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

Percentage Accurate: 99.8% → 99.9%

Time: 11.2s

Alternatives: 12

Speedup: 2.6×

• 29.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

Alternative 2: 33.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(\left(0.047619047619047616 \cdot {x}^{6} + 0.2 \cdot {x}^{4}\right) + \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right)}{{\left({\pi}^{0.25}\right)}^{2}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (*
   x
   (+
    (+ (* 0.047619047619047616 (pow x 6.0)) (* 0.2 (pow x 4.0)))
    (+ 2.0 (* 0.6666666666666666 (pow x 2.0)))))
  (pow (pow PI 0.25) 2.0)))

C

double code(double x) {
	return (x * (((0.047619047619047616 * pow(x, 6.0)) + (0.2 * pow(x, 4.0))) + (2.0 + (0.6666666666666666 * pow(x, 2.0))))) / pow(pow(((double) M_PI), 0.25), 2.0);
}

Java

public static double code(double x) {
	return (x * (((0.047619047619047616 * Math.pow(x, 6.0)) + (0.2 * Math.pow(x, 4.0))) + (2.0 + (0.6666666666666666 * Math.pow(x, 2.0))))) / Math.pow(Math.pow(Math.PI, 0.25), 2.0);
}

Python

def code(x):
	return (x * (((0.047619047619047616 * math.pow(x, 6.0)) + (0.2 * math.pow(x, 4.0))) + (2.0 + (0.6666666666666666 * math.pow(x, 2.0))))) / math.pow(math.pow(math.pi, 0.25), 2.0)

Julia

function code(x)
	return Float64(Float64(x * Float64(Float64(Float64(0.047619047619047616 * (x ^ 6.0)) + Float64(0.2 * (x ^ 4.0))) + Float64(2.0 + Float64(0.6666666666666666 * (x ^ 2.0))))) / ((pi ^ 0.25) ^ 2.0))
end

MATLAB

function tmp = code(x)
	tmp = (x * (((0.047619047619047616 * (x ^ 6.0)) + (0.2 * (x ^ 4.0))) + (2.0 + (0.6666666666666666 * (x ^ 2.0))))) / ((pi ^ 0.25) ^ 2.0);
end

Wolfram

code[x_] := N[(N[(x * N[(N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[Pi, 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

6. Applied egg-rr 34.5%

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

   1. fma-undefine 34.5%

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

8. Applied egg-rr 34.5%

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

   1. fma-undefine 34.5%

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

10. Applied egg-rr 34.5%

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

    1. add-sqr-sqrt 34.6%

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

    2. pow2 34.6%

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

    3. pow1/2 34.6%

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

    4. sqrt-pow1 34.6%

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

    5. metadata-eval 34.6%

       \[\leadsto \frac{x \cdot \left(\left(0.6666666666666666 \cdot {x}^{2} + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)}{{\left({\pi}^{\color{blue}{0.25}}\right)}^{2}} \]

12. Applied egg-rr 34.6%

    \[\leadsto \frac{x \cdot \left(\left(0.6666666666666666 \cdot {x}^{2} + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)}{\color{blue}{{\left({\pi}^{0.25}\right)}^{2}}} \]

13. Final simplification 34.6%

    \[\leadsto \frac{x \cdot \left(\left(0.047619047619047616 \cdot {x}^{6} + 0.2 \cdot {x}^{4}\right) + \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right)}{{\left({\pi}^{0.25}\right)}^{2}} \]
14. Add Preprocessing

Alternative 3: 33.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left({x}^{2}, 0.6666666666666666, 2\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(0.047619047619047616 \cdot {x}^{6} + 0.2 \cdot {x}^{4}\right) + 0.6666666666666666 \cdot {x}^{2}\right)}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.6)
   (* x (/ (fma (pow x 2.0) 0.6666666666666666 2.0) (sqrt PI)))
   (/
    (*
     x
     (+
      (+ (* 0.047619047619047616 (pow x 6.0)) (* 0.2 (pow x 4.0)))
      (* 0.6666666666666666 (pow x 2.0))))
    (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = x * (fma(pow(x, 2.0), 0.6666666666666666, 2.0) / sqrt(((double) M_PI)));
	} else {
		tmp = (x * (((0.047619047619047616 * pow(x, 6.0)) + (0.2 * pow(x, 4.0))) + (0.6666666666666666 * pow(x, 2.0)))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.6)
		tmp = Float64(x * Float64(fma((x ^ 2.0), 0.6666666666666666, 2.0) / sqrt(pi)));
	else
		tmp = Float64(Float64(x * Float64(Float64(Float64(0.047619047619047616 * (x ^ 6.0)) + Float64(0.2 * (x ^ 4.0))) + Float64(0.6666666666666666 * (x ^ 2.0)))) / sqrt(pi));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.6], N[(x * N[(N[(N[Power[x, 2.0], $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 34.5%

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

   7. Taylor expanded in x around 0 34.8%

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

      1. associate-*r* 34.8%

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

      2. distribute-rgt-out 34.8%

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

      3. *-commutative 34.8%

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

      4. fma-define 34.8%

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

   9. Simplified 34.8%

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

       1. *-commutative 34.8%

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

       2. sqrt-div 34.8%

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

       3. metadata-eval 34.8%

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

       4. un-div-inv 34.8%

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

   11. Applied egg-rr 34.8%

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

   if 1.6000000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 34.5%

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

      1. fma-undefine 34.5%

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

   8. Applied egg-rr 34.5%

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

   9. Taylor expanded in x around inf 3.6%

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

       1. *-commutative 3.6%

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

   11. Simplified 3.6%

       \[\leadsto \frac{x \cdot \left(\color{blue}{{x}^{2} \cdot 0.6666666666666666} + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)}{\sqrt{\pi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left({x}^{2}, 0.6666666666666666, 2\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(0.047619047619047616 \cdot {x}^{6} + 0.2 \cdot {x}^{4}\right) + 0.6666666666666666 \cdot {x}^{2}\right)}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 33.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left({x}^{2}, 0.6666666666666666, 2\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \left(\frac{0.2}{{x}^{2}} + \frac{0.6666666666666666}{{x}^{4}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.6)
   (* x (/ (fma (pow x 2.0) 0.6666666666666666 2.0) (sqrt PI)))
   (*
    (pow x 7.0)
    (*
     (sqrt (/ 1.0 PI))
     (+
      0.047619047619047616
      (+ (/ 0.2 (pow x 2.0)) (/ 0.6666666666666666 (pow x 4.0))))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = x * (fma(pow(x, 2.0), 0.6666666666666666, 2.0) / sqrt(((double) M_PI)));
	} else {
		tmp = pow(x, 7.0) * (sqrt((1.0 / ((double) M_PI))) * (0.047619047619047616 + ((0.2 / pow(x, 2.0)) + (0.6666666666666666 / pow(x, 4.0)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.6)
		tmp = Float64(x * Float64(fma((x ^ 2.0), 0.6666666666666666, 2.0) / sqrt(pi)));
	else
		tmp = Float64((x ^ 7.0) * Float64(sqrt(Float64(1.0 / pi)) * Float64(0.047619047619047616 + Float64(Float64(0.2 / (x ^ 2.0)) + Float64(0.6666666666666666 / (x ^ 4.0))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.6], N[(x * N[(N[(N[Power[x, 2.0], $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 7.0], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(0.047619047619047616 + N[(N[(0.2 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 34.5%

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

   7. Taylor expanded in x around 0 34.8%

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

      1. associate-*r* 34.8%

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

      2. distribute-rgt-out 34.8%

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

      3. *-commutative 34.8%

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

      4. fma-define 34.8%

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

   9. Simplified 34.8%

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

       1. *-commutative 34.8%

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

       2. sqrt-div 34.8%

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

       3. metadata-eval 34.8%

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

       4. un-div-inv 34.8%

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

   11. Applied egg-rr 34.8%

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

   if 1.6000000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 34.5%

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

   7. Taylor expanded in x around inf 0.8%

      \[\leadsto \color{blue}{{x}^{7} \cdot \left(0.047619047619047616 \cdot \sqrt{\frac{1}{\pi}} + \left(0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.6666666666666666 \cdot \left(\frac{1}{{x}^{4}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 0.8%

         \[\leadsto {x}^{7} \cdot \left(\color{blue}{\sqrt{\frac{1}{\pi}} \cdot 0.047619047619047616} + \left(0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.6666666666666666 \cdot \left(\frac{1}{{x}^{4}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) \]

      2. associate-*r* 0.8%

         \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.047619047619047616 + \left(\color{blue}{\left(0.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \sqrt{\frac{1}{\pi}}} + 0.6666666666666666 \cdot \left(\frac{1}{{x}^{4}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) \]

      3. associate-*r* 0.8%

         \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.047619047619047616 + \left(\left(0.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.6666666666666666 \cdot \frac{1}{{x}^{4}}\right) \cdot \sqrt{\frac{1}{\pi}}}\right)\right) \]

      4. distribute-rgt-out 0.8%

         \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 0.047619047619047616 + \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot \frac{1}{{x}^{2}} + 0.6666666666666666 \cdot \frac{1}{{x}^{4}}\right)}\right) \]

      5. distribute-lft-out 0.8%

         \[\leadsto {x}^{7} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \left(0.2 \cdot \frac{1}{{x}^{2}} + 0.6666666666666666 \cdot \frac{1}{{x}^{4}}\right)\right)\right)} \]

      6. associate-*r/ 0.8%

         \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \left(\color{blue}{\frac{0.2 \cdot 1}{{x}^{2}}} + 0.6666666666666666 \cdot \frac{1}{{x}^{4}}\right)\right)\right) \]

      7. metadata-eval 0.8%

         \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \left(\frac{\color{blue}{0.2}}{{x}^{2}} + 0.6666666666666666 \cdot \frac{1}{{x}^{4}}\right)\right)\right) \]

      8. associate-*r/ 0.8%

         \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \left(\frac{0.2}{{x}^{2}} + \color{blue}{\frac{0.6666666666666666 \cdot 1}{{x}^{4}}}\right)\right)\right) \]

      9. metadata-eval 0.8%

         \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \left(\frac{0.2}{{x}^{2}} + \frac{\color{blue}{0.6666666666666666}}{{x}^{4}}\right)\right)\right) \]

   9. Simplified 0.8%

      \[\leadsto \color{blue}{{x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \left(\frac{0.2}{{x}^{2}} + \frac{0.6666666666666666}{{x}^{4}}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 33.5% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

6. Applied egg-rr 34.5%

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

   1. fma-undefine 34.5%

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

8. Applied egg-rr 34.5%

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

   1. fma-undefine 34.5%

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

10. Applied egg-rr 34.5%

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

11. Final simplification 34.5%

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

Alternative 6: 33.6% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left({x}^{2}, 0.6666666666666666, 2\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (* x (/ (fma (pow x 2.0) 0.6666666666666666 2.0) (sqrt PI)))
   (*
    (pow x 7.0)
    (* (sqrt (/ 1.0 PI)) (+ 0.047619047619047616 (/ 0.2 (pow x 2.0)))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (fma(pow(x, 2.0), 0.6666666666666666, 2.0) / sqrt(((double) M_PI)));
	} else {
		tmp = pow(x, 7.0) * (sqrt((1.0 / ((double) M_PI))) * (0.047619047619047616 + (0.2 / pow(x, 2.0))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64(x * Float64(fma((x ^ 2.0), 0.6666666666666666, 2.0) / sqrt(pi)));
	else
		tmp = Float64((x ^ 7.0) * Float64(sqrt(Float64(1.0 / pi)) * Float64(0.047619047619047616 + Float64(0.2 / (x ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(x * N[(N[(N[Power[x, 2.0], $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 7.0], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(0.047619047619047616 + N[(0.2 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 34.5%

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

   7. Taylor expanded in x around 0 34.8%

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

      1. associate-*r* 34.8%

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

      2. distribute-rgt-out 34.8%

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

      3. *-commutative 34.8%

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

      4. fma-define 34.8%

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

   9. Simplified 34.8%

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

       1. *-commutative 34.8%

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

       2. sqrt-div 34.8%

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

       3. metadata-eval 34.8%

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

       4. un-div-inv 34.8%

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

   11. Applied egg-rr 34.8%

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

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 34.5%

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

      1. fma-undefine 34.5%

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

   8. Applied egg-rr 34.5%

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

   9. Taylor expanded in x around inf 1.6%

      \[\leadsto \color{blue}{{x}^{7} \cdot \left(0.047619047619047616 \cdot \sqrt{\frac{1}{\pi}} + 0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative 1.6%

          \[\leadsto {x}^{7} \cdot \color{blue}{\left(0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.047619047619047616 \cdot \sqrt{\frac{1}{\pi}}\right)} \]

       2. associate-*r* 1.6%

          \[\leadsto {x}^{7} \cdot \left(\color{blue}{\left(0.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \sqrt{\frac{1}{\pi}}} + 0.047619047619047616 \cdot \sqrt{\frac{1}{\pi}}\right) \]

       3. distribute-rgt-out 1.6%

          \[\leadsto {x}^{7} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot \frac{1}{{x}^{2}} + 0.047619047619047616\right)\right)} \]

       4. associate-*r/ 1.6%

          \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\frac{0.2 \cdot 1}{{x}^{2}}} + 0.047619047619047616\right)\right) \]

       5. metadata-eval 1.6%

          \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{\color{blue}{0.2}}{{x}^{2}} + 0.047619047619047616\right)\right) \]

   11. Simplified 1.6%

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

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left({x}^{2}, 0.6666666666666666, 2\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 33.6% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[(x * N[(N[(N[Power[x, 2.0], $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 / N[(N[Sqrt[Pi], $MachinePrecision] / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 34.5%

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

   7. Taylor expanded in x around 0 34.8%

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

      1. associate-*r* 34.8%

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

      2. distribute-rgt-out 34.8%

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

      3. *-commutative 34.8%

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

      4. fma-define 34.8%

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

   9. Simplified 34.8%

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

       1. *-commutative 34.8%

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

       2. sqrt-div 34.8%

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

       3. metadata-eval 34.8%

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

       4. un-div-inv 34.8%

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

   11. Applied egg-rr 34.8%

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

   if 2.2000000000000002 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 34.5%

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

      1. fma-undefine 34.5%

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

   8. Applied egg-rr 34.5%

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

   9. Taylor expanded in x around inf 3.5%

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

       1. associate-*r* 3.5%

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

       2. *-commutative 3.5%

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

   11. Simplified 3.5%

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

       1. *-commutative 3.5%

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

       2. sqrt-div 3.5%

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

       3. metadata-eval 3.5%

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

       4. div-inv 3.5%

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

   13. Applied egg-rr 3.5%

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

       1. associate-*l/ 3.5%

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

       2. associate-/r/ 3.5%

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

   15. Simplified 3.5%

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

Alternative 8: 33.6% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616}{\frac{\sqrt{\pi}}{{x}^{7}}}\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = x * (Math.sqrt((1.0 / Math.PI)) * (2.0 + (0.6666666666666666 * Math.pow(x, 2.0))));
	} else {
		tmp = 0.047619047619047616 / (Math.sqrt(Math.PI) / Math.pow(x, 7.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.2:
		tmp = x * (math.sqrt((1.0 / math.pi)) * (2.0 + (0.6666666666666666 * math.pow(x, 2.0))))
	else:
		tmp = 0.047619047619047616 / (math.sqrt(math.pi) / math.pow(x, 7.0))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.2)
		tmp = x * (sqrt((1.0 / pi)) * (2.0 + (0.6666666666666666 * (x ^ 2.0))));
	else
		tmp = 0.047619047619047616 / (sqrt(pi) / (x ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[(x * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(2.0 + N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 / N[(N[Sqrt[Pi], $MachinePrecision] / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 34.5%

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

   7. Taylor expanded in x around 0 34.8%

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

      1. associate-*r* 34.8%

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

      2. distribute-rgt-out 34.8%

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

      3. *-commutative 34.8%

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

      4. fma-define 34.8%

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

   9. Simplified 34.8%

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

       1. fma-undefine 34.8%

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

   11. Applied egg-rr 34.8%

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

   if 2.2000000000000002 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 34.5%

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

      1. fma-undefine 34.5%

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

   8. Applied egg-rr 34.5%

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

   9. Taylor expanded in x around inf 3.5%

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

       1. associate-*r* 3.5%

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

       2. *-commutative 3.5%

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

   11. Simplified 3.5%

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

       1. *-commutative 3.5%

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

       2. sqrt-div 3.5%

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

       3. metadata-eval 3.5%

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

       4. div-inv 3.5%

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

   13. Applied egg-rr 3.5%

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

       1. associate-*l/ 3.5%

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

       2. associate-/r/ 3.5%

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

   15. Simplified 3.5%

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

4. Final simplification 34.8%

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

Alternative 9: 33.5% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616}{\frac{\sqrt{\pi}}{{x}^{7}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.9)
   (* (sqrt (/ 1.0 PI)) (* x 2.0))
   (/ 0.047619047619047616 (/ (sqrt PI) (pow x 7.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = sqrt((1.0 / ((double) M_PI))) * (x * 2.0);
	} else {
		tmp = 0.047619047619047616 / (sqrt(((double) M_PI)) / pow(x, 7.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = Math.sqrt((1.0 / Math.PI)) * (x * 2.0);
	} else {
		tmp = 0.047619047619047616 / (Math.sqrt(Math.PI) / Math.pow(x, 7.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.9:
		tmp = math.sqrt((1.0 / math.pi)) * (x * 2.0)
	else:
		tmp = 0.047619047619047616 / (math.sqrt(math.pi) / math.pow(x, 7.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.9)
		tmp = Float64(sqrt(Float64(1.0 / pi)) * Float64(x * 2.0));
	else
		tmp = Float64(0.047619047619047616 / Float64(sqrt(pi) / (x ^ 7.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.9)
		tmp = sqrt((1.0 / pi)) * (x * 2.0);
	else
		tmp = 0.047619047619047616 / (sqrt(pi) / (x ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.9], N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 / N[(N[Sqrt[Pi], $MachinePrecision] / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 34.5%

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

      1. fma-undefine 34.5%

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

   8. Applied egg-rr 34.5%

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

   9. Taylor expanded in x around 0 34.8%

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

       1. associate-*r* 34.8%

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

   11. Simplified 34.8%

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

   if 1.8999999999999999 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 34.5%

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

      1. fma-undefine 34.5%

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

   8. Applied egg-rr 34.5%

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

   9. Taylor expanded in x around inf 3.5%

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

       1. associate-*r* 3.5%

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

       2. *-commutative 3.5%

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

   11. Simplified 3.5%

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

       1. *-commutative 3.5%

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

       2. sqrt-div 3.5%

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

       3. metadata-eval 3.5%

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

       4. div-inv 3.5%

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

   13. Applied egg-rr 3.5%

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

       1. associate-*l/ 3.5%

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

       2. associate-/r/ 3.5%

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

   15. Simplified 3.5%

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

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616}{\frac{\sqrt{\pi}}{{x}^{7}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 33.5% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{14} \cdot \frac{0.0022675736961451248}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.9)
   (* (sqrt (/ 1.0 PI)) (* x 2.0))
   (sqrt (* (pow x 14.0) (/ 0.0022675736961451248 PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = sqrt((1.0 / ((double) M_PI))) * (x * 2.0);
	} else {
		tmp = sqrt((pow(x, 14.0) * (0.0022675736961451248 / ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = Math.sqrt((1.0 / Math.PI)) * (x * 2.0);
	} else {
		tmp = Math.sqrt((Math.pow(x, 14.0) * (0.0022675736961451248 / Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.9:
		tmp = math.sqrt((1.0 / math.pi)) * (x * 2.0)
	else:
		tmp = math.sqrt((math.pow(x, 14.0) * (0.0022675736961451248 / math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.9)
		tmp = Float64(sqrt(Float64(1.0 / pi)) * Float64(x * 2.0));
	else
		tmp = sqrt(Float64((x ^ 14.0) * Float64(0.0022675736961451248 / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.9)
		tmp = sqrt((1.0 / pi)) * (x * 2.0);
	else
		tmp = sqrt(((x ^ 14.0) * (0.0022675736961451248 / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.9], N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[x, 14.0], $MachinePrecision] * N[(0.0022675736961451248 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 34.5%

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

      1. fma-undefine 34.5%

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

   8. Applied egg-rr 34.5%

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

   9. Taylor expanded in x around 0 34.8%

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

       1. associate-*r* 34.8%

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

   11. Simplified 34.8%

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

   if 1.8999999999999999 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 34.5%

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

      1. fma-undefine 34.5%

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

   8. Applied egg-rr 34.5%

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

   9. Taylor expanded in x around inf 3.5%

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

       1. associate-*r* 3.5%

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

       2. *-commutative 3.5%

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

   11. Simplified 3.5%

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

       1. add-sqr-sqrt 3.3%

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

       2. sqrt-unprod 34.5%

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

       3. *-commutative 34.5%

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

       4. sqrt-div 34.5%

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

       5. metadata-eval 34.5%

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

       6. div-inv 34.5%

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

       7. *-commutative 34.5%

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

       8. sqrt-div 34.5%

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

       9. metadata-eval 34.5%

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

       10. div-inv 34.5%

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

       11. frac-times 34.5%

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

   13. Applied egg-rr 34.5%

       \[\leadsto \color{blue}{\sqrt{\frac{{x}^{14} \cdot 0.0022675736961451248}{\pi}}} \]
   14. Step-by-step derivation

       1. associate-/l* 34.6%

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

   15. Simplified 34.6%

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

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{14} \cdot \frac{0.0022675736961451248}{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 33.5% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (sqrt (/ 1.0 PI)) (* x 2.0)))

C

double code(double x) {
	return sqrt((1.0 / ((double) M_PI))) * (x * 2.0);
}

Java

public static double code(double x) {
	return Math.sqrt((1.0 / Math.PI)) * (x * 2.0);
}

Python

def code(x):
	return math.sqrt((1.0 / math.pi)) * (x * 2.0)

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 / pi)) * Float64(x * 2.0))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 / pi)) * (x * 2.0);
end

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

6. Applied egg-rr 34.5%

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

   1. fma-undefine 34.5%

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

8. Applied egg-rr 34.5%

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

9. Taylor expanded in x around 0 34.8%

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

    1. associate-*r* 34.8%

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

11. Simplified 34.8%

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

12. Final simplification 34.8%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2\right) \]
13. Add Preprocessing

Alternative 12: 33.3% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

6. Applied egg-rr 34.5%

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

   1. fma-undefine 34.5%

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

8. Applied egg-rr 34.5%

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

9. Taylor expanded in x around 0 34.8%

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

    1. associate-*r* 34.8%

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

11. Simplified 34.8%

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

    1. sqrt-div 34.8%

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

    2. metadata-eval 34.8%

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

    3. un-div-inv 34.6%

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

    4. *-commutative 34.6%

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

13. Applied egg-rr 34.6%

    \[\leadsto \color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024089 
(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)))))))