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

Percentage Accurate: 99.8% → 99.9%

Time: 15.4s

Alternatives: 14

Speedup: 3.6×

• 28.6% 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, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[Abs[N[(N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * 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]), $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 99.4%

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

   1. div-inv 99.9%

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

   2. pow1/2 99.9%

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

   3. pow-flip 99.9%

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

   4. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

   1. expm1-log1p-u 99.6%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot {\pi}^{-0.5}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   2. expm1-udef 38.6%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\left|x\right| \cdot {\pi}^{-0.5}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   3. add-sqr-sqrt 2.5%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   4. fabs-sqr 2.5%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   5. add-sqr-sqrt 5.0%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{x} \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

8. Applied egg-rr 5.0%

   \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]
9. Step-by-step derivation

   1. expm1-def 65.9%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   2. expm1-log1p 99.9%

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

10. Simplified 99.9%

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

    1. fma-udef 99.9%

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

12. Applied egg-rr 99.9%

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

13. Final simplification 99.9%

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

Alternative 2: 99.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2:\\ \;\;\;\;\left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(0.2 \cdot {x}^{4} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \left(\frac{21}{{x}^{6}} + \frac{-88.2}{{x}^{8}}\right)\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 2.0)
   (fabs
    (*
     (* x (pow PI -0.5))
     (+ (* 0.2 (pow x 4.0)) (fma 0.6666666666666666 (* x x) 2.0))))
   (/
    (fabs x)
    (fabs (* (sqrt PI) (+ (/ 21.0 (pow x 6.0)) (/ -88.2 (pow x 8.0))))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 2.0) {
		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(x) / fabs((sqrt(((double) M_PI)) * ((21.0 / pow(x, 6.0)) + (-88.2 / pow(x, 8.0)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 2.0)
		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 = Float64(abs(x) / abs(Float64(sqrt(pi) * Float64(Float64(21.0 / (x ^ 6.0)) + Float64(-88.2 / (x ^ 8.0))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 2.0], 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[(N[Abs[x], $MachinePrecision] / N[Abs[N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[(21.0 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(-88.2 / N[Power[x, 8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2

   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 99.1%

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

      1. div-inv 99.9%

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

      2. pow1/2 99.9%

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

      3. pow-flip 99.9%

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

      4. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. expm1-log1p-u 99.9%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot {\pi}^{-0.5}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      2. expm1-udef 7.5%

         \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\left|x\right| \cdot {\pi}^{-0.5}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      3. add-sqr-sqrt 3.7%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      4. fabs-sqr 3.7%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      5. add-sqr-sqrt 7.6%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{x} \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   8. Applied egg-rr 7.6%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]
   9. Step-by-step derivation

      1. expm1-def 99.9%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      2. expm1-log1p 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in x around 0 99.4%

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

   if 2 < (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 99.9%

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

   5. Taylor expanded in x around inf 98.7%

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

      1. +-commutative 98.7%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right) + -88.2 \cdot \left(\frac{1}{{x}^{8}} \cdot \sqrt{\pi}\right)}\right|} \]

      2. associate-*r* 98.7%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\left(21 \cdot \frac{1}{{x}^{6}}\right) \cdot \sqrt{\pi}} + -88.2 \cdot \left(\frac{1}{{x}^{8}} \cdot \sqrt{\pi}\right)\right|} \]

      3. associate-*r* 98.7%

         \[\leadsto \frac{\left|x\right|}{\left|\left(21 \cdot \frac{1}{{x}^{6}}\right) \cdot \sqrt{\pi} + \color{blue}{\left(-88.2 \cdot \frac{1}{{x}^{8}}\right) \cdot \sqrt{\pi}}\right|} \]

      4. distribute-rgt-out 98.7%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot \left(21 \cdot \frac{1}{{x}^{6}} + -88.2 \cdot \frac{1}{{x}^{8}}\right)}\right|} \]

      5. associate-*r/ 98.7%

         \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \left(\color{blue}{\frac{21 \cdot 1}{{x}^{6}}} + -88.2 \cdot \frac{1}{{x}^{8}}\right)\right|} \]

      6. metadata-eval 98.7%

         \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \left(\frac{\color{blue}{21}}{{x}^{6}} + -88.2 \cdot \frac{1}{{x}^{8}}\right)\right|} \]

      7. associate-*r/ 98.7%

         \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \left(\frac{21}{{x}^{6}} + \color{blue}{\frac{-88.2 \cdot 1}{{x}^{8}}}\right)\right|} \]

      8. metadata-eval 98.7%

         \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \left(\frac{21}{{x}^{6}} + \frac{\color{blue}{-88.2}}{{x}^{8}}\right)\right|} \]

   7. Simplified 98.7%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot \left(\frac{21}{{x}^{6}} + \frac{-88.2}{{x}^{8}}\right)}\right|} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2:\\ \;\;\;\;\left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(0.2 \cdot {x}^{4} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \left(\frac{21}{{x}^{6}} + \frac{-88.2}{{x}^{8}}\right)\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2:\\ \;\;\;\;\left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(0.2 \cdot {x}^{4} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 2.0)
   (fabs
    (*
     (* x (pow PI -0.5))
     (+ (* 0.2 (pow x 4.0)) (fma 0.6666666666666666 (* x x) 2.0))))
   (fabs
    (*
     (sqrt (/ 1.0 PI))
     (+ (* 0.2 (pow x 5.0)) (* 0.047619047619047616 (pow x 7.0)))))))

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 2.0], 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[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2

   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 99.1%

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

      1. div-inv 99.9%

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

      2. pow1/2 99.9%

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

      3. pow-flip 99.9%

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

      4. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. expm1-log1p-u 99.9%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot {\pi}^{-0.5}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      2. expm1-udef 7.5%

         \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\left|x\right| \cdot {\pi}^{-0.5}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      3. add-sqr-sqrt 3.7%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      4. fabs-sqr 3.7%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      5. add-sqr-sqrt 7.6%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{x} \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   8. Applied egg-rr 7.6%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]
   9. Step-by-step derivation

      1. expm1-def 99.9%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      2. expm1-log1p 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in x around 0 99.4%

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

   if 2 < (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 99.9%

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

      1. div-inv 99.8%

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

      2. pow1/2 99.8%

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

      3. pow-flip 99.8%

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

      4. metadata-eval 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. expm1-log1p-u 99.1%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot {\pi}^{-0.5}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      2. expm1-udef 99.1%

         \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\left|x\right| \cdot {\pi}^{-0.5}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      3. add-sqr-sqrt 0.0%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      4. fabs-sqr 0.0%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{x} \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   8. Applied egg-rr 0.0%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]
   9. Step-by-step derivation

      1. expm1-def 0.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      2. expm1-log1p 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around inf 98.7%

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

       1. +-commutative 98.7%

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

       2. associate-*r* 98.7%

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

       3. associate-*r* 98.7%

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

       4. distribute-rgt-out 98.7%

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

   13. Simplified 98.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2:\\ \;\;\;\;\left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(0.2 \cdot {x}^{4} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2

   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 99.1%

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

      1. div-inv 99.9%

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

      2. pow1/2 99.9%

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

      3. pow-flip 99.9%

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

      4. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. expm1-log1p-u 99.9%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot {\pi}^{-0.5}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      2. expm1-udef 7.5%

         \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\left|x\right| \cdot {\pi}^{-0.5}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      3. add-sqr-sqrt 3.7%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      4. fabs-sqr 3.7%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      5. add-sqr-sqrt 7.6%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{x} \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   8. Applied egg-rr 7.6%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]
   9. Step-by-step derivation

      1. expm1-def 99.9%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      2. expm1-log1p 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in x around 0 99.4%

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

       1. +-commutative 99.4%

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

       2. associate-*r* 99.4%

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

       3. *-commutative 99.4%

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

       4. associate-*r* 99.4%

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

       5. distribute-rgt-out 99.4%

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

   13. Simplified 99.4%

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

   if 2 < (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 99.9%

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

      1. div-inv 99.8%

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

      2. pow1/2 99.8%

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

      3. pow-flip 99.8%

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

      4. metadata-eval 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. expm1-log1p-u 99.1%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot {\pi}^{-0.5}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      2. expm1-udef 99.1%

         \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\left|x\right| \cdot {\pi}^{-0.5}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      3. add-sqr-sqrt 0.0%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      4. fabs-sqr 0.0%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{x} \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   8. Applied egg-rr 0.0%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]
   9. Step-by-step derivation

      1. expm1-def 0.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      2. expm1-log1p 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around inf 98.7%

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

       1. +-commutative 98.7%

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

       2. associate-*r* 98.7%

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

       3. associate-*r* 98.7%

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

       4. distribute-rgt-out 98.7%

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

   13. Simplified 98.7%

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

4. Final simplification 99.2%

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

Alternative 5: 99.2% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Abs[N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

   1. div-inv 99.9%

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

   2. pow1/2 99.9%

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

   3. pow-flip 99.9%

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

   4. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around inf 98.9%

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

8. Final simplification 98.9%

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

Alternative 6: 98.7% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Abs[x], $MachinePrecision] / N[Abs[N[(N[Sqrt[Pi], $MachinePrecision] / N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $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 99.4%

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

5. Taylor expanded in x around inf 98.4%

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

6. Final simplification 98.4%

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

Alternative 7: 99.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x\right|}{\left|\frac{\sqrt{\pi} \cdot 21}{{x}^{6}}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 2.0)
   (fabs
    (* (sqrt (/ 1.0 PI)) (+ (* x 2.0) (* 0.6666666666666666 (pow x 3.0)))))
   (/ (fabs x) (fabs (/ (* (sqrt PI) 21.0) (pow x 6.0))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 2.0) {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * ((x * 2.0) + (0.6666666666666666 * pow(x, 3.0)))));
	} else {
		tmp = fabs(x) / fabs(((sqrt(((double) M_PI)) * 21.0) / pow(x, 6.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 2.0) {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * ((x * 2.0) + (0.6666666666666666 * Math.pow(x, 3.0)))));
	} else {
		tmp = Math.abs(x) / Math.abs(((Math.sqrt(Math.PI) * 21.0) / Math.pow(x, 6.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 2.0:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * ((x * 2.0) + (0.6666666666666666 * math.pow(x, 3.0)))))
	else:
		tmp = math.fabs(x) / math.fabs(((math.sqrt(math.pi) * 21.0) / math.pow(x, 6.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 2.0)
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(x * 2.0) + Float64(0.6666666666666666 * (x ^ 3.0)))));
	else
		tmp = Float64(abs(x) / abs(Float64(Float64(sqrt(pi) * 21.0) / (x ^ 6.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 2.0)
		tmp = abs((sqrt((1.0 / pi)) * ((x * 2.0) + (0.6666666666666666 * (x ^ 3.0)))));
	else
		tmp = abs(x) / abs(((sqrt(pi) * 21.0) / (x ^ 6.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 2.0], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(x * 2.0), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[x], $MachinePrecision] / N[Abs[N[(N[(N[Sqrt[Pi], $MachinePrecision] * 21.0), $MachinePrecision] / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2

   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 99.1%

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

      1. div-inv 99.9%

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

      2. pow1/2 99.9%

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

      3. pow-flip 99.9%

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

      4. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. expm1-log1p-u 99.9%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot {\pi}^{-0.5}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      2. expm1-udef 7.5%

         \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\left|x\right| \cdot {\pi}^{-0.5}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      3. add-sqr-sqrt 3.7%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      4. fabs-sqr 3.7%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      5. add-sqr-sqrt 7.6%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{x} \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   8. Applied egg-rr 7.6%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]
   9. Step-by-step derivation

      1. expm1-def 99.9%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      2. expm1-log1p 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in x around 0 99.4%

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

       1. +-commutative 99.4%

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

       2. associate-*r* 99.4%

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

       3. *-commutative 99.4%

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

       4. associate-*r* 99.4%

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

       5. distribute-rgt-out 99.4%

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

   13. Simplified 99.4%

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

   if 2 < (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 99.9%

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

   5. Taylor expanded in x around inf 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-*l/ 97.8%

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

      3. *-lft-identity 97.8%

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

      4. associate-*l/ 97.8%

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

   7. Simplified 97.8%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x\right|}{\left|\frac{\sqrt{\pi} \cdot 21}{{x}^{6}}\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\sqrt{\pi} \cdot 21\right) \cdot {x}^{-6}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 2.0)
   (fabs
    (* (sqrt (/ 1.0 PI)) (+ (* x 2.0) (* 0.6666666666666666 (pow x 3.0)))))
   (/ x (* (* (sqrt PI) 21.0) (pow x -6.0)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 2.0) {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * ((x * 2.0) + (0.6666666666666666 * pow(x, 3.0)))));
	} else {
		tmp = x / ((sqrt(((double) M_PI)) * 21.0) * pow(x, -6.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 2.0) {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * ((x * 2.0) + (0.6666666666666666 * Math.pow(x, 3.0)))));
	} else {
		tmp = x / ((Math.sqrt(Math.PI) * 21.0) * Math.pow(x, -6.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 2.0:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * ((x * 2.0) + (0.6666666666666666 * math.pow(x, 3.0)))))
	else:
		tmp = x / ((math.sqrt(math.pi) * 21.0) * math.pow(x, -6.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 2.0)
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(x * 2.0) + Float64(0.6666666666666666 * (x ^ 3.0)))));
	else
		tmp = Float64(x / Float64(Float64(sqrt(pi) * 21.0) * (x ^ -6.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 2.0)
		tmp = abs((sqrt((1.0 / pi)) * ((x * 2.0) + (0.6666666666666666 * (x ^ 3.0)))));
	else
		tmp = x / ((sqrt(pi) * 21.0) * (x ^ -6.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 2.0], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(x * 2.0), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(x / N[(N[(N[Sqrt[Pi], $MachinePrecision] * 21.0), $MachinePrecision] * N[Power[x, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2

   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 99.1%

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

      1. div-inv 99.9%

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

      2. pow1/2 99.9%

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

      3. pow-flip 99.9%

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

      4. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. expm1-log1p-u 99.9%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot {\pi}^{-0.5}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      2. expm1-udef 7.5%

         \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\left|x\right| \cdot {\pi}^{-0.5}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      3. add-sqr-sqrt 3.7%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      4. fabs-sqr 3.7%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      5. add-sqr-sqrt 7.6%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{x} \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   8. Applied egg-rr 7.6%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]
   9. Step-by-step derivation

      1. expm1-def 99.9%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

      2. expm1-log1p 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in x around 0 99.4%

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

       1. +-commutative 99.4%

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

       2. associate-*r* 99.4%

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

       3. *-commutative 99.4%

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

       4. associate-*r* 99.4%

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

       5. distribute-rgt-out 99.4%

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

   13. Simplified 99.4%

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

   if 2 < (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 99.9%

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

   5. Taylor expanded in x around inf 97.7%

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

      1. associate-*l/ 97.8%

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

      2. *-lft-identity 97.8%

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

   7. Simplified 97.8%

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

      1. div-inv 97.8%

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

      2. add-sqr-sqrt 0.0%

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

      3. fabs-sqr 0.0%

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

      4. add-sqr-sqrt 0.2%

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

      5. add-sqr-sqrt 0.2%

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

      6. fabs-sqr 0.2%

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

      7. add-sqr-sqrt 0.2%

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

      8. div-inv 0.2%

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

      9. pow-flip 0.2%

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

      10. metadata-eval 0.2%

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

   9. Applied egg-rr 0.2%

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

       1. un-div-inv 0.2%

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

       2. associate-*r* 0.2%

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

   11. Applied egg-rr 0.2%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\sqrt{\pi} \cdot 21\right) \cdot {x}^{-6}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.5% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \frac{1}{\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\sqrt{\pi} \cdot 21\right) \cdot {x}^{-6}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 2e-5)
   (* x (/ 1.0 (* (sqrt PI) (+ (* (pow x 2.0) -0.16666666666666666) 0.5))))
   (/ x (* (* (sqrt PI) 21.0) (pow x -6.0)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 2e-5) {
		tmp = x * (1.0 / (sqrt(((double) M_PI)) * ((pow(x, 2.0) * -0.16666666666666666) + 0.5)));
	} else {
		tmp = x / ((sqrt(((double) M_PI)) * 21.0) * pow(x, -6.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 2e-5) {
		tmp = x * (1.0 / (Math.sqrt(Math.PI) * ((Math.pow(x, 2.0) * -0.16666666666666666) + 0.5)));
	} else {
		tmp = x / ((Math.sqrt(Math.PI) * 21.0) * Math.pow(x, -6.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 2e-5:
		tmp = x * (1.0 / (math.sqrt(math.pi) * ((math.pow(x, 2.0) * -0.16666666666666666) + 0.5)))
	else:
		tmp = x / ((math.sqrt(math.pi) * 21.0) * math.pow(x, -6.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 2e-5)
		tmp = Float64(x * Float64(1.0 / Float64(sqrt(pi) * Float64(Float64((x ^ 2.0) * -0.16666666666666666) + 0.5))));
	else
		tmp = Float64(x / Float64(Float64(sqrt(pi) * 21.0) * (x ^ -6.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 2e-5)
		tmp = x * (1.0 / (sqrt(pi) * (((x ^ 2.0) * -0.16666666666666666) + 0.5)));
	else
		tmp = x / ((sqrt(pi) * 21.0) * (x ^ -6.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 2e-5], N[(x * N[(1.0 / N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[(N[Power[x, 2.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(N[Sqrt[Pi], $MachinePrecision] * 21.0), $MachinePrecision] * N[Power[x, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2.00000000000000016e-5

   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 99.1%

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

   5. Taylor expanded in x around 0 99.1%

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

      1. +-commutative 99.1%

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

      2. associate-*r* 99.1%

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

      3. distribute-rgt-out 99.1%

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

      4. *-commutative 99.1%

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

   7. Simplified 99.1%

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

      1. div-inv 99.9%

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

      2. add-sqr-sqrt 53.7%

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

      3. fabs-sqr 53.7%

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

      4. add-sqr-sqrt 55.9%

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

      5. add-sqr-sqrt 55.9%

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

      6. fabs-sqr 55.9%

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

      7. add-sqr-sqrt 55.9%

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

      8. +-commutative 55.9%

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

      9. pow2 55.9%

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

      10. fma-def 55.9%

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

      11. pow2 55.9%

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

   9. Applied egg-rr 55.9%

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

       1. fma-udef 55.9%

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

   11. Applied egg-rr 55.9%

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

   if 2.00000000000000016e-5 < (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 99.9%

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

   5. Taylor expanded in x around inf 96.8%

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

      1. associate-*l/ 96.8%

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

      2. *-lft-identity 96.8%

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

   7. Simplified 96.8%

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

      1. div-inv 96.9%

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

      2. add-sqr-sqrt 0.0%

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

      3. fabs-sqr 0.0%

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

      4. add-sqr-sqrt 0.2%

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

      5. add-sqr-sqrt 0.2%

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

      6. fabs-sqr 0.2%

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

      7. add-sqr-sqrt 0.2%

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

      8. div-inv 0.2%

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

      9. pow-flip 0.2%

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

      10. metadata-eval 0.2%

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

   9. Applied egg-rr 0.2%

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

       1. un-div-inv 0.2%

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

       2. associate-*r* 0.2%

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

   11. Applied egg-rr 0.2%

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

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \frac{1}{\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\sqrt{\pi} \cdot 21\right) \cdot {x}^{-6}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 33.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{\frac{x}{\sqrt{\pi}}}{{x}^{-6}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (* x (/ 2.0 (sqrt PI)))
   (* 0.047619047619047616 (/ (/ x (sqrt PI)) (pow x -6.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = 0.047619047619047616 * ((x / sqrt(((double) M_PI))) / pow(x, -6.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = 0.047619047619047616 * ((x / Math.sqrt(Math.PI)) / Math.pow(x, -6.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = 0.047619047619047616 * ((x / math.sqrt(math.pi)) / math.pow(x, -6.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(0.047619047619047616 * Float64(Float64(x / sqrt(pi)) / (x ^ -6.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = 0.047619047619047616 * ((x / sqrt(pi)) / (x ^ -6.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] / N[Power[x, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 67.0%

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

      1. *-commutative 67.0%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

   7. Simplified 67.0%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

   8. Taylor expanded in x around 0 67.0%

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

      1. *-commutative 67.0%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

      2. fabs-div 67.0%

         \[\leadsto \color{blue}{\left|\frac{x}{\sqrt{\pi} \cdot 0.5}\right|} \]

      3. rem-square-sqrt 35.1%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{x}{\sqrt{\pi} \cdot 0.5}} \cdot \sqrt{\frac{x}{\sqrt{\pi} \cdot 0.5}}}\right| \]

      4. fabs-sqr 35.1%

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

      5. rem-square-sqrt 36.5%

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

      6. *-rgt-identity 36.5%

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

      7. associate-*r/ 36.7%

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

      8. *-commutative 36.7%

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

      9. associate-/r* 36.7%

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

      10. metadata-eval 36.7%

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

   10. Simplified 36.7%

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

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around inf 36.9%

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

      1. associate-*l/ 37.0%

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

      2. *-lft-identity 37.0%

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

   7. Simplified 37.0%

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

      1. div-inv 37.0%

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

      2. add-sqr-sqrt 1.9%

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

      3. fabs-sqr 1.9%

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

      4. add-sqr-sqrt 3.6%

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

      5. add-sqr-sqrt 3.6%

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

      6. fabs-sqr 3.6%

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

      7. add-sqr-sqrt 3.6%

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

      8. div-inv 3.6%

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

      9. pow-flip 3.6%

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

      10. metadata-eval 3.6%

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

   9. Applied egg-rr 3.6%

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

       1. expm1-log1p-u 3.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{1}{21 \cdot \left(\sqrt{\pi} \cdot {x}^{-6}\right)}\right)\right)} \]

       2. expm1-udef 3.5%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{1}{21 \cdot \left(\sqrt{\pi} \cdot {x}^{-6}\right)}\right)} - 1} \]

       3. un-div-inv 3.5%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{21 \cdot \left(\sqrt{\pi} \cdot {x}^{-6}\right)}}\right)} - 1 \]

       4. *-un-lft-identity 3.5%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1 \cdot x}}{21 \cdot \left(\sqrt{\pi} \cdot {x}^{-6}\right)}\right)} - 1 \]

       5. times-frac 3.5%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{21} \cdot \frac{x}{\sqrt{\pi} \cdot {x}^{-6}}}\right)} - 1 \]

       6. metadata-eval 3.5%

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

   11. Applied egg-rr 3.5%

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

       1. expm1-def 3.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \frac{x}{\sqrt{\pi} \cdot {x}^{-6}}\right)\right)} \]

       2. expm1-log1p 3.6%

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

       3. associate-/r* 3.6%

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

   13. Simplified 3.6%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{\frac{x}{\sqrt{\pi}}}{{x}^{-6}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 33.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (* x (/ 2.0 (sqrt PI)))
   (* (/ 0.047619047619047616 (sqrt PI)) (/ x (pow x -6.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = (0.047619047619047616 / sqrt(((double) M_PI))) * (x / pow(x, -6.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = (0.047619047619047616 / Math.sqrt(Math.PI)) * (x / Math.pow(x, -6.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = (0.047619047619047616 / math.sqrt(math.pi)) * (x / math.pow(x, -6.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(Float64(0.047619047619047616 / sqrt(pi)) * Float64(x / (x ^ -6.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = (0.047619047619047616 / sqrt(pi)) * (x / (x ^ -6.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.047619047619047616 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(x / N[Power[x, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 67.0%

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

      1. *-commutative 67.0%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

   7. Simplified 67.0%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

   8. Taylor expanded in x around 0 67.0%

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

      1. *-commutative 67.0%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

      2. fabs-div 67.0%

         \[\leadsto \color{blue}{\left|\frac{x}{\sqrt{\pi} \cdot 0.5}\right|} \]

      3. rem-square-sqrt 35.1%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{x}{\sqrt{\pi} \cdot 0.5}} \cdot \sqrt{\frac{x}{\sqrt{\pi} \cdot 0.5}}}\right| \]

      4. fabs-sqr 35.1%

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

      5. rem-square-sqrt 36.5%

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

      6. *-rgt-identity 36.5%

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

      7. associate-*r/ 36.7%

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

      8. *-commutative 36.7%

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

      9. associate-/r* 36.7%

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

      10. metadata-eval 36.7%

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

   10. Simplified 36.7%

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

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around inf 36.9%

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

      1. associate-*l/ 37.0%

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

      2. *-lft-identity 37.0%

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

   7. Simplified 37.0%

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

      1. div-inv 37.0%

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

      2. add-sqr-sqrt 1.9%

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

      3. fabs-sqr 1.9%

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

      4. add-sqr-sqrt 3.6%

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

      5. add-sqr-sqrt 3.6%

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

      6. fabs-sqr 3.6%

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

      7. add-sqr-sqrt 3.6%

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

      8. div-inv 3.6%

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

      9. pow-flip 3.6%

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

      10. metadata-eval 3.6%

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

   9. Applied egg-rr 3.6%

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

       1. expm1-log1p-u 3.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{1}{21 \cdot \left(\sqrt{\pi} \cdot {x}^{-6}\right)}\right)\right)} \]

       2. expm1-udef 3.5%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{1}{21 \cdot \left(\sqrt{\pi} \cdot {x}^{-6}\right)}\right)} - 1} \]

       3. un-div-inv 3.5%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{21 \cdot \left(\sqrt{\pi} \cdot {x}^{-6}\right)}}\right)} - 1 \]

       4. *-un-lft-identity 3.5%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1 \cdot x}}{21 \cdot \left(\sqrt{\pi} \cdot {x}^{-6}\right)}\right)} - 1 \]

       5. times-frac 3.5%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{21} \cdot \frac{x}{\sqrt{\pi} \cdot {x}^{-6}}}\right)} - 1 \]

       6. metadata-eval 3.5%

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

   11. Applied egg-rr 3.5%

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

       1. expm1-def 3.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \frac{x}{\sqrt{\pi} \cdot {x}^{-6}}\right)\right)} \]

       2. expm1-log1p 3.6%

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

       3. associate-*r/ 3.6%

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

       4. times-frac 3.6%

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

   13. Simplified 3.6%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 33.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\sqrt{\pi} \cdot 21\right) \cdot {x}^{-6}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (* x (/ 2.0 (sqrt PI)))
   (/ x (* (* (sqrt PI) 21.0) (pow x -6.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = x / ((sqrt(((double) M_PI)) * 21.0) * pow(x, -6.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = x / ((Math.sqrt(Math.PI) * 21.0) * Math.pow(x, -6.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = x / ((math.sqrt(math.pi) * 21.0) * math.pow(x, -6.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(x / Float64(Float64(sqrt(pi) * 21.0) * (x ^ -6.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = x / ((sqrt(pi) * 21.0) * (x ^ -6.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(N[Sqrt[Pi], $MachinePrecision] * 21.0), $MachinePrecision] * N[Power[x, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 67.0%

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

      1. *-commutative 67.0%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

   7. Simplified 67.0%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

   8. Taylor expanded in x around 0 67.0%

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

      1. *-commutative 67.0%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

      2. fabs-div 67.0%

         \[\leadsto \color{blue}{\left|\frac{x}{\sqrt{\pi} \cdot 0.5}\right|} \]

      3. rem-square-sqrt 35.1%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{x}{\sqrt{\pi} \cdot 0.5}} \cdot \sqrt{\frac{x}{\sqrt{\pi} \cdot 0.5}}}\right| \]

      4. fabs-sqr 35.1%

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

      5. rem-square-sqrt 36.5%

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

      6. *-rgt-identity 36.5%

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

      7. associate-*r/ 36.7%

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

      8. *-commutative 36.7%

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

      9. associate-/r* 36.7%

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

      10. metadata-eval 36.7%

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

   10. Simplified 36.7%

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

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around inf 36.9%

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

      1. associate-*l/ 37.0%

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

      2. *-lft-identity 37.0%

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

   7. Simplified 37.0%

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

      1. div-inv 37.0%

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

      2. add-sqr-sqrt 1.9%

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

      3. fabs-sqr 1.9%

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

      4. add-sqr-sqrt 3.6%

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

      5. add-sqr-sqrt 3.6%

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

      6. fabs-sqr 3.6%

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

      7. add-sqr-sqrt 3.6%

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

      8. div-inv 3.6%

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

      9. pow-flip 3.6%

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

      10. metadata-eval 3.6%

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

   9. Applied egg-rr 3.6%

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

       1. un-div-inv 3.6%

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

       2. associate-*r* 3.6%

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

   11. Applied egg-rr 3.6%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\sqrt{\pi} \cdot 21\right) \cdot {x}^{-6}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 33.8% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+91}:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{2} \cdot \frac{4}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e+91) (* x (/ 2.0 (sqrt PI))) (sqrt (* (pow x 2.0) (/ 4.0 PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1e+91) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = sqrt((pow(x, 2.0) * (4.0 / ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1e+91) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.sqrt((Math.pow(x, 2.0) * (4.0 / Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1e+91:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.sqrt((math.pow(x, 2.0) * (4.0 / math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e+91)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = sqrt(Float64((x ^ 2.0) * Float64(4.0 / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1e+91)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = sqrt(((x ^ 2.0) * (4.0 / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1e+91], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[x, 2.0], $MachinePrecision] * N[(4.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000008e91

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 67.0%

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

      1. *-commutative 67.0%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

   7. Simplified 67.0%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

   8. Taylor expanded in x around 0 67.0%

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

      1. *-commutative 67.0%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

      2. fabs-div 67.0%

         \[\leadsto \color{blue}{\left|\frac{x}{\sqrt{\pi} \cdot 0.5}\right|} \]

      3. rem-square-sqrt 35.1%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{x}{\sqrt{\pi} \cdot 0.5}} \cdot \sqrt{\frac{x}{\sqrt{\pi} \cdot 0.5}}}\right| \]

      4. fabs-sqr 35.1%

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

      5. rem-square-sqrt 36.5%

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

      6. *-rgt-identity 36.5%

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

      7. associate-*r/ 36.7%

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

      8. *-commutative 36.7%

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

      9. associate-/r* 36.7%

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

      10. metadata-eval 36.7%

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

   10. Simplified 36.7%

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

   if 1.00000000000000008e91 < x

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 67.0%

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

      1. *-commutative 67.0%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

   7. Simplified 67.0%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

   8. Taylor expanded in x around 0 67.0%

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

      1. *-commutative 67.0%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

      2. fabs-div 67.0%

         \[\leadsto \color{blue}{\left|\frac{x}{\sqrt{\pi} \cdot 0.5}\right|} \]

      3. rem-square-sqrt 35.1%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{x}{\sqrt{\pi} \cdot 0.5}} \cdot \sqrt{\frac{x}{\sqrt{\pi} \cdot 0.5}}}\right| \]

      4. fabs-sqr 35.1%

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

      5. rem-square-sqrt 36.5%

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

      6. *-rgt-identity 36.5%

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

      7. associate-*r/ 36.7%

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

      8. *-commutative 36.7%

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

      9. associate-/r* 36.7%

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

      10. metadata-eval 36.7%

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

   10. Simplified 36.7%

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

       1. add-sqr-sqrt 35.2%

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

       2. sqrt-unprod 55.1%

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

       3. swap-sqr 55.0%

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

       4. pow2 55.0%

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

       5. frac-times 54.8%

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

       6. metadata-eval 54.8%

          \[\leadsto \sqrt{{x}^{2} \cdot \frac{\color{blue}{4}}{\sqrt{\pi} \cdot \sqrt{\pi}}} \]

       7. add-sqr-sqrt 55.0%

          \[\leadsto \sqrt{{x}^{2} \cdot \frac{4}{\color{blue}{\pi}}} \]

   12. Applied egg-rr 55.0%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+91}:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{2} \cdot \frac{4}{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 33.8% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 67.0%

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

   1. *-commutative 67.0%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

7. Simplified 67.0%

   \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

8. Taylor expanded in x around 0 67.0%

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

   1. *-commutative 67.0%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

   2. fabs-div 67.0%

      \[\leadsto \color{blue}{\left|\frac{x}{\sqrt{\pi} \cdot 0.5}\right|} \]

   3. rem-square-sqrt 35.1%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{x}{\sqrt{\pi} \cdot 0.5}} \cdot \sqrt{\frac{x}{\sqrt{\pi} \cdot 0.5}}}\right| \]

   4. fabs-sqr 35.1%

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

   5. rem-square-sqrt 36.5%

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

   6. *-rgt-identity 36.5%

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

   7. associate-*r/ 36.7%

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

   8. *-commutative 36.7%

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

   9. associate-/r* 36.7%

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

   10. metadata-eval 36.7%

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

10. Simplified 36.7%

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

11. Final simplification 36.7%

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

Reproduce

herbie shell --seed 2024014 
(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)))))))