| \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 11.0%

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

       2. fabs-sqr 11.0%

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

       3. add-sqr-sqrt 11.0%

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

       4. rem-exp-log 11.1%

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

       5. *-commutative 11.1%

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

       6. add-sqr-sqrt 3.8%

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

       7. fabs-sqr 3.8%

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

       8. add-sqr-sqrt 4.2%

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

   11. Applied egg-rr 4.2%

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

       1. add-sqr-sqrt 3.8%

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

       2. sqrt-unprod 46.9%

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

       3. *-commutative 46.9%

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

       4. *-commutative 46.9%

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

       5. swap-sqr 46.9%

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

       6. unpow2 46.9%

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

       7. pow-prod-up 46.9%

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

       8. metadata-eval 46.9%

          \[\leadsto \sqrt{{x}^{2} \cdot {\pi}^{\color{blue}{-1}}} \cdot 2 \]

       9. inv-pow 46.9%

          \[\leadsto \sqrt{{x}^{2} \cdot \color{blue}{\frac{1}{\pi}}} \cdot 2 \]

   13. Applied egg-rr 46.9%

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

       1. associate-*r/ 46.9%

          \[\leadsto \sqrt{\color{blue}{\frac{{x}^{2} \cdot 1}{\pi}}} \cdot 2 \]

       2. *-rgt-identity 46.9%

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

   15. Simplified 46.9%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-10}:\\ \;\;\;\;2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{\frac{{x}^{2}}{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.4% accurate, 17.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(2.0 * N[(x * N[Power[Pi, -0.5], $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 99.8%

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

5. Taylor expanded in x around 0 68.3%

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

   1. associate-*r* 68.3%

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

7. Simplified 68.3%

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

   1. associate-*r* 68.3%

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

   2. add-exp-log 62.6%

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

   3. inv-pow 62.6%

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

   4. sqrt-pow1 62.6%

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

   5. metadata-eval 62.6%

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

9. Applied egg-rr 62.6%

   \[\leadsto \left|\color{blue}{e^{\log \left(2 \cdot \left({\pi}^{-0.5} \cdot \left|x\right|\right)\right)}}\right| \]
10. Step-by-step derivation

    1. add-sqr-sqrt 62.6%

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

    2. fabs-sqr 62.6%

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

    3. add-sqr-sqrt 62.6%

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

    4. rem-exp-log 68.3%

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

    5. *-commutative 68.3%

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

    6. add-sqr-sqrt 30.4%

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

    7. fabs-sqr 30.4%

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

    8. add-sqr-sqrt 32.0%

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

11. Applied egg-rr 32.0%

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

12. Final simplification 32.0%

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

Reproduce

herbie shell --seed 2024188 
(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)))))))