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

Percentage Accurate: 99.8% → 99.8%

Time: 11.6s

Alternatives: 11

Speedup: 1.5×

• 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: 99.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot \left(x \cdot x\right)\\ t_1 := \left|x\right| \cdot \left(\left|x\right| \cdot t_0\right)\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + 0.6666666666666666 \cdot t_0\right) + 0.2 \cdot t_1\right) + 0.047619047619047616 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot t_1\right)\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (* x x))) (t_1 (* (fabs x) (* (fabs x) t_0))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* 0.6666666666666666 t_0)) (* 0.2 t_1))
      (* 0.047619047619047616 (* (fabs x) (* (fabs x) t_1))))))))

C

double code(double x) {
	double t_0 = fabs(x) * (x * x);
	double t_1 = fabs(x) * (fabs(x) * t_0);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (fabs(x) * (fabs(x) * t_1))))));
}

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * (x * x);
	double t_1 = Math.abs(x) * (Math.abs(x) * t_0);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (Math.abs(x) * (Math.abs(x) * t_1))))));
}

Python

def code(x):
	t_0 = math.fabs(x) * (x * x)
	t_1 = math.fabs(x) * (math.fabs(x) * t_0)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (math.fabs(x) * (math.fabs(x) * t_1))))))

Julia

function code(x)
	t_0 = Float64(abs(x) * Float64(x * x))
	t_1 = Float64(abs(x) * Float64(abs(x) * t_0))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(0.6666666666666666 * t_0)) + Float64(0.2 * t_1)) + Float64(0.047619047619047616 * Float64(abs(x) * Float64(abs(x) * t_1))))))
end

MATLAB

function tmp = code(x)
	t_0 = abs(x) * (x * x);
	t_1 = abs(x) * (abs(x) * t_0);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (abs(x) * (abs(x) * t_1))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$0), $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[(0.6666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.2 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 34.9% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return (x * Math.pow(Math.PI, -0.5)) * (((0.2 * Math.pow(x, 4.0)) + (0.047619047619047616 * Math.pow(x, 6.0))) + (2.0 + (0.6666666666666666 * Math.pow(x, 2.0))));
}

Python

def code(x):
	return (x * math.pow(math.pi, -0.5)) * (((0.2 * math.pow(x, 4.0)) + (0.047619047619047616 * math.pow(x, 6.0))) + (2.0 + (0.6666666666666666 * math.pow(x, 2.0))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = (x * (pi ^ -0.5)) * (((0.2 * (x ^ 4.0)) + (0.047619047619047616 * (x ^ 6.0))) + (2.0 + (0.6666666666666666 * (x ^ 2.0))));
end

Wolfram

code[x_] := N[(N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 99.4%

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

   1. fabs-neg 99.4%

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

   2. associate-*r/ 99.4%

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

   3. *-rgt-identity 99.4%

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

   4. +-commutative 99.4%

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

   5. fabs-div 99.4%

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

   6. associate-/r/ 99.4%

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

   7. associate-*l/ 99.4%

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

   8. associate-*r/ 99.8%

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

   9. distribute-lft-neg-in 99.8%

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

   10. fabs-neg 99.8%

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

7. Simplified 36.0%

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

   1. fma-udef 36.0%

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

9. Applied egg-rr 36.0%

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

    1. fma-udef 35.7%

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

11. Applied egg-rr 36.0%

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

    1. clear-num 36.0%

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

    2. associate-/r/ 36.2%

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

    3. pow1/2 36.2%

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

    4. pow-flip 36.2%

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

    5. metadata-eval 36.2%

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

13. Applied egg-rr 36.2%

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

14. Final simplification 36.2%

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

Alternative 3: 34.7% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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 99.4%

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

   1. fabs-neg 99.4%

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

   2. associate-*r/ 99.4%

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

   3. *-rgt-identity 99.4%

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

   4. +-commutative 99.4%

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

   5. fabs-div 99.4%

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

   6. associate-/r/ 99.4%

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

   7. associate-*l/ 99.4%

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

   8. associate-*r/ 99.8%

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

   9. distribute-lft-neg-in 99.8%

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

   10. fabs-neg 99.8%

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

7. Simplified 36.0%

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

   1. fma-udef 36.0%

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

9. Applied egg-rr 36.0%

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

    1. fma-udef 35.7%

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

11. Applied egg-rr 36.0%

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

12. Final simplification 36.0%

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

Alternative 4: 34.4% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * 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]), $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 99.4%

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

   1. fabs-neg 99.4%

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

   2. associate-*r/ 99.4%

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

   3. *-rgt-identity 99.4%

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

   4. +-commutative 99.4%

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

   5. fabs-div 99.4%

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

   6. associate-/r/ 99.4%

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

   7. associate-*l/ 99.4%

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

   8. associate-*r/ 99.8%

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

   9. distribute-lft-neg-in 99.8%

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

   10. fabs-neg 99.8%

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

7. Simplified 36.0%

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

8. Taylor expanded in x around 0 35.7%

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

   1. fma-udef 35.7%

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

10. Applied egg-rr 35.7%

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

11. Final simplification 35.7%

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

Alternative 5: 34.4% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Taylor expanded in x around 0 98.7%

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

6. Taylor expanded in x around inf 98.5%

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

7. Taylor expanded in x around 0 98.4%

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

   1. rem-square-sqrt 34.5%

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

   2. fabs-sqr 34.5%

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

   3. rem-square-sqrt 35.7%

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

   4. associate-*r/ 35.7%

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

   5. *-rgt-identity 35.7%

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

   6. +-commutative 35.7%

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

   7. fma-udef 35.7%

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

   8. rem-square-sqrt 35.8%

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

   9. fabs-sqr 35.8%

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

   10. rem-square-sqrt 35.7%

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

   11. associate-/l* 35.7%

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

9. Simplified 35.7%

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

10. Final simplification 35.7%

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

Alternative 6: 34.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left(2 \cdot x\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (if (<= x 1.85)
     (* (* 2.0 x) t_0)
     (* t_0 (* 0.047619047619047616 (pow x 7.0))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double tmp;
	if (x <= 1.85) {
		tmp = (2.0 * x) * t_0;
	} else {
		tmp = t_0 * (0.047619047619047616 * pow(x, 7.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	double tmp;
	if (x <= 1.85) {
		tmp = (2.0 * x) * t_0;
	} else {
		tmp = t_0 * (0.047619047619047616 * Math.pow(x, 7.0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / math.pi))
	tmp = 0
	if x <= 1.85:
		tmp = (2.0 * x) * t_0
	else:
		tmp = t_0 * (0.047619047619047616 * math.pow(x, 7.0))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / pi))
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64(Float64(2.0 * x) * t_0);
	else
		tmp = Float64(t_0 * Float64(0.047619047619047616 * (x ^ 7.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 / pi));
	tmp = 0.0;
	if (x <= 1.85)
		tmp = (2.0 * x) * t_0;
	else
		tmp = t_0 * (0.047619047619047616 * (x ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.85], N[(N[(2.0 * x), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.4%

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

      1. fabs-neg 99.4%

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

      2. associate-*r/ 99.4%

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

      3. *-rgt-identity 99.4%

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

      4. +-commutative 99.4%

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

      5. fabs-div 99.4%

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

      6. associate-/r/ 99.4%

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

      7. associate-*l/ 99.4%

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

      8. associate-*r/ 99.8%

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

      9. distribute-lft-neg-in 99.8%

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

      10. fabs-neg 99.8%

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

   7. Simplified 36.0%

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

   8. Taylor expanded in x around 0 35.7%

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

   9. Taylor expanded in x around inf 35.7%

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

   10. Taylor expanded in x around 0 36.0%

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

       1. associate-*r* 36.0%

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

   12. Simplified 36.0%

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

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.4%

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

      1. fabs-neg 99.4%

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

      2. associate-*r/ 99.4%

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

      3. *-rgt-identity 99.4%

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

      4. +-commutative 99.4%

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

      5. fabs-div 99.4%

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

      6. associate-/r/ 99.4%

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

      7. associate-*l/ 99.4%

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

      8. associate-*r/ 99.8%

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

      9. distribute-lft-neg-in 99.8%

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

      10. fabs-neg 99.8%

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

   7. Simplified 36.0%

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

   8. Taylor expanded in x around 0 35.7%

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

   9. Taylor expanded in x around inf 35.7%

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

   10. Taylor expanded in x around inf 3.5%

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

       1. associate-*r* 3.5%

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

   12. Simplified 3.5%

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

4. Final simplification 36.0%

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

Alternative 7: 34.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (* (* 2.0 x) (sqrt (/ 1.0 PI)))
   (/ (* x (* 0.047619047619047616 (pow x 6.0))) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = (2.0 * x) * sqrt((1.0 / ((double) M_PI)));
	} else {
		tmp = (x * (0.047619047619047616 * pow(x, 6.0))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = (2.0 * x) * Math.sqrt((1.0 / Math.PI));
	} else {
		tmp = (x * (0.047619047619047616 * Math.pow(x, 6.0))) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = (2.0 * x) * math.sqrt((1.0 / math.pi))
	else:
		tmp = (x * (0.047619047619047616 * math.pow(x, 6.0))) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64(Float64(2.0 * x) * sqrt(Float64(1.0 / pi)));
	else
		tmp = Float64(Float64(x * Float64(0.047619047619047616 * (x ^ 6.0))) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = (2.0 * x) * sqrt((1.0 / pi));
	else
		tmp = (x * (0.047619047619047616 * (x ^ 6.0))) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(N[(2.0 * x), $MachinePrecision] * N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.4%

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

      1. fabs-neg 99.4%

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

      2. associate-*r/ 99.4%

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

      3. *-rgt-identity 99.4%

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

      4. +-commutative 99.4%

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

      5. fabs-div 99.4%

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

      6. associate-/r/ 99.4%

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

      7. associate-*l/ 99.4%

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

      8. associate-*r/ 99.8%

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

      9. distribute-lft-neg-in 99.8%

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

      10. fabs-neg 99.8%

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

   7. Simplified 36.0%

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

   8. Taylor expanded in x around 0 35.7%

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

   9. Taylor expanded in x around inf 35.7%

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

   10. Taylor expanded in x around 0 36.0%

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

       1. associate-*r* 36.0%

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

   12. Simplified 36.0%

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

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 98.7%

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

   6. Taylor expanded in x around inf 98.5%

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

   7. Taylor expanded in x around 0 98.4%

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

      1. rem-square-sqrt 34.5%

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

      2. fabs-sqr 34.5%

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

      3. rem-square-sqrt 35.7%

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

      4. associate-*r/ 35.7%

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

      5. *-rgt-identity 35.7%

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

      6. +-commutative 35.7%

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

      7. fma-udef 35.7%

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

      8. rem-square-sqrt 35.8%

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

      9. fabs-sqr 35.8%

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

      10. rem-square-sqrt 35.7%

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

      11. associate-/l* 35.7%

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

   9. Simplified 35.7%

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

   10. Taylor expanded in x around inf 3.5%

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

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.4% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 99.4%

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

   1. fabs-neg 99.4%

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

   2. associate-*r/ 99.4%

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

   3. *-rgt-identity 99.4%

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

   4. +-commutative 99.4%

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

   5. fabs-div 99.4%

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

   6. associate-/r/ 99.4%

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

   7. associate-*l/ 99.4%

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

   8. associate-*r/ 99.8%

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

   9. distribute-lft-neg-in 99.8%

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

   10. fabs-neg 99.8%

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

7. Simplified 36.0%

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

8. Taylor expanded in x around 0 35.7%

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

9. Taylor expanded in x around inf 35.7%

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

10. Final simplification 35.7%

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

Alternative 9: 34.7% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = (2.0 * x) * Math.sqrt((1.0 / Math.PI));
	} else {
		tmp = (0.047619047619047616 * Math.pow(x, 7.0)) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = (2.0 * x) * math.sqrt((1.0 / math.pi))
	else:
		tmp = (0.047619047619047616 * math.pow(x, 7.0)) / math.sqrt(math.pi)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = (2.0 * x) * sqrt((1.0 / pi));
	else
		tmp = (0.047619047619047616 * (x ^ 7.0)) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.4%

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

      1. fabs-neg 99.4%

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

      2. associate-*r/ 99.4%

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

      3. *-rgt-identity 99.4%

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

      4. +-commutative 99.4%

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

      5. fabs-div 99.4%

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

      6. associate-/r/ 99.4%

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

      7. associate-*l/ 99.4%

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

      8. associate-*r/ 99.8%

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

      9. distribute-lft-neg-in 99.8%

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

      10. fabs-neg 99.8%

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

   7. Simplified 36.0%

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

   8. Taylor expanded in x around 0 35.7%

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

   9. Taylor expanded in x around inf 35.7%

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

   10. Taylor expanded in x around 0 36.0%

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

       1. associate-*r* 36.0%

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

   12. Simplified 36.0%

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

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 98.7%

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

   6. Taylor expanded in x around inf 98.5%

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

   7. Taylor expanded in x around 0 98.4%

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

      1. rem-square-sqrt 34.5%

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

      2. fabs-sqr 34.5%

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

      3. rem-square-sqrt 35.7%

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

      4. associate-*r/ 35.7%

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

      5. *-rgt-identity 35.7%

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

      6. +-commutative 35.7%

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

      7. fma-udef 35.7%

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

      8. rem-square-sqrt 35.8%

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

      9. fabs-sqr 35.8%

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

      10. rem-square-sqrt 35.7%

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

      11. associate-/l* 35.7%

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

   9. Simplified 35.7%

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

   10. Taylor expanded in x around inf 3.5%

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

4. Final simplification 36.0%

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

Alternative 10: 34.7% accurate, 9.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(2.0 * x), $MachinePrecision] * N[Sqrt[N[(1.0 / 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 99.4%

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

   1. fabs-neg 99.4%

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

   2. associate-*r/ 99.4%

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

   3. *-rgt-identity 99.4%

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

   4. +-commutative 99.4%

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

   5. fabs-div 99.4%

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

   6. associate-/r/ 99.4%

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

   7. associate-*l/ 99.4%

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

   8. associate-*r/ 99.8%

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

   9. distribute-lft-neg-in 99.8%

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

   10. fabs-neg 99.8%

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

7. Simplified 36.0%

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

8. Taylor expanded in x around 0 35.7%

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

9. Taylor expanded in x around inf 35.7%

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

10. Taylor expanded in x around 0 36.0%

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

    1. associate-*r* 36.0%

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

12. Simplified 36.0%

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

13. Final simplification 36.0%

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

Alternative 11: 34.5% accurate, 9.5× speedup

Math

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

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(Float64(2.0 * x) / sqrt(pi))
end

MATLAB

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

Wolfram

code[x_] := N[(N[(2.0 * x), $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 99.4%

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

5. Taylor expanded in x around 0 98.7%

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

6. Taylor expanded in x around inf 98.5%

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

7. Taylor expanded in x around 0 98.4%

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

   1. rem-square-sqrt 34.5%

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

   2. fabs-sqr 34.5%

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

   3. rem-square-sqrt 35.7%

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

   4. associate-*r/ 35.7%

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

   5. *-rgt-identity 35.7%

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

   6. +-commutative 35.7%

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

   7. fma-udef 35.7%

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

   8. rem-square-sqrt 35.8%

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

   9. fabs-sqr 35.8%

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

   10. rem-square-sqrt 35.7%

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

   11. associate-/l* 35.7%

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

9. Simplified 35.7%

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

10. Taylor expanded in x around 0 35.8%

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

11. Final simplification 35.8%

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

Reproduce

herbie shell --seed 2024013 
(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)))))))