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

Percentage Accurate: 99.8% → 99.9%

Time: 12.4s

Alternatives: 7

Speedup: 3.0×

• 28.7% 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, 2.3× speedup

Math

\[\begin{array}{l} \\ \left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left({x}^{4}, 0.2, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fabs x)
  (fabs
   (*
    (sqrt (/ 1.0 PI))
    (+
     (fma 0.6666666666666666 (pow x 2.0) 2.0)
     (fma (pow x 4.0) 0.2 (* 0.047619047619047616 (pow x 6.0))))))))

C

double code(double x) {
	return fabs(x) * fabs((sqrt((1.0 / ((double) M_PI))) * (fma(0.6666666666666666, pow(x, 2.0), 2.0) + fma(pow(x, 4.0), 0.2, (0.047619047619047616 * pow(x, 6.0))))));
}

Julia

function code(x)
	return Float64(abs(x) * abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(fma(0.6666666666666666, (x ^ 2.0), 2.0) + fma((x ^ 4.0), 0.2, Float64(0.047619047619047616 * (x ^ 6.0)))))))
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 99.8%

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

   1. +-commutative 99.8%

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

   2. +-commutative 99.8%

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

   3. associate-+l+ 99.8%

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

   4. fma-undefine 99.8%

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

   5. associate-+l+ 99.9%

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

   6. +-commutative 99.9%

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

   7. fma-define 99.9%

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

   8. fma-undefine 99.9%

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

   9. *-commutative 99.9%

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

   10. fma-define 99.9%

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

7. Simplified 99.9%

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

Alternative 2: 99.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. pow1/2 99.8%

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

   3. inv-pow 99.8%

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

   4. pow-pow 99.8%

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

   5. metadata-eval 99.8%

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

7. Applied egg-rr 99.8%

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

   1. *-lft-identity 99.8%

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

9. Simplified 99.8%

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

10. Final simplification 99.8%

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

Alternative 3: 99.0% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 98.6%

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

6. Taylor expanded in x around 0 98.6%

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

7. Final simplification 98.6%

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

Alternative 4: 99.0% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 98.6%

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

   1. mul-fabs 98.6%

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

   2. +-commutative 98.6%

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

7. Applied egg-rr 98.6%

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

Alternative 5: 98.8% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 98.7%

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

      1. *-commutative 98.7%

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

      2. associate-*l* 98.7%

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

      3. *-commutative 98.7%

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

   7. Simplified 98.7%

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

      1. *-un-lft-identity 99.8%

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

      2. pow1/2 99.8%

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

      3. inv-pow 99.8%

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

      4. pow-pow 99.8%

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

      5. metadata-eval 99.8%

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

   9. Applied egg-rr 98.7%

      \[\leadsto \left|\color{blue}{\left(1 \cdot {\pi}^{-0.5}\right)} \cdot \left(2 \cdot \left|x\right|\right)\right| \]
   10. Step-by-step derivation

       1. *-lft-identity 99.8%

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

   11. Simplified 98.7%

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

   12. Taylor expanded in x around 0 98.7%

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

       1. *-commutative 98.7%

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

       2. *-commutative 98.7%

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

       3. rem-exp-log 98.7%

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

       4. exp-neg 98.7%

          \[\leadsto \left|2 \cdot \left(\sqrt{\color{blue}{e^{-\log \pi}}} \cdot \left|x\right|\right)\right| \]

       5. unpow1/2 98.7%

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

       6. exp-prod 98.7%

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

       7. distribute-lft-neg-out 98.7%

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

       8. exp-neg 98.7%

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

       9. associate-*l/ 98.3%

          \[\leadsto \left|2 \cdot \color{blue}{\frac{1 \cdot \left|x\right|}{e^{\log \pi \cdot 0.5}}}\right| \]

       10. *-lft-identity 98.3%

           \[\leadsto \left|2 \cdot \frac{\color{blue}{\left|x\right|}}{e^{\log \pi \cdot 0.5}}\right| \]

       11. exp-to-pow 98.0%

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

       12. unpow1/2 98.0%

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

   14. Simplified 98.0%

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

       1. fabs-mul 98.0%

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

       2. metadata-eval 98.0%

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

   16. Applied egg-rr 98.0%

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

       1. rem-square-sqrt 97.7%

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

       2. fabs-sqr 97.7%

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

       3. rem-square-sqrt 98.0%

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

       4. associate-*r/ 98.0%

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

       5. associate-*l/ 98.7%

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

       6. *-commutative 98.7%

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

   18. Simplified 98.7%

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

   if 2 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 97.6%

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

      1. associate-*r* 97.6%

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

      2. *-commutative 97.6%

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

      3. *-commutative 97.6%

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

   7. Simplified 97.6%

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

   8. Taylor expanded in x around 0 97.6%

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

      1. *-commutative 97.6%

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

      2. metadata-eval 97.6%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\left(\left|x\right| \cdot {x}^{\color{blue}{\left(2 \cdot 3\right)}}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      3. pow-sqr 97.5%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\left(\left|x\right| \cdot \color{blue}{\left({x}^{3} \cdot {x}^{3}\right)}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      4. cube-prod 97.6%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\left(\left|x\right| \cdot \color{blue}{{\left(x \cdot x\right)}^{3}}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      5. unpow2 97.6%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\left(\left|x\right| \cdot {\color{blue}{\left({x}^{2}\right)}}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      6. cube-mult 97.5%

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

      7. pow-sqr 97.6%

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

      8. metadata-eval 97.6%

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

      9. associate-*l* 97.6%

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

      10. unpow-1 97.6%

          \[\leadsto \left|0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot {x}^{2}\right) \cdot {x}^{4}\right) \cdot \sqrt{\color{blue}{{\pi}^{-1}}}\right)\right| \]

      11. metadata-eval 97.6%

          \[\leadsto \left|0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot {x}^{2}\right) \cdot {x}^{4}\right) \cdot \sqrt{{\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right)\right| \]

      12. pow-sqr 97.6%

          \[\leadsto \left|0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot {x}^{2}\right) \cdot {x}^{4}\right) \cdot \sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}}\right)\right| \]

      13. rem-sqrt-square 97.6%

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

      14. rem-square-sqrt 97.6%

          \[\leadsto \left|0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot {x}^{2}\right) \cdot {x}^{4}\right) \cdot \left|\color{blue}{\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}}\right|\right)\right| \]

      15. fabs-sqr 97.6%

          \[\leadsto \left|0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot {x}^{2}\right) \cdot {x}^{4}\right) \cdot \color{blue}{\left(\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}\right)}\right)\right| \]

      16. rem-square-sqrt 97.6%

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

      17. *-commutative 97.6%

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

   10. Simplified 97.7%

       \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {\left(\left|x\right|\right)}^{7}\right)}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.9% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 98.6%

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

6. Taylor expanded in x around inf 98.3%

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

   1. pow1 98.3%

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

   2. mul-fabs 98.3%

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

   3. fma-define 98.3%

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

8. Applied egg-rr 98.3%

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

   1. unpow1 98.3%

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

   2. fabs-mul 98.3%

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

   3. rem-square-sqrt 97.2%

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

   4. fabs-sqr 97.2%

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

   5. rem-square-sqrt 98.3%

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

10. Simplified 98.3%

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

Alternative 7: 67.7% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 69.4%

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

   1. *-commutative 69.4%

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

   2. associate-*l* 69.4%

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

   3. *-commutative 69.4%

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

7. Simplified 69.4%

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

   1. *-un-lft-identity 99.8%

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

   2. pow1/2 99.8%

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

   3. inv-pow 99.8%

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

   4. pow-pow 99.8%

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

   5. metadata-eval 99.8%

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

9. Applied egg-rr 69.4%

   \[\leadsto \left|\color{blue}{\left(1 \cdot {\pi}^{-0.5}\right)} \cdot \left(2 \cdot \left|x\right|\right)\right| \]
10. Step-by-step derivation

    1. *-lft-identity 99.8%

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

11. Simplified 69.4%

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

12. Taylor expanded in x around 0 69.4%

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

    1. *-commutative 69.4%

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

    2. *-commutative 69.4%

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

    3. rem-exp-log 69.4%

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

    4. exp-neg 69.4%

       \[\leadsto \left|2 \cdot \left(\sqrt{\color{blue}{e^{-\log \pi}}} \cdot \left|x\right|\right)\right| \]

    5. unpow1/2 69.4%

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

    6. exp-prod 69.4%

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

    7. distribute-lft-neg-out 69.4%

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

    8. exp-neg 69.4%

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

    9. associate-*l/ 69.2%

       \[\leadsto \left|2 \cdot \color{blue}{\frac{1 \cdot \left|x\right|}{e^{\log \pi \cdot 0.5}}}\right| \]

    10. *-lft-identity 69.2%

        \[\leadsto \left|2 \cdot \frac{\color{blue}{\left|x\right|}}{e^{\log \pi \cdot 0.5}}\right| \]

    11. exp-to-pow 68.9%

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

    12. unpow1/2 68.9%

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

14. Simplified 68.9%

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

    1. fabs-mul 68.9%

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

    2. metadata-eval 68.9%

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

16. Applied egg-rr 68.9%

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

    1. rem-square-sqrt 68.7%

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

    2. fabs-sqr 68.7%

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

    3. rem-square-sqrt 68.9%

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

    4. associate-*r/ 68.9%

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

    5. associate-*l/ 69.4%

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

    6. *-commutative 69.4%

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

18. Simplified 69.4%

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

Reproduce

herbie shell --seed 2024131 
(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)))))))