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

Percentage Accurate: 99.8% → 99.8%

Time: 12.2s

Alternatives: 12

Speedup: 3.5×

• 28.2% 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: 99.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $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. Final simplification 99.9%

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

Alternative 3: 99.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(2.0 + N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. 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. unpow2 35.8%

      \[\leadsto \frac{x}{\sqrt{\pi} \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot -0.16666666666666666\right)} \]

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 4: 34.2% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

   1. add-sqr-sqrt 34.1%

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

   2. fabs-sqr 34.1%

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

   3. add-sqr-sqrt 35.5%

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

   4. add-sqr-sqrt 34.9%

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

   5. fabs-sqr 34.9%

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

   6. add-sqr-sqrt 35.5%

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

   7. clear-num 35.5%

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

   8. un-div-inv 35.2%

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

   9. +-commutative 35.2%

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

   10. pow2 35.2%

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

6. Applied egg-rr 35.2%

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

   1. unpow2 35.8%

      \[\leadsto \frac{x}{\sqrt{\pi} \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot -0.16666666666666666\right)} \]

8. Applied egg-rr 35.2%

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

9. Final simplification 35.2%

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

Alternative 5: 35.1% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;x \cdot \frac{\frac{1}{\sqrt{\pi}}}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{\pi} \cdot \left(21 + \frac{-88.2}{{x}^{2}}\right)}{{x}^{6}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.75)
   (* x (/ (/ 1.0 (sqrt PI)) (fma (pow x 2.0) -0.16666666666666666 0.5)))
   (/ x (/ (* (sqrt PI) (+ 21.0 (/ -88.2 (pow x 2.0)))) (pow x 6.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.75) {
		tmp = x * ((1.0 / sqrt(((double) M_PI))) / fma(pow(x, 2.0), -0.16666666666666666, 0.5));
	} else {
		tmp = x / ((sqrt(((double) M_PI)) * (21.0 + (-88.2 / pow(x, 2.0)))) / pow(x, 6.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.75)
		tmp = Float64(x * Float64(Float64(1.0 / sqrt(pi)) / fma((x ^ 2.0), -0.16666666666666666, 0.5)));
	else
		tmp = Float64(x / Float64(Float64(sqrt(pi) * Float64(21.0 + Float64(-88.2 / (x ^ 2.0)))) / (x ^ 6.0)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.75], N[(x * N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, 2.0], $MachinePrecision] * -0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[(21.0 + N[(-88.2 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.75

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. add-sqr-sqrt 34.1%

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

      2. fabs-sqr 34.1%

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

      3. add-sqr-sqrt 35.5%

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

      4. add-sqr-sqrt 34.9%

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

      5. fabs-sqr 34.9%

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

      6. add-sqr-sqrt 35.5%

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

      7. clear-num 35.5%

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

      8. un-div-inv 35.2%

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

      9. +-commutative 35.2%

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

      10. pow2 35.2%

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

   6. Applied egg-rr 35.2%

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

   7. Taylor expanded in x around 0 35.8%

      \[\leadsto \frac{x}{\color{blue}{-0.16666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\pi}\right) + 0.5 \cdot \sqrt{\pi}}} \]
   8. Step-by-step derivation

      1. +-commutative 35.8%

         \[\leadsto \frac{x}{\color{blue}{0.5 \cdot \sqrt{\pi} + -0.16666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\pi}\right)}} \]

      2. associate-*r* 35.8%

         \[\leadsto \frac{x}{0.5 \cdot \sqrt{\pi} + \color{blue}{\left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot \sqrt{\pi}}} \]

      3. distribute-rgt-out 35.8%

         \[\leadsto \frac{x}{\color{blue}{\sqrt{\pi} \cdot \left(0.5 + -0.16666666666666666 \cdot {x}^{2}\right)}} \]

      4. *-commutative 35.8%

         \[\leadsto \frac{x}{\sqrt{\pi} \cdot \left(0.5 + \color{blue}{{x}^{2} \cdot -0.16666666666666666}\right)} \]

   9. Simplified 35.8%

      \[\leadsto \frac{x}{\color{blue}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)}} \]
   10. Step-by-step derivation

       1. div-inv 36.0%

          \[\leadsto \color{blue}{x \cdot \frac{1}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)}} \]

       2. +-commutative 36.0%

          \[\leadsto x \cdot \frac{1}{\sqrt{\pi} \cdot \color{blue}{\left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)}} \]

       3. fma-define 36.0%

          \[\leadsto x \cdot \frac{1}{\sqrt{\pi} \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]

   11. Applied egg-rr 36.0%

       \[\leadsto \color{blue}{x \cdot \frac{1}{\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]
   12. Step-by-step derivation

       1. associate-/r* 36.0%

          \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{\sqrt{\pi}}}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]

   13. Simplified 36.0%

       \[\leadsto \color{blue}{x \cdot \frac{\frac{1}{\sqrt{\pi}}}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]

   if 1.75 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. add-sqr-sqrt 34.1%

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

      2. fabs-sqr 34.1%

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

      3. add-sqr-sqrt 35.5%

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

      4. add-sqr-sqrt 34.9%

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

      5. fabs-sqr 34.9%

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

      6. add-sqr-sqrt 35.5%

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

      7. clear-num 35.5%

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

      8. un-div-inv 35.2%

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

      9. +-commutative 35.2%

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

      10. pow2 35.2%

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

   6. Applied egg-rr 35.2%

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

   7. Taylor expanded in x around inf 3.5%

      \[\leadsto \frac{x}{\color{blue}{\frac{-88.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\pi}\right) + 21 \cdot \sqrt{\pi}}{{x}^{6}}}} \]
   8. Step-by-step derivation

      1. associate-*r* 3.5%

         \[\leadsto \frac{x}{\frac{\color{blue}{\left(-88.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \sqrt{\pi}} + 21 \cdot \sqrt{\pi}}{{x}^{6}}} \]

      2. distribute-rgt-out 3.5%

         \[\leadsto \frac{x}{\frac{\color{blue}{\sqrt{\pi} \cdot \left(-88.2 \cdot \frac{1}{{x}^{2}} + 21\right)}}{{x}^{6}}} \]

      3. associate-*r/ 3.5%

         \[\leadsto \frac{x}{\frac{\sqrt{\pi} \cdot \left(\color{blue}{\frac{-88.2 \cdot 1}{{x}^{2}}} + 21\right)}{{x}^{6}}} \]

      4. metadata-eval 3.5%

         \[\leadsto \frac{x}{\frac{\sqrt{\pi} \cdot \left(\frac{\color{blue}{-88.2}}{{x}^{2}} + 21\right)}{{x}^{6}}} \]

   9. Simplified 3.5%

      \[\leadsto \frac{x}{\color{blue}{\frac{\sqrt{\pi} \cdot \left(\frac{-88.2}{{x}^{2}} + 21\right)}{{x}^{6}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;x \cdot \frac{\frac{1}{\sqrt{\pi}}}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{\pi} \cdot \left(21 + \frac{-88.2}{{x}^{2}}\right)}{{x}^{6}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 35.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.65:\\ \;\;\;\;x \cdot \frac{\frac{1}{\sqrt{\pi}}}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{7}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.65)
   (* x (/ (/ 1.0 (sqrt PI)) (fma (pow x 2.0) -0.16666666666666666 0.5)))
   (fabs (* 0.047619047619047616 (* (sqrt (/ 1.0 PI)) (pow x 7.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.65) {
		tmp = x * ((1.0 / sqrt(((double) M_PI))) / fma(pow(x, 2.0), -0.16666666666666666, 0.5));
	} else {
		tmp = fabs((0.047619047619047616 * (sqrt((1.0 / ((double) M_PI))) * pow(x, 7.0))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.65)
		tmp = Float64(x * Float64(Float64(1.0 / sqrt(pi)) / fma((x ^ 2.0), -0.16666666666666666, 0.5)));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64(sqrt(Float64(1.0 / pi)) * (x ^ 7.0))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.65], N[(x * N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, 2.0], $MachinePrecision] * -0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6499999999999999

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. add-sqr-sqrt 34.1%

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

      2. fabs-sqr 34.1%

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

      3. add-sqr-sqrt 35.5%

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

      4. add-sqr-sqrt 34.9%

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

      5. fabs-sqr 34.9%

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

      6. add-sqr-sqrt 35.5%

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

      7. clear-num 35.5%

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

      8. un-div-inv 35.2%

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

      9. +-commutative 35.2%

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

      10. pow2 35.2%

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

   6. Applied egg-rr 35.2%

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

   7. Taylor expanded in x around 0 35.8%

      \[\leadsto \frac{x}{\color{blue}{-0.16666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\pi}\right) + 0.5 \cdot \sqrt{\pi}}} \]
   8. Step-by-step derivation

      1. +-commutative 35.8%

         \[\leadsto \frac{x}{\color{blue}{0.5 \cdot \sqrt{\pi} + -0.16666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\pi}\right)}} \]

      2. associate-*r* 35.8%

         \[\leadsto \frac{x}{0.5 \cdot \sqrt{\pi} + \color{blue}{\left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot \sqrt{\pi}}} \]

      3. distribute-rgt-out 35.8%

         \[\leadsto \frac{x}{\color{blue}{\sqrt{\pi} \cdot \left(0.5 + -0.16666666666666666 \cdot {x}^{2}\right)}} \]

      4. *-commutative 35.8%

         \[\leadsto \frac{x}{\sqrt{\pi} \cdot \left(0.5 + \color{blue}{{x}^{2} \cdot -0.16666666666666666}\right)} \]

   9. Simplified 35.8%

      \[\leadsto \frac{x}{\color{blue}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)}} \]
   10. Step-by-step derivation

       1. div-inv 36.0%

          \[\leadsto \color{blue}{x \cdot \frac{1}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)}} \]

       2. +-commutative 36.0%

          \[\leadsto x \cdot \frac{1}{\sqrt{\pi} \cdot \color{blue}{\left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)}} \]

       3. fma-define 36.0%

          \[\leadsto x \cdot \frac{1}{\sqrt{\pi} \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]

   11. Applied egg-rr 36.0%

       \[\leadsto \color{blue}{x \cdot \frac{1}{\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]
   12. Step-by-step derivation

       1. associate-/r* 36.0%

          \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{\sqrt{\pi}}}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]

   13. Simplified 36.0%

       \[\leadsto \color{blue}{x \cdot \frac{\frac{1}{\sqrt{\pi}}}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]

   if 1.6499999999999999 < 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 99.4%

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

   5. Taylor expanded in x around 0 98.9%

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

      1. associate-+r+ 98.9%

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

      2. distribute-lft-in 98.9%

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

      3. fma-define 98.9%

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

      4. rem-square-sqrt 33.5%

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

      5. fabs-sqr 33.5%

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

      6. rem-square-sqrt 70.6%

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

      7. +-commutative 70.6%

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

      8. fma-define 70.6%

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

      9. rem-square-sqrt 33.7%

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

      10. fabs-sqr 33.7%

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

      11. rem-square-sqrt 70.6%

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

      12. rem-square-sqrt 33.7%

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

      13. fabs-sqr 33.7%

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

      14. rem-square-sqrt 98.9%

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

      15. *-commutative 98.9%

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

      16. rem-square-sqrt 33.7%

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

      17. fabs-sqr 33.7%

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

      18. rem-square-sqrt 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in x around inf 40.0%

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

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65:\\ \;\;\;\;x \cdot \frac{\frac{1}{\sqrt{\pi}}}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{7}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(t\_0 \cdot {x}^{7}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (if (<= x 2.2)
     (* x (* (fma 0.6666666666666666 (* x x) 2.0) t_0))
     (fabs (* 0.047619047619047616 (* t_0 (pow x 7.0)))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double tmp;
	if (x <= 2.2) {
		tmp = x * (fma(0.6666666666666666, (x * x), 2.0) * t_0);
	} else {
		tmp = fabs((0.047619047619047616 * (t_0 * pow(x, 7.0))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / pi))
	tmp = 0.0
	if (x <= 2.2)
		tmp = Float64(x * Float64(fma(0.6666666666666666, Float64(x * x), 2.0) * t_0));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64(t_0 * (x ^ 7.0))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 2.2], N[(x * N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(t$95$0 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. add-sqr-sqrt 34.1%

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

      2. fabs-sqr 34.1%

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

      3. add-sqr-sqrt 35.5%

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

      4. add-sqr-sqrt 34.9%

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

      5. fabs-sqr 34.9%

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

      6. add-sqr-sqrt 35.5%

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

      7. clear-num 35.5%

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

      8. un-div-inv 35.2%

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

      9. +-commutative 35.2%

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

      10. pow2 35.2%

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

   6. Applied egg-rr 35.2%

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

   7. Taylor expanded in x around 0 35.2%

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

      1. associate-*r* 35.2%

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

      2. distribute-rgt-out 35.2%

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

      3. fma-undefine 35.2%

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

   9. Simplified 35.2%

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

       1. unpow2 35.8%

          \[\leadsto \frac{x}{\sqrt{\pi} \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot -0.16666666666666666\right)} \]

   11. Applied egg-rr 35.2%

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

   if 2.2000000000000002 < 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 99.4%

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

   5. Taylor expanded in x around 0 98.9%

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

      1. associate-+r+ 98.9%

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

      2. distribute-lft-in 98.9%

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

      3. fma-define 98.9%

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

      4. rem-square-sqrt 33.5%

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

      5. fabs-sqr 33.5%

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

      6. rem-square-sqrt 70.6%

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

      7. +-commutative 70.6%

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

      8. fma-define 70.6%

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

      9. rem-square-sqrt 33.7%

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

      10. fabs-sqr 33.7%

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

      11. rem-square-sqrt 70.6%

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

      12. rem-square-sqrt 33.7%

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

      13. fabs-sqr 33.7%

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

      14. rem-square-sqrt 98.9%

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

      15. *-commutative 98.9%

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

      16. rem-square-sqrt 33.7%

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

      17. fabs-sqr 33.7%

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

      18. rem-square-sqrt 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in x around inf 40.0%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) \cdot \sqrt{\frac{1}{\pi}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{7}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.3% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) \cdot \sqrt{\frac{1}{\pi}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot \frac{x}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (* x (* (fma 0.6666666666666666 (* x x) 2.0) (sqrt (/ 1.0 PI))))
   (* (* 0.047619047619047616 (pow x 6.0)) (/ x (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = x * (fma(0.6666666666666666, (x * x), 2.0) * sqrt((1.0 / ((double) M_PI))));
	} else {
		tmp = (0.047619047619047616 * pow(x, 6.0)) * (x / sqrt(((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = Float64(x * Float64(fma(0.6666666666666666, Float64(x * x), 2.0) * sqrt(Float64(1.0 / pi))));
	else
		tmp = Float64(Float64(0.047619047619047616 * (x ^ 6.0)) * Float64(x / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[(x * N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] * N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. add-sqr-sqrt 34.1%

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

      2. fabs-sqr 34.1%

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

      3. add-sqr-sqrt 35.5%

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

      4. add-sqr-sqrt 34.9%

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

      5. fabs-sqr 34.9%

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

      6. add-sqr-sqrt 35.5%

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

      7. clear-num 35.5%

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

      8. un-div-inv 35.2%

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

      9. +-commutative 35.2%

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

      10. pow2 35.2%

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

   6. Applied egg-rr 35.2%

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

   7. Taylor expanded in x around 0 35.2%

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

      1. associate-*r* 35.2%

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

      2. distribute-rgt-out 35.2%

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

      3. fma-undefine 35.2%

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

   9. Simplified 35.2%

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

       1. unpow2 35.8%

          \[\leadsto \frac{x}{\sqrt{\pi} \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot -0.16666666666666666\right)} \]

   11. Applied egg-rr 35.2%

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

   if 2.2000000000000002 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. add-sqr-sqrt 34.1%

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

      2. fabs-sqr 34.1%

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

      3. add-sqr-sqrt 35.5%

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

      4. add-sqr-sqrt 34.9%

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

      5. fabs-sqr 34.9%

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

      6. add-sqr-sqrt 35.5%

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

      7. clear-num 35.5%

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

      8. un-div-inv 35.2%

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

      9. +-commutative 35.2%

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

      10. pow2 35.2%

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

   6. Applied egg-rr 35.2%

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

   7. Taylor expanded in x around inf 3.5%

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

      1. associate-*r* 3.5%

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

      2. *-commutative 3.5%

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

      3. associate-*r/ 3.5%

         \[\leadsto \frac{x}{\sqrt{\pi} \cdot \color{blue}{\frac{21 \cdot 1}{{x}^{6}}}} \]

      4. metadata-eval 3.5%

         \[\leadsto \frac{x}{\sqrt{\pi} \cdot \frac{\color{blue}{21}}{{x}^{6}}} \]

   9. Simplified 3.5%

      \[\leadsto \frac{x}{\color{blue}{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}}} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 3.5%

          \[\leadsto \frac{\color{blue}{1 \cdot x}}{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}} \]

       2. *-commutative 3.5%

          \[\leadsto \frac{1 \cdot x}{\color{blue}{\frac{21}{{x}^{6}} \cdot \sqrt{\pi}}} \]

       3. times-frac 3.5%

          \[\leadsto \color{blue}{\frac{1}{\frac{21}{{x}^{6}}} \cdot \frac{x}{\sqrt{\pi}}} \]

       4. clear-num 3.5%

          \[\leadsto \color{blue}{\frac{{x}^{6}}{21}} \cdot \frac{x}{\sqrt{\pi}} \]

       5. div-inv 3.5%

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

       6. metadata-eval 3.5%

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

       7. *-commutative 3.5%

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

   11. Applied egg-rr 3.5%

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

       1. *-commutative 3.5%

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

   13. Simplified 3.5%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) \cdot \sqrt{\frac{1}{\pi}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot \frac{x}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.65:\\ \;\;\;\;\frac{x}{\sqrt{\pi} \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.65)
   (/ x (* (sqrt PI) (+ 0.5 (* (* x x) -0.16666666666666666))))
   (* (sqrt (/ 1.0 PI)) (* 0.047619047619047616 (pow x 7.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.65) {
		tmp = x / (sqrt(((double) M_PI)) * (0.5 + ((x * x) * -0.16666666666666666)));
	} else {
		tmp = sqrt((1.0 / ((double) M_PI))) * (0.047619047619047616 * pow(x, 7.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.65) {
		tmp = x / (Math.sqrt(Math.PI) * (0.5 + ((x * x) * -0.16666666666666666)));
	} else {
		tmp = Math.sqrt((1.0 / Math.PI)) * (0.047619047619047616 * Math.pow(x, 7.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.65:
		tmp = x / (math.sqrt(math.pi) * (0.5 + ((x * x) * -0.16666666666666666)))
	else:
		tmp = math.sqrt((1.0 / math.pi)) * (0.047619047619047616 * math.pow(x, 7.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.65)
		tmp = Float64(x / Float64(sqrt(pi) * Float64(0.5 + Float64(Float64(x * x) * -0.16666666666666666))));
	else
		tmp = Float64(sqrt(Float64(1.0 / pi)) * Float64(0.047619047619047616 * (x ^ 7.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.65)
		tmp = x / (sqrt(pi) * (0.5 + ((x * x) * -0.16666666666666666)));
	else
		tmp = sqrt((1.0 / pi)) * (0.047619047619047616 * (x ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.65], N[(x / N[(N[Sqrt[Pi], $MachinePrecision] * N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6499999999999999

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. add-sqr-sqrt 34.1%

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

      2. fabs-sqr 34.1%

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

      3. add-sqr-sqrt 35.5%

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

      4. add-sqr-sqrt 34.9%

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

      5. fabs-sqr 34.9%

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

      6. add-sqr-sqrt 35.5%

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

      7. clear-num 35.5%

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

      8. un-div-inv 35.2%

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

      9. +-commutative 35.2%

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

      10. pow2 35.2%

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

   6. Applied egg-rr 35.2%

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

   7. Taylor expanded in x around 0 35.8%

      \[\leadsto \frac{x}{\color{blue}{-0.16666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\pi}\right) + 0.5 \cdot \sqrt{\pi}}} \]
   8. Step-by-step derivation

      1. +-commutative 35.8%

         \[\leadsto \frac{x}{\color{blue}{0.5 \cdot \sqrt{\pi} + -0.16666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\pi}\right)}} \]

      2. associate-*r* 35.8%

         \[\leadsto \frac{x}{0.5 \cdot \sqrt{\pi} + \color{blue}{\left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot \sqrt{\pi}}} \]

      3. distribute-rgt-out 35.8%

         \[\leadsto \frac{x}{\color{blue}{\sqrt{\pi} \cdot \left(0.5 + -0.16666666666666666 \cdot {x}^{2}\right)}} \]

      4. *-commutative 35.8%

         \[\leadsto \frac{x}{\sqrt{\pi} \cdot \left(0.5 + \color{blue}{{x}^{2} \cdot -0.16666666666666666}\right)} \]

   9. Simplified 35.8%

      \[\leadsto \frac{x}{\color{blue}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)}} \]
   10. Step-by-step derivation

       1. unpow2 35.8%

          \[\leadsto \frac{x}{\sqrt{\pi} \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot -0.16666666666666666\right)} \]

   11. Applied egg-rr 35.8%

       \[\leadsto \frac{x}{\sqrt{\pi} \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot -0.16666666666666666\right)} \]

   if 1.6499999999999999 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. add-sqr-sqrt 34.1%

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

      2. fabs-sqr 34.1%

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

      3. add-sqr-sqrt 35.5%

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

      4. add-sqr-sqrt 34.9%

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

      5. fabs-sqr 34.9%

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

      6. add-sqr-sqrt 35.5%

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

      7. clear-num 35.5%

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

      8. un-div-inv 35.2%

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

      9. +-commutative 35.2%

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

      10. pow2 35.2%

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

   6. Applied egg-rr 35.2%

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

      1. unpow2 35.8%

         \[\leadsto \frac{x}{\sqrt{\pi} \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot -0.16666666666666666\right)} \]

   8. Applied egg-rr 35.2%

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

   9. Taylor expanded in x around inf 3.5%

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

       1. associate-*r* 3.5%

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

       2. *-commutative 3.5%

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

   11. Simplified 3.5%

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

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65:\\ \;\;\;\;\frac{x}{\sqrt{\pi} \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 34.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.65:\\ \;\;\;\;\frac{x}{\sqrt{\pi} \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot \frac{x}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.65)
   (/ x (* (sqrt PI) (+ 0.5 (* (* x x) -0.16666666666666666))))
   (* (* 0.047619047619047616 (pow x 6.0)) (/ x (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.65) {
		tmp = x / (sqrt(((double) M_PI)) * (0.5 + ((x * x) * -0.16666666666666666)));
	} else {
		tmp = (0.047619047619047616 * pow(x, 6.0)) * (x / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.65) {
		tmp = x / (Math.sqrt(Math.PI) * (0.5 + ((x * x) * -0.16666666666666666)));
	} else {
		tmp = (0.047619047619047616 * Math.pow(x, 6.0)) * (x / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.65:
		tmp = x / (math.sqrt(math.pi) * (0.5 + ((x * x) * -0.16666666666666666)))
	else:
		tmp = (0.047619047619047616 * math.pow(x, 6.0)) * (x / math.sqrt(math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.65)
		tmp = Float64(x / Float64(sqrt(pi) * Float64(0.5 + Float64(Float64(x * x) * -0.16666666666666666))));
	else
		tmp = Float64(Float64(0.047619047619047616 * (x ^ 6.0)) * Float64(x / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.65)
		tmp = x / (sqrt(pi) * (0.5 + ((x * x) * -0.16666666666666666)));
	else
		tmp = (0.047619047619047616 * (x ^ 6.0)) * (x / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.65], N[(x / N[(N[Sqrt[Pi], $MachinePrecision] * N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] * N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6499999999999999

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. add-sqr-sqrt 34.1%

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

      2. fabs-sqr 34.1%

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

      3. add-sqr-sqrt 35.5%

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

      4. add-sqr-sqrt 34.9%

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

      5. fabs-sqr 34.9%

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

      6. add-sqr-sqrt 35.5%

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

      7. clear-num 35.5%

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

      8. un-div-inv 35.2%

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

      9. +-commutative 35.2%

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

      10. pow2 35.2%

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

   6. Applied egg-rr 35.2%

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

   7. Taylor expanded in x around 0 35.8%

      \[\leadsto \frac{x}{\color{blue}{-0.16666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\pi}\right) + 0.5 \cdot \sqrt{\pi}}} \]
   8. Step-by-step derivation

      1. +-commutative 35.8%

         \[\leadsto \frac{x}{\color{blue}{0.5 \cdot \sqrt{\pi} + -0.16666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\pi}\right)}} \]

      2. associate-*r* 35.8%

         \[\leadsto \frac{x}{0.5 \cdot \sqrt{\pi} + \color{blue}{\left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot \sqrt{\pi}}} \]

      3. distribute-rgt-out 35.8%

         \[\leadsto \frac{x}{\color{blue}{\sqrt{\pi} \cdot \left(0.5 + -0.16666666666666666 \cdot {x}^{2}\right)}} \]

      4. *-commutative 35.8%

         \[\leadsto \frac{x}{\sqrt{\pi} \cdot \left(0.5 + \color{blue}{{x}^{2} \cdot -0.16666666666666666}\right)} \]

   9. Simplified 35.8%

      \[\leadsto \frac{x}{\color{blue}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)}} \]
   10. Step-by-step derivation

       1. unpow2 35.8%

          \[\leadsto \frac{x}{\sqrt{\pi} \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot -0.16666666666666666\right)} \]

   11. Applied egg-rr 35.8%

       \[\leadsto \frac{x}{\sqrt{\pi} \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot -0.16666666666666666\right)} \]

   if 1.6499999999999999 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. add-sqr-sqrt 34.1%

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

      2. fabs-sqr 34.1%

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

      3. add-sqr-sqrt 35.5%

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

      4. add-sqr-sqrt 34.9%

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

      5. fabs-sqr 34.9%

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

      6. add-sqr-sqrt 35.5%

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

      7. clear-num 35.5%

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

      8. un-div-inv 35.2%

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

      9. +-commutative 35.2%

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

      10. pow2 35.2%

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

   6. Applied egg-rr 35.2%

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

   7. Taylor expanded in x around inf 3.5%

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

      1. associate-*r* 3.5%

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

      2. *-commutative 3.5%

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

      3. associate-*r/ 3.5%

         \[\leadsto \frac{x}{\sqrt{\pi} \cdot \color{blue}{\frac{21 \cdot 1}{{x}^{6}}}} \]

      4. metadata-eval 3.5%

         \[\leadsto \frac{x}{\sqrt{\pi} \cdot \frac{\color{blue}{21}}{{x}^{6}}} \]

   9. Simplified 3.5%

      \[\leadsto \frac{x}{\color{blue}{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}}} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 3.5%

          \[\leadsto \frac{\color{blue}{1 \cdot x}}{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}} \]

       2. *-commutative 3.5%

          \[\leadsto \frac{1 \cdot x}{\color{blue}{\frac{21}{{x}^{6}} \cdot \sqrt{\pi}}} \]

       3. times-frac 3.5%

          \[\leadsto \color{blue}{\frac{1}{\frac{21}{{x}^{6}}} \cdot \frac{x}{\sqrt{\pi}}} \]

       4. clear-num 3.5%

          \[\leadsto \color{blue}{\frac{{x}^{6}}{21}} \cdot \frac{x}{\sqrt{\pi}} \]

       5. div-inv 3.5%

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

       6. metadata-eval 3.5%

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

       7. *-commutative 3.5%

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

   11. Applied egg-rr 3.5%

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

       1. *-commutative 3.5%

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

   13. Simplified 3.5%

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

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65:\\ \;\;\;\;\frac{x}{\sqrt{\pi} \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.047619047619047616 \cdot {x}^{6}\right) \cdot \frac{x}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 34.8% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.65:\\ \;\;\;\;\frac{x}{\sqrt{\pi} \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.65)
   (/ x (* (sqrt PI) (+ 0.5 (* (* x x) -0.16666666666666666))))
   (* 0.047619047619047616 (/ (pow x 7.0) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.65) {
		tmp = x / (sqrt(((double) M_PI)) * (0.5 + ((x * x) * -0.16666666666666666)));
	} else {
		tmp = 0.047619047619047616 * (pow(x, 7.0) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.65) {
		tmp = x / (Math.sqrt(Math.PI) * (0.5 + ((x * x) * -0.16666666666666666)));
	} else {
		tmp = 0.047619047619047616 * (Math.pow(x, 7.0) / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.65:
		tmp = x / (math.sqrt(math.pi) * (0.5 + ((x * x) * -0.16666666666666666)))
	else:
		tmp = 0.047619047619047616 * (math.pow(x, 7.0) / math.sqrt(math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.65)
		tmp = Float64(x / Float64(sqrt(pi) * Float64(0.5 + Float64(Float64(x * x) * -0.16666666666666666))));
	else
		tmp = Float64(0.047619047619047616 * Float64((x ^ 7.0) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.65)
		tmp = x / (sqrt(pi) * (0.5 + ((x * x) * -0.16666666666666666)));
	else
		tmp = 0.047619047619047616 * ((x ^ 7.0) / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.65], N[(x / N[(N[Sqrt[Pi], $MachinePrecision] * N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[x, 7.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6499999999999999

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. add-sqr-sqrt 34.1%

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

      2. fabs-sqr 34.1%

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

      3. add-sqr-sqrt 35.5%

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

      4. add-sqr-sqrt 34.9%

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

      5. fabs-sqr 34.9%

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

      6. add-sqr-sqrt 35.5%

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

      7. clear-num 35.5%

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

      8. un-div-inv 35.2%

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

      9. +-commutative 35.2%

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

      10. pow2 35.2%

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

   6. Applied egg-rr 35.2%

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

   7. Taylor expanded in x around 0 35.8%

      \[\leadsto \frac{x}{\color{blue}{-0.16666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\pi}\right) + 0.5 \cdot \sqrt{\pi}}} \]
   8. Step-by-step derivation

      1. +-commutative 35.8%

         \[\leadsto \frac{x}{\color{blue}{0.5 \cdot \sqrt{\pi} + -0.16666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\pi}\right)}} \]

      2. associate-*r* 35.8%

         \[\leadsto \frac{x}{0.5 \cdot \sqrt{\pi} + \color{blue}{\left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot \sqrt{\pi}}} \]

      3. distribute-rgt-out 35.8%

         \[\leadsto \frac{x}{\color{blue}{\sqrt{\pi} \cdot \left(0.5 + -0.16666666666666666 \cdot {x}^{2}\right)}} \]

      4. *-commutative 35.8%

         \[\leadsto \frac{x}{\sqrt{\pi} \cdot \left(0.5 + \color{blue}{{x}^{2} \cdot -0.16666666666666666}\right)} \]

   9. Simplified 35.8%

      \[\leadsto \frac{x}{\color{blue}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)}} \]
   10. Step-by-step derivation

       1. unpow2 35.8%

          \[\leadsto \frac{x}{\sqrt{\pi} \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot -0.16666666666666666\right)} \]

   11. Applied egg-rr 35.8%

       \[\leadsto \frac{x}{\sqrt{\pi} \cdot \left(0.5 + \color{blue}{\left(x \cdot x\right)} \cdot -0.16666666666666666\right)} \]

   if 1.6499999999999999 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. add-sqr-sqrt 34.1%

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

      2. fabs-sqr 34.1%

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

      3. add-sqr-sqrt 35.5%

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

      4. add-sqr-sqrt 34.9%

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

      5. fabs-sqr 34.9%

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

      6. add-sqr-sqrt 35.5%

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

      7. clear-num 35.5%

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

      8. un-div-inv 35.2%

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

      9. +-commutative 35.2%

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

      10. pow2 35.2%

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

   6. Applied egg-rr 35.2%

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

   7. Taylor expanded in x around inf 3.5%

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

      1. associate-*r* 3.5%

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

      2. *-commutative 3.5%

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

      3. associate-*r/ 3.5%

         \[\leadsto \frac{x}{\sqrt{\pi} \cdot \color{blue}{\frac{21 \cdot 1}{{x}^{6}}}} \]

      4. metadata-eval 3.5%

         \[\leadsto \frac{x}{\sqrt{\pi} \cdot \frac{\color{blue}{21}}{{x}^{6}}} \]

   9. Simplified 3.5%

      \[\leadsto \frac{x}{\color{blue}{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}}} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 3.5%

          \[\leadsto \frac{\color{blue}{1 \cdot x}}{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}} \]

       2. *-commutative 3.5%

          \[\leadsto \frac{1 \cdot x}{\color{blue}{\frac{21}{{x}^{6}} \cdot \sqrt{\pi}}} \]

       3. times-frac 3.5%

          \[\leadsto \color{blue}{\frac{1}{\frac{21}{{x}^{6}}} \cdot \frac{x}{\sqrt{\pi}}} \]

       4. clear-num 3.5%

          \[\leadsto \color{blue}{\frac{{x}^{6}}{21}} \cdot \frac{x}{\sqrt{\pi}} \]

       5. div-inv 3.5%

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

       6. metadata-eval 3.5%

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

       7. *-commutative 3.5%

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

   11. Applied egg-rr 3.5%

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

       1. associate-*l* 3.5%

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

       2. associate-*r/ 3.5%

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

       3. pow-plus 3.5%

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

       4. metadata-eval 3.5%

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

   13. Simplified 3.5%

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

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65:\\ \;\;\;\;\frac{x}{\sqrt{\pi} \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 34.2% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

   1. add-sqr-sqrt 34.1%

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

   2. fabs-sqr 34.1%

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

   3. add-sqr-sqrt 35.5%

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

   4. add-sqr-sqrt 34.9%

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

   5. fabs-sqr 34.9%

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

   6. add-sqr-sqrt 35.5%

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

   7. clear-num 35.5%

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

   8. un-div-inv 35.2%

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

   9. +-commutative 35.2%

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

   10. pow2 35.2%

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

6. Applied egg-rr 35.2%

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

7. Taylor expanded in x around 0 34.7%

   \[\leadsto \frac{x}{\color{blue}{0.5 \cdot \sqrt{\pi}}} \]
8. Step-by-step derivation

   1. div-inv 34.9%

      \[\leadsto \color{blue}{x \cdot \frac{1}{0.5 \cdot \sqrt{\pi}}} \]

9. Applied egg-rr 34.9%

   \[\leadsto \color{blue}{x \cdot \frac{1}{0.5 \cdot \sqrt{\pi}}} \]
10. Step-by-step derivation

    1. associate-/r* 34.9%

       \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{0.5}}{\sqrt{\pi}}} \]

    2. metadata-eval 34.9%

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

11. Simplified 34.9%

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

12. Final simplification 34.9%

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

Reproduce

herbie shell --seed 2024080 
(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)))))))