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

Percentage Accurate: 99.8% → 99.9%

Time: 14.7s

Alternatives: 7

Speedup: 3.0×

• 29.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 3.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fabs x)
  (fabs
   (/
    (+
     (+ (* 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(((((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(Float64(Float64(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[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $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 99.8%

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

3. Simplified 99.9%

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

   1. fma-undefine 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 99.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.02:\\ \;\;\;\;\left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(0.2 \cdot {x}^{4} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.2}{x} + \left|x\right| \cdot 0.047619047619047616\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.02)
   (fabs
    (*
     (* x (pow PI -0.5))
     (+ (* 0.2 (pow x 4.0)) (fma 0.6666666666666666 (* x x) 2.0))))
   (fabs
    (*
     (pow x 6.0)
     (* (sqrt (/ 1.0 PI)) (+ (/ 0.2 x) (* (fabs x) 0.047619047619047616)))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.02) {
		tmp = fabs(((x * pow(((double) M_PI), -0.5)) * ((0.2 * pow(x, 4.0)) + fma(0.6666666666666666, (x * x), 2.0))));
	} else {
		tmp = fabs((pow(x, 6.0) * (sqrt((1.0 / ((double) M_PI))) * ((0.2 / x) + (fabs(x) * 0.047619047619047616)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.02)
		tmp = abs(Float64(Float64(x * (pi ^ -0.5)) * Float64(Float64(0.2 * (x ^ 4.0)) + fma(0.6666666666666666, Float64(x * x), 2.0))));
	else
		tmp = abs(Float64((x ^ 6.0) * Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(0.2 / x) + Float64(abs(x) * 0.047619047619047616)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.02], N[Abs[N[(N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Power[x, 6.0], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(0.2 / x), $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] * 0.047619047619047616), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0200000000000000004

   1. Initial program 99.8%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in x around 0 99.1%

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

      1. div-inv 99.7%

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

      2. pow1/2 99.7%

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

      3. pow-flip 99.7%

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

      4. metadata-eval 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. pow1 99.7%

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

      2. add-sqr-sqrt 43.7%

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

      3. fabs-sqr 43.7%

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

      4. add-sqr-sqrt 99.7%

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

   9. Applied egg-rr 99.7%

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

       1. unpow1 99.7%

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

   11. Simplified 99.7%

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

   if 0.0200000000000000004 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-*r* 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l* 99.9%

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

      6. distribute-lft-out 99.9%

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

   7. Simplified 99.9%

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

      1. expm1-log1p-u 99.9%

         \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.2 \cdot \frac{\left|x\right|}{{x}^{2}}\right)\right)} + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

      2. expm1-undefine 99.9%

         \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(0.2 \cdot \frac{\left|x\right|}{{x}^{2}}\right)} - 1\right)} + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

      3. pow2 99.9%

         \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(e^{\mathsf{log1p}\left(0.2 \cdot \frac{\left|x\right|}{\color{blue}{x \cdot x}}\right)} - 1\right) + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

      4. div-inv 99.9%

         \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(e^{\mathsf{log1p}\left(0.2 \cdot \color{blue}{\left(\left|x\right| \cdot \frac{1}{x \cdot x}\right)}\right)} - 1\right) + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(e^{\mathsf{log1p}\left(0.2 \cdot \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \frac{1}{x \cdot x}\right)\right)} - 1\right) + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

      6. fabs-sqr 0.0%

         \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(e^{\mathsf{log1p}\left(0.2 \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \frac{1}{x \cdot x}\right)\right)} - 1\right) + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

      7. add-sqr-sqrt 99.8%

         \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(e^{\mathsf{log1p}\left(0.2 \cdot \left(\color{blue}{x} \cdot \frac{1}{x \cdot x}\right)\right)} - 1\right) + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

      8. pow2 99.8%

         \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(e^{\mathsf{log1p}\left(0.2 \cdot \left(x \cdot \frac{1}{\color{blue}{{x}^{2}}}\right)\right)} - 1\right) + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

      9. pow-flip 99.8%

         \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(e^{\mathsf{log1p}\left(0.2 \cdot \left(x \cdot \color{blue}{{x}^{\left(-2\right)}}\right)\right)} - 1\right) + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

      10. metadata-eval 99.8%

          \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(e^{\mathsf{log1p}\left(0.2 \cdot \left(x \cdot {x}^{\color{blue}{-2}}\right)\right)} - 1\right) + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

   9. Applied egg-rr 99.8%

      \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(0.2 \cdot \left(x \cdot {x}^{-2}\right)\right)} - 1\right)} + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]
   10. Step-by-step derivation

       1. sub-neg 99.8%

          \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(0.2 \cdot \left(x \cdot {x}^{-2}\right)\right)} + \left(-1\right)\right)} + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

       2. metadata-eval 99.8%

          \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(e^{\mathsf{log1p}\left(0.2 \cdot \left(x \cdot {x}^{-2}\right)\right)} + \color{blue}{-1}\right) + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

       3. +-commutative 99.8%

          \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\left(-1 + e^{\mathsf{log1p}\left(0.2 \cdot \left(x \cdot {x}^{-2}\right)\right)}\right)} + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

       4. log1p-undefine 99.8%

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

       5. rem-exp-log 99.8%

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

       6. associate-+r+ 99.8%

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

       7. metadata-eval 99.8%

          \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(\color{blue}{0} + 0.2 \cdot \left(x \cdot {x}^{-2}\right)\right) + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

       8. metadata-eval 99.8%

          \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(\color{blue}{0.2 \cdot 0} + 0.2 \cdot \left(x \cdot {x}^{-2}\right)\right) + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

       9. distribute-lft-in 99.8%

          \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{0.2 \cdot \left(0 + x \cdot {x}^{-2}\right)} + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

       10. +-lft-identity 99.8%

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

       11. *-commutative 99.8%

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

       12. pow-plus 99.8%

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

       13. metadata-eval 99.8%

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

       14. unpow-1 99.8%

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

       15. *-inverses 99.8%

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

       16. associate-/r* 99.8%

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

       17. unpow2 99.8%

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

       18. remove-double-neg 99.8%

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

       19. distribute-frac-neg2 99.8%

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

       20. distribute-frac-neg2 99.8%

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

       21. remove-double-neg 99.8%

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

       22. unpow2 99.8%

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

       23. associate-/r* 99.8%

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

   11. Simplified 99.8%

       \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\frac{0.2}{x}} + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.02:\\ \;\;\;\;\left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(0.2 \cdot {x}^{4} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.2}{x} + \left|x\right| \cdot 0.047619047619047616\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \mathbf{if}\;\left|x\right| \leq 0.02:\\ \;\;\;\;\left|t\_0 \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{6} \cdot \left(t\_0 \cdot \left(\frac{0.2}{x} + \left|x\right| \cdot 0.047619047619047616\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (if (<= (fabs x) 0.02)
     (fabs (* t_0 (* x (fma 0.6666666666666666 (* x x) 2.0))))
     (fabs
      (*
       (pow x 6.0)
       (* t_0 (+ (/ 0.2 x) (* (fabs x) 0.047619047619047616))))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double tmp;
	if (fabs(x) <= 0.02) {
		tmp = fabs((t_0 * (x * fma(0.6666666666666666, (x * x), 2.0))));
	} else {
		tmp = fabs((pow(x, 6.0) * (t_0 * ((0.2 / x) + (fabs(x) * 0.047619047619047616)))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / pi))
	tmp = 0.0
	if (abs(x) <= 0.02)
		tmp = abs(Float64(t_0 * Float64(x * fma(0.6666666666666666, Float64(x * x), 2.0))));
	else
		tmp = abs(Float64((x ^ 6.0) * Float64(t_0 * Float64(Float64(0.2 / x) + Float64(abs(x) * 0.047619047619047616)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.02], N[Abs[N[(t$95$0 * N[(x * N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Power[x, 6.0], $MachinePrecision] * N[(t$95$0 * N[(N[(0.2 / x), $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] * 0.047619047619047616), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0200000000000000004

   1. Initial program 99.8%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in x around 0 99.1%

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

      1. div-inv 99.7%

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

      2. pow1/2 99.7%

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

      3. pow-flip 99.7%

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

      4. metadata-eval 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in x around 0 99.6%

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

      1. associate-*r* 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*r* 99.6%

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

      4. distribute-rgt-out 99.6%

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

      5. associate-*r* 99.6%

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

      6. rem-square-sqrt 43.9%

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

      7. fabs-sqr 43.9%

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

      8. rem-square-sqrt 99.4%

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

      9. rem-square-sqrt 43.6%

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

      10. fabs-sqr 43.6%

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

      11. rem-square-sqrt 99.6%

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

      12. distribute-rgt-in 99.6%

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

      13. fma-undefine 99.6%

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

   10. Simplified 99.6%

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

       1. pow2 99.6%

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

   12. Applied egg-rr 99.6%

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

   if 0.0200000000000000004 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-*r* 99.8%

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

      3. *-commutative 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-*l* 99.9%

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

      6. distribute-lft-out 99.9%

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

   7. Simplified 99.9%

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

      1. expm1-log1p-u 99.9%

         \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.2 \cdot \frac{\left|x\right|}{{x}^{2}}\right)\right)} + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

      2. expm1-undefine 99.9%

         \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(0.2 \cdot \frac{\left|x\right|}{{x}^{2}}\right)} - 1\right)} + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

      3. pow2 99.9%

         \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(e^{\mathsf{log1p}\left(0.2 \cdot \frac{\left|x\right|}{\color{blue}{x \cdot x}}\right)} - 1\right) + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

      4. div-inv 99.9%

         \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(e^{\mathsf{log1p}\left(0.2 \cdot \color{blue}{\left(\left|x\right| \cdot \frac{1}{x \cdot x}\right)}\right)} - 1\right) + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(e^{\mathsf{log1p}\left(0.2 \cdot \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \frac{1}{x \cdot x}\right)\right)} - 1\right) + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

      6. fabs-sqr 0.0%

         \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(e^{\mathsf{log1p}\left(0.2 \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \frac{1}{x \cdot x}\right)\right)} - 1\right) + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

      7. add-sqr-sqrt 99.8%

         \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(e^{\mathsf{log1p}\left(0.2 \cdot \left(\color{blue}{x} \cdot \frac{1}{x \cdot x}\right)\right)} - 1\right) + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

      8. pow2 99.8%

         \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(e^{\mathsf{log1p}\left(0.2 \cdot \left(x \cdot \frac{1}{\color{blue}{{x}^{2}}}\right)\right)} - 1\right) + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

      9. pow-flip 99.8%

         \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(e^{\mathsf{log1p}\left(0.2 \cdot \left(x \cdot \color{blue}{{x}^{\left(-2\right)}}\right)\right)} - 1\right) + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

      10. metadata-eval 99.8%

          \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(e^{\mathsf{log1p}\left(0.2 \cdot \left(x \cdot {x}^{\color{blue}{-2}}\right)\right)} - 1\right) + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

   9. Applied egg-rr 99.8%

      \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(0.2 \cdot \left(x \cdot {x}^{-2}\right)\right)} - 1\right)} + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]
   10. Step-by-step derivation

       1. sub-neg 99.8%

          \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(0.2 \cdot \left(x \cdot {x}^{-2}\right)\right)} + \left(-1\right)\right)} + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

       2. metadata-eval 99.8%

          \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(e^{\mathsf{log1p}\left(0.2 \cdot \left(x \cdot {x}^{-2}\right)\right)} + \color{blue}{-1}\right) + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

       3. +-commutative 99.8%

          \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\left(-1 + e^{\mathsf{log1p}\left(0.2 \cdot \left(x \cdot {x}^{-2}\right)\right)}\right)} + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

       4. log1p-undefine 99.8%

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

       5. rem-exp-log 99.8%

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

       6. associate-+r+ 99.8%

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

       7. metadata-eval 99.8%

          \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(\color{blue}{0} + 0.2 \cdot \left(x \cdot {x}^{-2}\right)\right) + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

       8. metadata-eval 99.8%

          \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(\color{blue}{0.2 \cdot 0} + 0.2 \cdot \left(x \cdot {x}^{-2}\right)\right) + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

       9. distribute-lft-in 99.8%

          \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{0.2 \cdot \left(0 + x \cdot {x}^{-2}\right)} + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]

       10. +-lft-identity 99.8%

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

       11. *-commutative 99.8%

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

       12. pow-plus 99.8%

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

       13. metadata-eval 99.8%

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

       14. unpow-1 99.8%

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

       15. *-inverses 99.8%

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

       16. associate-/r* 99.8%

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

       17. unpow2 99.8%

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

       18. remove-double-neg 99.8%

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

       19. distribute-frac-neg2 99.8%

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

       20. distribute-frac-neg2 99.8%

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

       21. remove-double-neg 99.8%

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

       22. unpow2 99.8%

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

       23. associate-/r* 99.8%

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

   11. Simplified 99.8%

       \[\leadsto \left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\frac{0.2}{x}} + \left|x\right| \cdot 0.047619047619047616\right)\right)\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.02:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{6} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.2}{x} + \left|x\right| \cdot 0.047619047619047616\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 3.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	return fabs(x) * fabs((((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(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[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around inf 99.7%

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

6. Final simplification 99.7%

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

Alternative 5: 99.1% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.02:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{0.047619047619047616}{\sqrt{\pi}} \cdot {x}^{7}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.02)
   (fabs (* (sqrt (/ 1.0 PI)) (* x (fma 0.6666666666666666 (* x x) 2.0))))
   (fabs (* (/ 0.047619047619047616 (sqrt PI)) (pow x 7.0)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.02) {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * (x * fma(0.6666666666666666, (x * x), 2.0))));
	} else {
		tmp = fabs(((0.047619047619047616 / sqrt(((double) M_PI))) * pow(x, 7.0)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.02)
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(x * fma(0.6666666666666666, Float64(x * x), 2.0))));
	else
		tmp = abs(Float64(Float64(0.047619047619047616 / sqrt(pi)) * (x ^ 7.0)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.02], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x * N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(0.047619047619047616 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0200000000000000004

   1. Initial program 99.8%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in x around 0 99.1%

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

      1. div-inv 99.7%

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

      2. pow1/2 99.7%

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

      3. pow-flip 99.7%

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

      4. metadata-eval 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in x around 0 99.6%

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

      1. associate-*r* 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*r* 99.6%

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

      4. distribute-rgt-out 99.6%

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

      5. associate-*r* 99.6%

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

      6. rem-square-sqrt 43.9%

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

      7. fabs-sqr 43.9%

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

      8. rem-square-sqrt 99.4%

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

      9. rem-square-sqrt 43.6%

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

      10. fabs-sqr 43.6%

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

      11. rem-square-sqrt 99.6%

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

      12. distribute-rgt-in 99.6%

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

      13. fma-undefine 99.6%

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

   10. Simplified 99.6%

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

       1. pow2 99.6%

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

   12. Applied egg-rr 99.6%

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

   if 0.0200000000000000004 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 99.8%

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

      1. *-commutative 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-*r* 99.8%

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

      4. associate-*l* 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-*l* 99.8%

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

      7. *-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. pow1 99.8%

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

      2. *-commutative 99.8%

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

      3. add-sqr-sqrt 0.0%

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

      4. fabs-sqr 0.0%

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

      5. add-sqr-sqrt 99.8%

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

      6. sqrt-div 99.8%

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

      7. metadata-eval 99.8%

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

      8. un-div-inv 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. unpow1 99.8%

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

       2. *-commutative 99.8%

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

       3. associate-*l/ 99.8%

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

       4. associate-*r/ 99.9%

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

   11. Simplified 99.9%

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

       1. associate-*r/ 99.8%

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

       2. clear-num 99.8%

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

       3. associate-*r* 99.8%

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

   13. Applied egg-rr 99.8%

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

       1. associate-/r/ 99.9%

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

       2. associate-*l* 99.8%

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

       3. associate-*r* 99.8%

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

       4. associate-*l/ 99.8%

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

       5. metadata-eval 99.8%

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

       6. distribute-lft-neg-in 99.8%

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

       7. neg-mul-1 99.8%

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

       8. remove-double-neg 99.8%

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

       9. /-rgt-identity 99.8%

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

       10. associate-*r/ 99.8%

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

       11. times-frac 99.8%

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

       12. *-rgt-identity 99.8%

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

       13. associate-*r/ 99.9%

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

       14. associate-*r/ 99.8%

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

       15. *-commutative 99.8%

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

       16. associate-*r/ 99.9%

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

       17. associate-*l/ 99.8%

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

       18. associate-*l* 99.9%

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

       19. pow-plus 99.9%

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

       20. metadata-eval 99.9%

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

   15. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.02:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{0.047619047619047616}{\sqrt{\pi}} \cdot {x}^{7}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.02:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{0.047619047619047616}{\sqrt{\pi}} \cdot {x}^{7}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.02)
   (fabs (* x (/ 2.0 (sqrt PI))))
   (fabs (* (/ 0.047619047619047616 (sqrt PI)) (pow x 7.0)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.02) {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs(((0.047619047619047616 / sqrt(((double) M_PI))) * pow(x, 7.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.02) {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs(((0.047619047619047616 / Math.sqrt(Math.PI)) * Math.pow(x, 7.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.02:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs(((0.047619047619047616 / math.sqrt(math.pi)) * math.pow(x, 7.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.02)
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(Float64(Float64(0.047619047619047616 / sqrt(pi)) * (x ^ 7.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.02)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = abs(((0.047619047619047616 / sqrt(pi)) * (x ^ 7.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.02], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(0.047619047619047616 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0200000000000000004

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.1%

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

      1. associate-*r* 99.1%

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

   7. Simplified 99.1%

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

      1. pow1 99.1%

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

      2. *-commutative 99.1%

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

      3. add-sqr-sqrt 43.3%

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

      4. fabs-sqr 43.3%

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

      5. add-sqr-sqrt 99.1%

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

      6. sqrt-div 99.1%

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

      7. metadata-eval 99.1%

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

      8. un-div-inv 99.1%

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

   9. Applied egg-rr 99.1%

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

       1. unpow1 99.1%

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

   11. Simplified 99.1%

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

   if 0.0200000000000000004 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 99.8%

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

      1. *-commutative 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-*r* 99.8%

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

      4. associate-*l* 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-*l* 99.8%

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

      7. *-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. pow1 99.8%

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

      2. *-commutative 99.8%

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

      3. add-sqr-sqrt 0.0%

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

      4. fabs-sqr 0.0%

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

      5. add-sqr-sqrt 99.8%

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

      6. sqrt-div 99.8%

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

      7. metadata-eval 99.8%

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

      8. un-div-inv 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. unpow1 99.8%

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

       2. *-commutative 99.8%

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

       3. associate-*l/ 99.8%

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

       4. associate-*r/ 99.9%

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

   11. Simplified 99.9%

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

       1. associate-*r/ 99.8%

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

       2. clear-num 99.8%

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

       3. associate-*r* 99.8%

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

   13. Applied egg-rr 99.8%

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

       1. associate-/r/ 99.9%

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

       2. associate-*l* 99.8%

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

       3. associate-*r* 99.8%

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

       4. associate-*l/ 99.8%

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

       5. metadata-eval 99.8%

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

       6. distribute-lft-neg-in 99.8%

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

       7. neg-mul-1 99.8%

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

       8. remove-double-neg 99.8%

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

       9. /-rgt-identity 99.8%

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

       10. associate-*r/ 99.8%

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

       11. times-frac 99.8%

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

       12. *-rgt-identity 99.8%

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

       13. associate-*r/ 99.9%

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

       14. associate-*r/ 99.8%

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

       15. *-commutative 99.8%

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

       16. associate-*r/ 99.9%

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

       17. associate-*l/ 99.8%

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

       18. associate-*l* 99.9%

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

       19. pow-plus 99.9%

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

       20. metadata-eval 99.9%

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

   15. Simplified 99.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.02:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{0.047619047619047616}{\sqrt{\pi}} \cdot {x}^{7}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.2% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 65.2%

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

   1. associate-*r* 65.2%

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

7. Simplified 65.2%

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

   1. pow1 65.2%

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

   2. *-commutative 65.2%

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

   3. add-sqr-sqrt 27.6%

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

   4. fabs-sqr 27.6%

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

   5. add-sqr-sqrt 65.2%

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

   6. sqrt-div 65.2%

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

   7. metadata-eval 65.2%

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

   8. un-div-inv 65.2%

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

9. Applied egg-rr 65.2%

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

    1. unpow1 65.2%

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

11. Simplified 65.2%

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

12. Final simplification 65.2%

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

Reproduce

herbie shell --seed 2024053 
(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)))))))