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

Percentage Accurate: 99.8% → 99.9%

Time: 11.9s

Alternatives: 10

Speedup: 3.0×

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

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 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 2: 99.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := 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[(x * N[Power[Pi, -0.5], $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 99.5%

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

   1. add-sqr-sqrt 29.0%

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

   2. fabs-sqr 29.0%

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

   3. add-sqr-sqrt 99.5%

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

   4. div-inv 99.9%

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

   5. pow1/2 99.9%

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

   6. pow-flip 99.9%

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

   7. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 3: 33.4% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

   1. add-sqr-sqrt 29.0%

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

   2. fabs-sqr 29.0%

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

   3. add-sqr-sqrt 28.8%

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

   4. fabs-sqr 28.8%

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

   5. add-sqr-sqrt 29.9%

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

   6. add-sqr-sqrt 30.4%

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

   7. clear-num 30.4%

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

   8. un-div-inv 30.2%

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

6. Applied egg-rr 30.2%

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

   1. fma-undefine 30.2%

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

8. Applied egg-rr 30.2%

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

   1. fma-undefine 30.2%

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

   2. *-commutative 30.2%

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

10. Applied egg-rr 30.2%

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

11. Final simplification 30.2%

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

Alternative 4: 33.2% accurate, 4.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/
  x
  (/
   (sqrt PI)
   (+ (fma 0.2 (pow x 4.0) (* 0.047619047619047616 (pow x 6.0))) 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))) + 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))) + 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] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

   1. add-sqr-sqrt 29.0%

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

   2. fabs-sqr 29.0%

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

   3. add-sqr-sqrt 28.8%

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

   4. fabs-sqr 28.8%

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

   5. add-sqr-sqrt 29.9%

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

   6. add-sqr-sqrt 30.4%

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

   7. clear-num 30.4%

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

   8. un-div-inv 30.2%

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

6. Applied egg-rr 30.2%

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

7. Taylor expanded in x around 0 30.2%

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

8. Final simplification 30.2%

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

Alternative 5: 33.2% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

   1. add-sqr-sqrt 29.0%

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

   2. fabs-sqr 29.0%

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

   3. add-sqr-sqrt 28.8%

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

   4. fabs-sqr 28.8%

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

   5. add-sqr-sqrt 29.9%

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

   6. add-sqr-sqrt 30.4%

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

   7. clear-num 30.4%

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

   8. un-div-inv 30.2%

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

6. Applied egg-rr 30.2%

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

7. Taylor expanded in x around 0 30.2%

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

8. Taylor expanded in x around 0 30.2%

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

9. Final simplification 30.2%

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

Alternative 6: 33.5% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.86:\\ \;\;\;\;2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.0022675736961451248 \cdot {x}^{14}}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.86)
   (* 2.0 (* x (pow PI -0.5)))
   (sqrt (/ (* 0.0022675736961451248 (pow x 14.0)) PI))))

C

double code(double x) {
	double tmp;
	if (x <= 1.86) {
		tmp = 2.0 * (x * pow(((double) M_PI), -0.5));
	} else {
		tmp = sqrt(((0.0022675736961451248 * pow(x, 14.0)) / ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.86) {
		tmp = 2.0 * (x * Math.pow(Math.PI, -0.5));
	} else {
		tmp = Math.sqrt(((0.0022675736961451248 * Math.pow(x, 14.0)) / Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.86:
		tmp = 2.0 * (x * math.pow(math.pi, -0.5))
	else:
		tmp = math.sqrt(((0.0022675736961451248 * math.pow(x, 14.0)) / math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.86)
		tmp = Float64(2.0 * Float64(x * (pi ^ -0.5)));
	else
		tmp = sqrt(Float64(Float64(0.0022675736961451248 * (x ^ 14.0)) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.86)
		tmp = 2.0 * (x * (pi ^ -0.5));
	else
		tmp = sqrt(((0.0022675736961451248 * (x ^ 14.0)) / pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.86], N[(2.0 * N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(0.0022675736961451248 * N[Power[x, 14.0], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8600000000000001

   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 29.0%

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

      2. fabs-sqr 29.0%

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

      3. add-sqr-sqrt 28.8%

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

      4. fabs-sqr 28.8%

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

      5. add-sqr-sqrt 29.9%

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

      6. add-sqr-sqrt 30.4%

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

      7. clear-num 30.4%

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

      8. un-div-inv 30.2%

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

   6. Applied egg-rr 30.2%

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

   7. Taylor expanded in x around 0 30.5%

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

      1. *-commutative 30.5%

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

   9. Simplified 30.5%

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

       1. sqrt-div 30.5%

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

       2. metadata-eval 30.5%

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

       3. div-inv 30.3%

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

   11. Applied egg-rr 30.3%

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

       1. div-inv 30.5%

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

       2. pow1/2 30.5%

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

       3. pow-flip 30.5%

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

       4. metadata-eval 30.5%

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

   13. Applied egg-rr 30.5%

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

   if 1.8600000000000001 < 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 29.0%

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

      2. fabs-sqr 29.0%

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

      3. add-sqr-sqrt 28.8%

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

      4. fabs-sqr 28.8%

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

      5. add-sqr-sqrt 29.9%

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

      6. add-sqr-sqrt 30.4%

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

      7. clear-num 30.4%

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

      8. un-div-inv 30.2%

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

   6. Applied egg-rr 30.2%

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

   7. Taylor expanded in x around inf 3.4%

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

      1. associate-*r* 3.4%

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

   9. Simplified 3.4%

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

       1. add-sqr-sqrt 3.2%

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

       2. sqrt-unprod 40.0%

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

       3. *-commutative 40.0%

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

       4. *-commutative 40.0%

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

       5. swap-sqr 40.0%

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

       6. add-sqr-sqrt 40.0%

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

       7. *-commutative 40.0%

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

       8. *-commutative 40.0%

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

       9. swap-sqr 40.0%

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

       10. pow-prod-up 40.0%

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

       11. metadata-eval 40.0%

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

       12. metadata-eval 40.0%

           \[\leadsto \sqrt{\frac{1}{\pi} \cdot \left({x}^{14} \cdot \color{blue}{0.0022675736961451248}\right)} \]

   11. Applied egg-rr 40.0%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi} \cdot \left({x}^{14} \cdot 0.0022675736961451248\right)}} \]
   12. Step-by-step derivation

       1. associate-*l/ 40.0%

          \[\leadsto \sqrt{\color{blue}{\frac{1 \cdot \left({x}^{14} \cdot 0.0022675736961451248\right)}{\pi}}} \]

       2. *-lft-identity 40.0%

          \[\leadsto \sqrt{\frac{\color{blue}{{x}^{14} \cdot 0.0022675736961451248}}{\pi}} \]

       3. *-commutative 40.0%

          \[\leadsto \sqrt{\frac{\color{blue}{0.0022675736961451248 \cdot {x}^{14}}}{\pi}} \]

   13. Simplified 40.0%

       \[\leadsto \color{blue}{\sqrt{\frac{0.0022675736961451248 \cdot {x}^{14}}{\pi}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.86:\\ \;\;\;\;2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.0022675736961451248 \cdot {x}^{14}}{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 33.5% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.86:\\ \;\;\;\;2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.86)
   (* 2.0 (* x (pow PI -0.5)))
   (* (pow x 7.0) (/ 0.047619047619047616 (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.86) {
		tmp = 2.0 * (x * pow(((double) M_PI), -0.5));
	} else {
		tmp = pow(x, 7.0) * (0.047619047619047616 / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.86) {
		tmp = 2.0 * (x * Math.pow(Math.PI, -0.5));
	} else {
		tmp = Math.pow(x, 7.0) * (0.047619047619047616 / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.86:
		tmp = 2.0 * (x * math.pow(math.pi, -0.5))
	else:
		tmp = math.pow(x, 7.0) * (0.047619047619047616 / math.sqrt(math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.86)
		tmp = Float64(2.0 * Float64(x * (pi ^ -0.5)));
	else
		tmp = Float64((x ^ 7.0) * Float64(0.047619047619047616 / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.86)
		tmp = 2.0 * (x * (pi ^ -0.5));
	else
		tmp = (x ^ 7.0) * (0.047619047619047616 / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.86], N[(2.0 * N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 7.0], $MachinePrecision] * N[(0.047619047619047616 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8600000000000001

   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 29.0%

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

      2. fabs-sqr 29.0%

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

      3. add-sqr-sqrt 28.8%

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

      4. fabs-sqr 28.8%

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

      5. add-sqr-sqrt 29.9%

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

      6. add-sqr-sqrt 30.4%

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

      7. clear-num 30.4%

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

      8. un-div-inv 30.2%

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

   6. Applied egg-rr 30.2%

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

   7. Taylor expanded in x around 0 30.5%

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

      1. *-commutative 30.5%

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

   9. Simplified 30.5%

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

       1. sqrt-div 30.5%

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

       2. metadata-eval 30.5%

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

       3. div-inv 30.3%

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

   11. Applied egg-rr 30.3%

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

       1. div-inv 30.5%

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

       2. pow1/2 30.5%

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

       3. pow-flip 30.5%

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

       4. metadata-eval 30.5%

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

   13. Applied egg-rr 30.5%

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

   if 1.8600000000000001 < 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 29.0%

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

      2. fabs-sqr 29.0%

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

      3. add-sqr-sqrt 28.8%

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

      4. fabs-sqr 28.8%

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

      5. add-sqr-sqrt 29.9%

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

      6. add-sqr-sqrt 30.4%

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

      7. clear-num 30.4%

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

      8. un-div-inv 30.2%

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

   6. Applied egg-rr 30.2%

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

   7. Taylor expanded in x around inf 3.4%

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

      1. associate-*r* 3.4%

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

   9. Simplified 3.4%

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

       1. sqrt-div 3.4%

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

       2. metadata-eval 3.4%

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

       3. un-div-inv 3.4%

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

   11. Applied egg-rr 3.4%

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

       1. associate-*l/ 3.4%

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

       2. *-commutative 3.4%

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

   13. Simplified 3.4%

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

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.86:\\ \;\;\;\;2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 33.2% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{\sqrt{\pi}}{0.047619047619047616 \cdot {x}^{6} + 2}} \end{array} \]

FPCore

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

C

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

Java

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

Python

def code(x):
	return x / (math.sqrt(math.pi) / ((0.047619047619047616 * math.pow(x, 6.0)) + 2.0))

Julia

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

MATLAB

function tmp = code(x)
	tmp = x / (sqrt(pi) / ((0.047619047619047616 * (x ^ 6.0)) + 2.0));
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

   1. add-sqr-sqrt 29.0%

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

   2. fabs-sqr 29.0%

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

   3. add-sqr-sqrt 28.8%

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

   4. fabs-sqr 28.8%

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

   5. add-sqr-sqrt 29.9%

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

   6. add-sqr-sqrt 30.4%

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

   7. clear-num 30.4%

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

   8. un-div-inv 30.2%

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

6. Applied egg-rr 30.2%

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

7. Taylor expanded in x around 0 30.2%

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

8. Taylor expanded in x around inf 30.2%

   \[\leadsto \frac{x}{\frac{\sqrt{\pi}}{2 + \color{blue}{0.047619047619047616 \cdot {x}^{6}}}} \]

9. Final simplification 30.2%

   \[\leadsto \frac{x}{\frac{\sqrt{\pi}}{0.047619047619047616 \cdot {x}^{6} + 2}} \]
10. Add Preprocessing

Alternative 9: 33.5% accurate, 17.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

   1. add-sqr-sqrt 29.0%

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

   2. fabs-sqr 29.0%

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

   3. add-sqr-sqrt 28.8%

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

   4. fabs-sqr 28.8%

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

   5. add-sqr-sqrt 29.9%

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

   6. add-sqr-sqrt 30.4%

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

   7. clear-num 30.4%

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

   8. un-div-inv 30.2%

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

6. Applied egg-rr 30.2%

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

7. Taylor expanded in x around 0 30.5%

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

   1. *-commutative 30.5%

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

9. Simplified 30.5%

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

    1. sqrt-div 30.5%

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

    2. metadata-eval 30.5%

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

    3. div-inv 30.3%

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

11. Applied egg-rr 30.3%

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

    1. div-inv 30.5%

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

    2. pow1/2 30.5%

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

    3. pow-flip 30.5%

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

    4. metadata-eval 30.5%

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

13. Applied egg-rr 30.5%

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

14. Final simplification 30.5%

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

Alternative 10: 33.3% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(2.0 * N[(x / 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 29.0%

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

   2. fabs-sqr 29.0%

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

   3. add-sqr-sqrt 28.8%

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

   4. fabs-sqr 28.8%

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

   5. add-sqr-sqrt 29.9%

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

   6. add-sqr-sqrt 30.4%

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

   7. clear-num 30.4%

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

   8. un-div-inv 30.2%

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

6. Applied egg-rr 30.2%

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

7. Taylor expanded in x around 0 30.5%

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

   1. *-commutative 30.5%

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

9. Simplified 30.5%

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

    1. sqrt-div 30.5%

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

    2. metadata-eval 30.5%

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

    3. div-inv 30.3%

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

11. Applied egg-rr 30.3%

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

12. Final simplification 30.3%

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

Reproduce

herbie shell --seed 2024060 
(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)))))))