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

Percentage Accurate: 99.8% → 99.9%

Time: 13.3s

Alternatives: 13

Speedup: 3.6×

• 28.5% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

6. Applied egg-rr 99.9%

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

   1. fma-udef 99.9%

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

8. Applied egg-rr 99.9%

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

   1. fma-udef 99.9%

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

10. Applied egg-rr 99.9%

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

11. Final simplification 99.9%

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

Alternative 2: 67.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;x \cdot \frac{{\pi}^{-0.5}}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.4)
   (* x (/ (pow PI -0.5) (fma (pow x 2.0) -0.16666666666666666 0.5)))
   (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) <= 0.4) {
		tmp = x * (pow(((double) M_PI), -0.5) / fma(pow(x, 2.0), -0.16666666666666666, 0.5));
	} 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) <= 0.4)
		tmp = Float64(x * Float64((pi ^ -0.5) / fma((x ^ 2.0), -0.16666666666666666, 0.5)));
	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], 0.4], N[(x * N[(N[Power[Pi, -0.5], $MachinePrecision] / N[(N[Power[x, 2.0], $MachinePrecision] * -0.16666666666666666 + 0.5), $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) < 0.40000000000000002

   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 99.2%

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

   5. Taylor expanded in x around 0 98.9%

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

      1. *-un-lft-identity 98.9%

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

      2. *-commutative 98.9%

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

   7. Applied egg-rr 55.7%

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

      1. clear-num 55.7%

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

      2. associate-/r/ 56.1%

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

      3. associate-/r* 56.1%

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

      4. pow1/2 56.1%

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

      5. pow-flip 56.1%

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

      6. metadata-eval 56.1%

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

      7. fma-def 56.1%

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

   9. Applied egg-rr 56.1%

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

   if 0.40000000000000002 < (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%

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

   6. Applied egg-rr 99.9%

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

      1. fma-udef 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around 0 99.9%

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

   10. Taylor expanded in x around inf 99.9%

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

       1. +-commutative 99.9%

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

       2. associate-*r* 99.9%

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

       3. associate-*r* 99.8%

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

       4. distribute-rgt-out 99.9%

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

   12. Simplified 99.9%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;x \cdot \frac{{\pi}^{-0.5}}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 2 + 0.6666666666666666 \cdot {x}^{2}\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.4)
   (*
    x
    (/
     (fma 0.2 (pow x 4.0) (+ 2.0 (* 0.6666666666666666 (pow x 2.0))))
     (sqrt PI)))
   (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) <= 0.4) {
		tmp = x * (fma(0.2, pow(x, 4.0), (2.0 + (0.6666666666666666 * pow(x, 2.0)))) / sqrt(((double) M_PI)));
	} 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) <= 0.4)
		tmp = Float64(x * Float64(fma(0.2, (x ^ 4.0), Float64(2.0 + Float64(0.6666666666666666 * (x ^ 2.0)))) / sqrt(pi)));
	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], 0.4], N[(x * N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision] + N[(2.0 + N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $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) < 0.40000000000000002

   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 99.2%

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

   5. Taylor expanded in x around 0 98.9%

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

      1. clear-num 98.9%

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

      2. add-sqr-sqrt 99.1%

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

      3. fabs-sqr 99.1%

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

      4. add-sqr-sqrt 98.9%

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

      5. add-sqr-sqrt 53.6%

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

      6. fabs-sqr 53.6%

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

      7. add-sqr-sqrt 55.7%

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

      8. associate-/r/ 56.1%

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

   7. Applied egg-rr 56.1%

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

      1. fma-udef 99.9%

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

   9. Applied egg-rr 56.1%

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

   if 0.40000000000000002 < (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%

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

   6. Applied egg-rr 99.9%

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

      1. fma-udef 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around 0 99.9%

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

   10. Taylor expanded in x around inf 99.9%

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

       1. +-commutative 99.9%

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

       2. associate-*r* 99.9%

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

       3. associate-*r* 99.8%

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

       4. distribute-rgt-out 99.9%

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

   12. Simplified 99.9%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 2 + 0.6666666666666666 \cdot {x}^{2}\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

6. Applied egg-rr 99.9%

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

   1. fma-udef 99.9%

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

8. Applied egg-rr 99.9%

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

9. Taylor expanded in x around 0 99.4%

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

10. Final simplification 99.4%

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

Alternative 5: 67.3% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;x \cdot \frac{{\pi}^{-0.5}}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{7}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.4)
   (* x (/ (pow PI -0.5) (fma (pow x 2.0) -0.16666666666666666 0.5)))
   (fabs (* 0.047619047619047616 (* (sqrt (/ 1.0 PI)) (pow x 7.0))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.4) {
		tmp = x * (pow(((double) M_PI), -0.5) / fma(pow(x, 2.0), -0.16666666666666666, 0.5));
	} else {
		tmp = fabs((0.047619047619047616 * (sqrt((1.0 / ((double) M_PI))) * pow(x, 7.0))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.4)
		tmp = Float64(x * Float64((pi ^ -0.5) / fma((x ^ 2.0), -0.16666666666666666, 0.5)));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64(sqrt(Float64(1.0 / pi)) * (x ^ 7.0))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.4], N[(x * N[(N[Power[Pi, -0.5], $MachinePrecision] / N[(N[Power[x, 2.0], $MachinePrecision] * -0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.40000000000000002

   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 99.2%

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

   5. Taylor expanded in x around 0 98.9%

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

      1. *-un-lft-identity 98.9%

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

      2. *-commutative 98.9%

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

   7. Applied egg-rr 55.7%

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

      1. clear-num 55.7%

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

      2. associate-/r/ 56.1%

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

      3. associate-/r* 56.1%

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

      4. pow1/2 56.1%

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

      5. pow-flip 56.1%

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

      6. metadata-eval 56.1%

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

      7. fma-def 56.1%

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

   9. Applied egg-rr 56.1%

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

   if 0.40000000000000002 < (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%

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

   6. Applied egg-rr 99.9%

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

      1. fma-udef 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around 0 99.9%

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

   10. Taylor expanded in x around inf 99.0%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;x \cdot \frac{{\pi}^{-0.5}}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{7}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.3% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;x \cdot \frac{1}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{7}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.4)
   (* x (/ 1.0 (* (sqrt PI) (+ 0.5 (* (pow x 2.0) -0.16666666666666666)))))
   (fabs (* 0.047619047619047616 (* (sqrt (/ 1.0 PI)) (pow x 7.0))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.4) {
		tmp = x * (1.0 / (sqrt(((double) M_PI)) * (0.5 + (pow(x, 2.0) * -0.16666666666666666))));
	} else {
		tmp = fabs((0.047619047619047616 * (sqrt((1.0 / ((double) M_PI))) * pow(x, 7.0))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.4) {
		tmp = x * (1.0 / (Math.sqrt(Math.PI) * (0.5 + (Math.pow(x, 2.0) * -0.16666666666666666))));
	} else {
		tmp = Math.abs((0.047619047619047616 * (Math.sqrt((1.0 / Math.PI)) * Math.pow(x, 7.0))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.4:
		tmp = x * (1.0 / (math.sqrt(math.pi) * (0.5 + (math.pow(x, 2.0) * -0.16666666666666666))))
	else:
		tmp = math.fabs((0.047619047619047616 * (math.sqrt((1.0 / math.pi)) * math.pow(x, 7.0))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.4)
		tmp = Float64(x * Float64(1.0 / Float64(sqrt(pi) * Float64(0.5 + Float64((x ^ 2.0) * -0.16666666666666666)))));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64(sqrt(Float64(1.0 / pi)) * (x ^ 7.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.4)
		tmp = x * (1.0 / (sqrt(pi) * (0.5 + ((x ^ 2.0) * -0.16666666666666666))));
	else
		tmp = abs((0.047619047619047616 * (sqrt((1.0 / pi)) * (x ^ 7.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.4], N[(x * N[(1.0 / N[(N[Sqrt[Pi], $MachinePrecision] * N[(0.5 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.40000000000000002

   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 99.2%

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

   5. Taylor expanded in x around 0 98.9%

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

      1. div-inv 99.6%

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

      2. add-sqr-sqrt 53.7%

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

      3. fabs-sqr 53.7%

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

      4. add-sqr-sqrt 56.1%

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

      5. *-commutative 56.1%

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

   7. Applied egg-rr 56.1%

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

   if 0.40000000000000002 < (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%

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

   6. Applied egg-rr 99.9%

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

      1. fma-udef 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around 0 99.9%

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

   10. Taylor expanded in x around inf 99.0%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;x \cdot \frac{1}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{7}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.2% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{\sqrt{\pi} \cdot 0.5}\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log1p (expm1 (/ x (* (sqrt PI) 0.5)))))

C

double code(double x) {
	return log1p(expm1((x / (sqrt(((double) M_PI)) * 0.5))));
}

Java

public static double code(double x) {
	return Math.log1p(Math.expm1((x / (Math.sqrt(Math.PI) * 0.5))));
}

Python

def code(x):
	return math.log1p(math.expm1((x / (math.sqrt(math.pi) * 0.5))))

Julia

function code(x)
	return log1p(expm1(Float64(x / Float64(sqrt(pi) * 0.5))))
end

Wolfram

code[x_] := N[Log[1 + N[(Exp[N[(x / N[(N[Sqrt[Pi], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]] - 1), $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 70.3%

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

   1. *-commutative 70.3%

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

7. Simplified 70.3%

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

   1. add-sqr-sqrt 37.1%

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

   2. fabs-sqr 37.1%

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

   3. *-commutative 37.1%

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

   4. add-sqr-sqrt 37.1%

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

   5. fabs-sqr 37.1%

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

   6. add-sqr-sqrt 38.8%

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

   7. add-sqr-sqrt 38.6%

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

   8. log1p-expm1-u 38.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{0.5 \cdot \sqrt{\pi}}\right)\right)} \]

   9. *-commutative 38.6%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{\color{blue}{\sqrt{\pi} \cdot 0.5}}\right)\right) \]

9. Applied egg-rr 38.6%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{\sqrt{\pi} \cdot 0.5}\right)\right)} \]

10. Final simplification 38.6%

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{\sqrt{\pi} \cdot 0.5}\right)\right) \]
11. Add Preprocessing

Alternative 8: 35.4% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.66:\\ \;\;\;\;x \cdot \frac{1}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{\frac{{x}^{8} \cdot 0.04}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.66)
   (* x (/ 1.0 (* (sqrt PI) (+ 0.5 (* (pow x 2.0) -0.16666666666666666)))))
   (* x (sqrt (/ (* (pow x 8.0) 0.04) PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.66) {
		tmp = x * (1.0 / (sqrt(((double) M_PI)) * (0.5 + (pow(x, 2.0) * -0.16666666666666666))));
	} else {
		tmp = x * sqrt(((pow(x, 8.0) * 0.04) / ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.66) {
		tmp = x * (1.0 / (Math.sqrt(Math.PI) * (0.5 + (Math.pow(x, 2.0) * -0.16666666666666666))));
	} else {
		tmp = x * Math.sqrt(((Math.pow(x, 8.0) * 0.04) / Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.66:
		tmp = x * (1.0 / (math.sqrt(math.pi) * (0.5 + (math.pow(x, 2.0) * -0.16666666666666666))))
	else:
		tmp = x * math.sqrt(((math.pow(x, 8.0) * 0.04) / math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.66)
		tmp = Float64(x * Float64(1.0 / Float64(sqrt(pi) * Float64(0.5 + Float64((x ^ 2.0) * -0.16666666666666666)))));
	else
		tmp = Float64(x * sqrt(Float64(Float64((x ^ 8.0) * 0.04) / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.66)
		tmp = x * (1.0 / (sqrt(pi) * (0.5 + ((x ^ 2.0) * -0.16666666666666666))));
	else
		tmp = x * sqrt((((x ^ 8.0) * 0.04) / pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.66], N[(x * N[(1.0 / N[(N[Sqrt[Pi], $MachinePrecision] * N[(0.5 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Sqrt[N[(N[(N[Power[x, 8.0], $MachinePrecision] * 0.04), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.65999999999999992

   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 69.4%

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

      1. div-inv 69.9%

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

      2. add-sqr-sqrt 37.3%

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

      3. fabs-sqr 37.3%

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

      4. add-sqr-sqrt 39.4%

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

      5. *-commutative 39.4%

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

   7. Applied egg-rr 39.7%

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

   if 1.65999999999999992 < 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 94.2%

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

      1. clear-num 94.2%

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

      2. add-sqr-sqrt 94.4%

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

      3. fabs-sqr 94.4%

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

      4. add-sqr-sqrt 94.2%

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

      5. add-sqr-sqrt 37.3%

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

      6. fabs-sqr 37.3%

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

      7. add-sqr-sqrt 38.8%

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

      8. associate-/r/ 39.1%

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

   7. Applied egg-rr 39.1%

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

   8. Taylor expanded in x around inf 4.1%

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

      1. associate-*r* 4.1%

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

      2. *-commutative 4.1%

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

   10. Simplified 4.1%

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

       1. add-sqr-sqrt 4.1%

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

       2. sqrt-unprod 4.1%

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

       3. swap-sqr 4.1%

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

       4. add-sqr-sqrt 4.1%

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

       5. *-commutative 4.1%

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

       6. *-commutative 4.1%

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

       7. swap-sqr 4.1%

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

       8. pow-prod-up 4.1%

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

       9. metadata-eval 4.1%

          \[\leadsto \sqrt{\frac{1}{\pi} \cdot \left({x}^{\color{blue}{8}} \cdot \left(0.2 \cdot 0.2\right)\right)} \cdot x \]

       10. metadata-eval 4.1%

           \[\leadsto \sqrt{\frac{1}{\pi} \cdot \left({x}^{8} \cdot \color{blue}{0.04}\right)} \cdot x \]

   12. Applied egg-rr 4.1%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi} \cdot \left({x}^{8} \cdot 0.04\right)}} \cdot x \]
   13. Step-by-step derivation

       1. associate-*l/ 4.1%

          \[\leadsto \sqrt{\color{blue}{\frac{1 \cdot \left({x}^{8} \cdot 0.04\right)}{\pi}}} \cdot x \]

       2. *-lft-identity 4.1%

          \[\leadsto \sqrt{\frac{\color{blue}{{x}^{8} \cdot 0.04}}{\pi}} \cdot x \]

   14. Simplified 4.1%

       \[\leadsto \color{blue}{\sqrt{\frac{{x}^{8} \cdot 0.04}{\pi}}} \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.66:\\ \;\;\;\;x \cdot \frac{1}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{\frac{{x}^{8} \cdot 0.04}{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.5% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.76:\\ \;\;\;\;2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{\frac{{x}^{8} \cdot 0.04}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.76)
   (* 2.0 (* x (pow PI -0.5)))
   (* x (sqrt (/ (* (pow x 8.0) 0.04) PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.76) {
		tmp = 2.0 * (x * pow(((double) M_PI), -0.5));
	} else {
		tmp = x * sqrt(((pow(x, 8.0) * 0.04) / ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.76) {
		tmp = 2.0 * (x * Math.pow(Math.PI, -0.5));
	} else {
		tmp = x * Math.sqrt(((Math.pow(x, 8.0) * 0.04) / Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.76:
		tmp = 2.0 * (x * math.pow(math.pi, -0.5))
	else:
		tmp = x * math.sqrt(((math.pow(x, 8.0) * 0.04) / math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.76)
		tmp = Float64(2.0 * Float64(x * (pi ^ -0.5)));
	else
		tmp = Float64(x * sqrt(Float64(Float64((x ^ 8.0) * 0.04) / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.76)
		tmp = 2.0 * (x * (pi ^ -0.5));
	else
		tmp = x * sqrt((((x ^ 8.0) * 0.04) / pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.76], N[(2.0 * N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Sqrt[N[(N[(N[Power[x, 8.0], $MachinePrecision] * 0.04), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.76000000000000001

   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 70.3%

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

      1. *-commutative 70.3%

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

   7. Simplified 70.3%

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

   8. Taylor expanded in x around 0 70.3%

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

      1. *-commutative 70.3%

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

      2. fabs-div 70.3%

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

      3. rem-square-sqrt 37.1%

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

      4. fabs-sqr 37.1%

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

      5. rem-square-sqrt 38.6%

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

      6. *-rgt-identity 38.6%

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

      7. times-frac 38.6%

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

      8. metadata-eval 38.6%

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

      9. *-commutative 38.6%

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

   10. Simplified 38.6%

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

       1. clear-num 38.6%

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

       2. associate-/r/ 38.9%

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

       3. pow1/2 38.9%

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

       4. pow-flip 38.9%

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

       5. metadata-eval 38.9%

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

   12. Applied egg-rr 38.9%

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

   if 1.76000000000000001 < 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 94.2%

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

      1. clear-num 94.2%

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

      2. add-sqr-sqrt 94.4%

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

      3. fabs-sqr 94.4%

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

      4. add-sqr-sqrt 94.2%

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

      5. add-sqr-sqrt 37.3%

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

      6. fabs-sqr 37.3%

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

      7. add-sqr-sqrt 38.8%

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

      8. associate-/r/ 39.1%

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

   7. Applied egg-rr 39.1%

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

   8. Taylor expanded in x around inf 4.1%

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

      1. associate-*r* 4.1%

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

      2. *-commutative 4.1%

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

   10. Simplified 4.1%

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

       1. add-sqr-sqrt 4.1%

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

       2. sqrt-unprod 4.1%

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

       3. swap-sqr 4.1%

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

       4. add-sqr-sqrt 4.1%

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

       5. *-commutative 4.1%

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

       6. *-commutative 4.1%

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

       7. swap-sqr 4.1%

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

       8. pow-prod-up 4.1%

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

       9. metadata-eval 4.1%

          \[\leadsto \sqrt{\frac{1}{\pi} \cdot \left({x}^{\color{blue}{8}} \cdot \left(0.2 \cdot 0.2\right)\right)} \cdot x \]

       10. metadata-eval 4.1%

           \[\leadsto \sqrt{\frac{1}{\pi} \cdot \left({x}^{8} \cdot \color{blue}{0.04}\right)} \cdot x \]

   12. Applied egg-rr 4.1%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi} \cdot \left({x}^{8} \cdot 0.04\right)}} \cdot x \]
   13. Step-by-step derivation

       1. associate-*l/ 4.1%

          \[\leadsto \sqrt{\color{blue}{\frac{1 \cdot \left({x}^{8} \cdot 0.04\right)}{\pi}}} \cdot x \]

       2. *-lft-identity 4.1%

          \[\leadsto \sqrt{\frac{\color{blue}{{x}^{8} \cdot 0.04}}{\pi}} \cdot x \]

   14. Simplified 4.1%

       \[\leadsto \color{blue}{\sqrt{\frac{{x}^{8} \cdot 0.04}{\pi}}} \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.76:\\ \;\;\;\;2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{\frac{{x}^{8} \cdot 0.04}{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 34.5% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-31}:\\ \;\;\;\;2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{x}^{2}}{\pi \cdot 0.25}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e-31)
   (* 2.0 (* x (pow PI -0.5)))
   (sqrt (/ (pow x 2.0) (* PI 0.25)))))

C

double code(double x) {
	double tmp;
	if (x <= 1e-31) {
		tmp = 2.0 * (x * pow(((double) M_PI), -0.5));
	} else {
		tmp = sqrt((pow(x, 2.0) / (((double) M_PI) * 0.25)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1e-31) {
		tmp = 2.0 * (x * Math.pow(Math.PI, -0.5));
	} else {
		tmp = Math.sqrt((Math.pow(x, 2.0) / (Math.PI * 0.25)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1e-31:
		tmp = 2.0 * (x * math.pow(math.pi, -0.5))
	else:
		tmp = math.sqrt((math.pow(x, 2.0) / (math.pi * 0.25)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e-31)
		tmp = Float64(2.0 * Float64(x * (pi ^ -0.5)));
	else
		tmp = sqrt(Float64((x ^ 2.0) / Float64(pi * 0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1e-31)
		tmp = 2.0 * (x * (pi ^ -0.5));
	else
		tmp = sqrt(((x ^ 2.0) / (pi * 0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1e-31], N[(2.0 * N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[x, 2.0], $MachinePrecision] / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e-31

   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 69.9%

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

      1. *-commutative 69.9%

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

   7. Simplified 69.9%

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

   8. Taylor expanded in x around 0 69.9%

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

      1. *-commutative 69.9%

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

      2. fabs-div 69.9%

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

      3. rem-square-sqrt 35.8%

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

      4. fabs-sqr 35.8%

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

      5. rem-square-sqrt 37.4%

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

      6. *-rgt-identity 37.4%

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

      7. times-frac 37.4%

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

      8. metadata-eval 37.4%

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

      9. *-commutative 37.4%

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

   10. Simplified 37.4%

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

       1. clear-num 37.4%

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

       2. associate-/r/ 37.6%

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

       3. pow1/2 37.6%

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

       4. pow-flip 37.6%

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

       5. metadata-eval 37.6%

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

   12. Applied egg-rr 37.6%

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

   if 1e-31 < 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 99.6%

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

   5. Taylor expanded in x around 0 84.0%

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

      1. *-commutative 84.0%

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

   7. Simplified 84.0%

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

      1. add-sqr-sqrt 83.5%

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

      2. sqrt-unprod 84.0%

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

      3. div-fabs 84.0%

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

      4. div-fabs 84.0%

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

      5. sqr-abs 84.0%

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

      6. *-commutative 84.0%

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

      7. *-commutative 84.0%

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

      8. frac-times 83.8%

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

      9. pow2 83.8%

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

      10. *-commutative 83.8%

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

      11. *-commutative 83.8%

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

      12. swap-sqr 83.8%

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

      13. add-sqr-sqrt 84.0%

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

      14. metadata-eval 84.0%

          \[\leadsto \sqrt{\frac{{x}^{2}}{\pi \cdot \color{blue}{0.25}}} \]

   9. Applied egg-rr 84.0%

      \[\leadsto \color{blue}{\sqrt{\frac{{x}^{2}}{\pi \cdot 0.25}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-31}:\\ \;\;\;\;2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{x}^{2}}{\pi \cdot 0.25}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 34.5% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.76:\\ \;\;\;\;2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{5} \cdot \frac{0.2}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.76)
   (* 2.0 (* x (pow PI -0.5)))
   (* (pow x 5.0) (/ 0.2 (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.76) {
		tmp = 2.0 * (x * pow(((double) M_PI), -0.5));
	} else {
		tmp = pow(x, 5.0) * (0.2 / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.76) {
		tmp = 2.0 * (x * Math.pow(Math.PI, -0.5));
	} else {
		tmp = Math.pow(x, 5.0) * (0.2 / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.76:
		tmp = 2.0 * (x * math.pow(math.pi, -0.5))
	else:
		tmp = math.pow(x, 5.0) * (0.2 / math.sqrt(math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.76)
		tmp = Float64(2.0 * Float64(x * (pi ^ -0.5)));
	else
		tmp = Float64((x ^ 5.0) * Float64(0.2 / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.76)
		tmp = 2.0 * (x * (pi ^ -0.5));
	else
		tmp = (x ^ 5.0) * (0.2 / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.76], N[(2.0 * N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 5.0], $MachinePrecision] * N[(0.2 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.76000000000000001

   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 70.3%

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

      1. *-commutative 70.3%

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

   7. Simplified 70.3%

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

   8. Taylor expanded in x around 0 70.3%

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

      1. *-commutative 70.3%

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

      2. fabs-div 70.3%

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

      3. rem-square-sqrt 37.1%

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

      4. fabs-sqr 37.1%

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

      5. rem-square-sqrt 38.6%

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

      6. *-rgt-identity 38.6%

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

      7. times-frac 38.6%

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

      8. metadata-eval 38.6%

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

      9. *-commutative 38.6%

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

   10. Simplified 38.6%

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

       1. clear-num 38.6%

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

       2. associate-/r/ 38.9%

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

       3. pow1/2 38.9%

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

       4. pow-flip 38.9%

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

       5. metadata-eval 38.9%

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

   12. Applied egg-rr 38.9%

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

   if 1.76000000000000001 < 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 94.2%

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

      1. clear-num 94.2%

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

      2. add-sqr-sqrt 94.4%

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

      3. fabs-sqr 94.4%

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

      4. add-sqr-sqrt 94.2%

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

      5. add-sqr-sqrt 37.3%

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

      6. fabs-sqr 37.3%

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

      7. add-sqr-sqrt 38.8%

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

      8. associate-/r/ 39.1%

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

   7. Applied egg-rr 39.1%

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

   8. Taylor expanded in x around inf 4.1%

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

      1. associate-*r* 4.1%

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

      2. *-commutative 4.1%

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

   10. Simplified 4.1%

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

       1. expm1-log1p-u 4.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{4}\right)\right) \cdot x\right)\right)} \]

       2. expm1-udef 4.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{4}\right)\right) \cdot x\right)} - 1} \]

       3. *-commutative 4.0%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{4}\right)\right)}\right)} - 1 \]

       4. *-commutative 4.0%

          \[\leadsto e^{\mathsf{log1p}\left(x \cdot \color{blue}{\left(\left(0.2 \cdot {x}^{4}\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right)} - 1 \]

       5. sqrt-div 4.0%

          \[\leadsto e^{\mathsf{log1p}\left(x \cdot \left(\left(0.2 \cdot {x}^{4}\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)\right)} - 1 \]

       6. metadata-eval 4.0%

          \[\leadsto e^{\mathsf{log1p}\left(x \cdot \left(\left(0.2 \cdot {x}^{4}\right) \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)\right)} - 1 \]

       7. un-div-inv 4.0%

          \[\leadsto e^{\mathsf{log1p}\left(x \cdot \color{blue}{\frac{0.2 \cdot {x}^{4}}{\sqrt{\pi}}}\right)} - 1 \]

   12. Applied egg-rr 4.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{0.2 \cdot {x}^{4}}{\sqrt{\pi}}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 4.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \frac{0.2 \cdot {x}^{4}}{\sqrt{\pi}}\right)\right)} \]

       2. expm1-log1p 4.1%

          \[\leadsto \color{blue}{x \cdot \frac{0.2 \cdot {x}^{4}}{\sqrt{\pi}}} \]

       3. associate-*r/ 4.1%

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

       4. *-commutative 4.1%

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

       5. associate-*l* 4.1%

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

       6. pow-plus 4.1%

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

       7. metadata-eval 4.1%

          \[\leadsto \frac{0.2 \cdot {x}^{\color{blue}{5}}}{\sqrt{\pi}} \]

       8. associate-/l* 4.1%

          \[\leadsto \color{blue}{\frac{0.2}{\frac{\sqrt{\pi}}{{x}^{5}}}} \]

       9. associate-/r/ 4.1%

          \[\leadsto \color{blue}{\frac{0.2}{\sqrt{\pi}} \cdot {x}^{5}} \]

   14. Simplified 4.1%

       \[\leadsto \color{blue}{\frac{0.2}{\sqrt{\pi}} \cdot {x}^{5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.76:\\ \;\;\;\;2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{5} \cdot \frac{0.2}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 34.5% accurate, 17.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 70.3%

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

   1. *-commutative 70.3%

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

7. Simplified 70.3%

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

8. Taylor expanded in x around 0 70.3%

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

   1. *-commutative 70.3%

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

   2. fabs-div 70.3%

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

   3. rem-square-sqrt 37.1%

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

   4. fabs-sqr 37.1%

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

   5. rem-square-sqrt 38.6%

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

   6. *-rgt-identity 38.6%

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

   7. times-frac 38.6%

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

   8. metadata-eval 38.6%

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

   9. *-commutative 38.6%

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

10. Simplified 38.6%

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

    1. clear-num 38.6%

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

    2. associate-/r/ 38.9%

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

    3. pow1/2 38.9%

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

    4. pow-flip 38.9%

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

    5. metadata-eval 38.9%

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

12. Applied egg-rr 38.9%

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

13. Final simplification 38.9%

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

Alternative 13: 34.3% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 70.3%

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

   1. *-commutative 70.3%

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

7. Simplified 70.3%

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

8. Taylor expanded in x around 0 70.3%

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

   1. *-commutative 70.3%

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

   2. fabs-div 70.3%

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

   3. rem-square-sqrt 37.1%

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

   4. fabs-sqr 37.1%

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

   5. rem-square-sqrt 38.6%

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

   6. *-rgt-identity 38.6%

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

   7. times-frac 38.6%

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

   8. metadata-eval 38.6%

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

   9. *-commutative 38.6%

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

10. Simplified 38.6%

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

11. Final simplification 38.6%

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

Reproduce

herbie shell --seed 2024019 
(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)))))))