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

Percentage Accurate: 99.8% → 99.9%

Time: 11.7s

Alternatives: 7

Speedup: 3.6×

• 27.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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. Step-by-step derivation

   1. pow1 99.8%

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

   2. mul-fabs 99.8%

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

   3. +-commutative 99.8%

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

   4. pow2 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow1 99.8%

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

8. Simplified 99.8%

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

   1. fma-undefine 99.8%

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

10. Applied egg-rr 99.8%

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

    1. fma-undefine 99.8%

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

12. Applied egg-rr 99.8%

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

13. Final simplification 99.8%

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

Alternative 2: 99.1% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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. Step-by-step derivation

   1. pow1 99.8%

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

   2. mul-fabs 99.8%

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

   3. +-commutative 99.8%

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

   4. pow2 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow1 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in x around 0 99.0%

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

    1. fma-undefine 99.8%

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

11. Applied egg-rr 99.0%

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

12. Final simplification 99.0%

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

Alternative 3: 68.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\left|2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{{x}^{14}}{\pi} \cdot 0.0022675736961451248}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.9)
   (fabs (* 2.0 (* x (pow PI -0.5))))
   (fabs (sqrt (* (/ (pow x 14.0) PI) 0.0022675736961451248)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = fabs((2.0 * (x * pow(((double) M_PI), -0.5))));
	} else {
		tmp = fabs(sqrt(((pow(x, 14.0) / ((double) M_PI)) * 0.0022675736961451248)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = Math.abs((2.0 * (x * Math.pow(Math.PI, -0.5))));
	} else {
		tmp = Math.abs(Math.sqrt(((Math.pow(x, 14.0) / Math.PI) * 0.0022675736961451248)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.9:
		tmp = math.fabs((2.0 * (x * math.pow(math.pi, -0.5))))
	else:
		tmp = math.fabs(math.sqrt(((math.pow(x, 14.0) / math.pi) * 0.0022675736961451248)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.9)
		tmp = abs(Float64(2.0 * Float64(x * (pi ^ -0.5))));
	else
		tmp = abs(sqrt(Float64(Float64((x ^ 14.0) / pi) * 0.0022675736961451248)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.9)
		tmp = abs((2.0 * (x * (pi ^ -0.5))));
	else
		tmp = abs(sqrt((((x ^ 14.0) / pi) * 0.0022675736961451248)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.9], N[Abs[N[(2.0 * N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[Sqrt[N[(N[(N[Power[x, 14.0], $MachinePrecision] / Pi), $MachinePrecision] * 0.0022675736961451248), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   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 67.6%

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

      1. *-commutative 67.6%

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

      2. unpow1 67.6%

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

      3. sqr-pow 35.2%

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

      4. fabs-sqr 35.2%

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

      5. sqr-pow 67.6%

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

      6. unpow1 67.6%

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

   7. Simplified 67.6%

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

      1. sqrt-div 67.6%

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

      2. metadata-eval 67.6%

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

      3. un-div-inv 67.1%

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

   9. Applied egg-rr 67.1%

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

       1. div-inv 67.6%

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

       2. pow1/2 67.6%

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

       3. pow-flip 67.6%

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

       4. metadata-eval 67.6%

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

   11. Applied egg-rr 67.6%

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

   if 1.8999999999999999 < 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%

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

      1. pow1 99.8%

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

      2. mul-fabs 99.8%

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

      3. +-commutative 99.8%

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

      4. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow1 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around inf 36.7%

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

       1. add-sqr-sqrt 3.5%

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

       2. sqrt-unprod 34.1%

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

       3. *-commutative 34.1%

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

       4. *-commutative 34.1%

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

       5. swap-sqr 34.1%

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

       6. *-commutative 34.1%

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

       7. *-commutative 34.1%

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

       8. swap-sqr 34.1%

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

       9. add-sqr-sqrt 34.1%

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

       10. pow-prod-up 34.1%

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

       11. metadata-eval 34.1%

           \[\leadsto \left|\sqrt{\left(\frac{1}{\pi} \cdot {x}^{\color{blue}{14}}\right) \cdot \left(0.047619047619047616 \cdot 0.047619047619047616\right)}\right| \]

       12. metadata-eval 34.1%

           \[\leadsto \left|\sqrt{\left(\frac{1}{\pi} \cdot {x}^{14}\right) \cdot \color{blue}{0.0022675736961451248}}\right| \]

   11. Applied egg-rr 34.1%

       \[\leadsto \left|\color{blue}{\sqrt{\left(\frac{1}{\pi} \cdot {x}^{14}\right) \cdot 0.0022675736961451248}}\right| \]
   12. Step-by-step derivation

       1. metadata-eval 34.1%

          \[\leadsto \left|\sqrt{\left(\frac{1}{\pi} \cdot {x}^{\color{blue}{\left(2 \cdot 7\right)}}\right) \cdot 0.0022675736961451248}\right| \]

       2. pow-sqr 34.1%

          \[\leadsto \left|\sqrt{\left(\frac{1}{\pi} \cdot \color{blue}{\left({x}^{7} \cdot {x}^{7}\right)}\right) \cdot 0.0022675736961451248}\right| \]

       3. associate-*l/ 34.1%

          \[\leadsto \left|\sqrt{\color{blue}{\frac{1 \cdot \left({x}^{7} \cdot {x}^{7}\right)}{\pi}} \cdot 0.0022675736961451248}\right| \]

       4. *-lft-identity 34.1%

          \[\leadsto \left|\sqrt{\frac{\color{blue}{{x}^{7} \cdot {x}^{7}}}{\pi} \cdot 0.0022675736961451248}\right| \]

       5. pow-sqr 34.1%

          \[\leadsto \left|\sqrt{\frac{\color{blue}{{x}^{\left(2 \cdot 7\right)}}}{\pi} \cdot 0.0022675736961451248}\right| \]

       6. metadata-eval 34.1%

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

   13. Simplified 34.1%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\left|2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{{x}^{14}}{\pi} \cdot 0.0022675736961451248}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\left|2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.9)
   (fabs (* 2.0 (* x (pow PI -0.5))))
   (fabs (* (pow x 7.0) (/ 0.047619047619047616 (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = fabs((2.0 * (x * pow(((double) M_PI), -0.5))));
	} else {
		tmp = fabs((pow(x, 7.0) * (0.047619047619047616 / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = Math.abs((2.0 * (x * Math.pow(Math.PI, -0.5))));
	} else {
		tmp = Math.abs((Math.pow(x, 7.0) * (0.047619047619047616 / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.9:
		tmp = math.fabs((2.0 * (x * math.pow(math.pi, -0.5))))
	else:
		tmp = math.fabs((math.pow(x, 7.0) * (0.047619047619047616 / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.9)
		tmp = abs(Float64(2.0 * Float64(x * (pi ^ -0.5))));
	else
		tmp = abs(Float64((x ^ 7.0) * Float64(0.047619047619047616 / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.9)
		tmp = abs((2.0 * (x * (pi ^ -0.5))));
	else
		tmp = abs(((x ^ 7.0) * (0.047619047619047616 / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.9], N[Abs[N[(2.0 * N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Power[x, 7.0], $MachinePrecision] * N[(0.047619047619047616 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   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 67.6%

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

      1. *-commutative 67.6%

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

      2. unpow1 67.6%

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

      3. sqr-pow 35.2%

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

      4. fabs-sqr 35.2%

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

      5. sqr-pow 67.6%

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

      6. unpow1 67.6%

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

   7. Simplified 67.6%

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

      1. sqrt-div 67.6%

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

      2. metadata-eval 67.6%

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

      3. un-div-inv 67.1%

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

   9. Applied egg-rr 67.1%

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

       1. div-inv 67.6%

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

       2. pow1/2 67.6%

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

       3. pow-flip 67.6%

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

       4. metadata-eval 67.6%

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

   11. Applied egg-rr 67.6%

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

   if 1.8999999999999999 < 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%

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

      1. pow1 99.8%

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

      2. mul-fabs 99.8%

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

      3. +-commutative 99.8%

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

      4. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow1 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around inf 36.7%

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

       1. associate-*r* 36.6%

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

       2. sqrt-div 36.6%

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

       3. metadata-eval 36.6%

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

       4. un-div-inv 36.7%

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

       5. *-commutative 36.7%

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

   11. Applied egg-rr 36.7%

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

       1. associate-*r/ 36.7%

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

   13. Simplified 36.7%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\left|2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(x * N[(N[(2.0 + N[(0.047619047619047616 * N[Power[x, 6.0], $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. Step-by-step derivation

   1. pow1 99.8%

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

   2. mul-fabs 99.8%

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

   3. +-commutative 99.8%

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

   4. pow2 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow1 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in x around 0 99.0%

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

10. Taylor expanded in x around inf 98.6%

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

11. Final simplification 98.6%

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

Alternative 6: 68.9% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Taylor expanded in x around 0 67.6%

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

   1. *-commutative 67.6%

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

   2. unpow1 67.6%

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

   3. sqr-pow 35.2%

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

   4. fabs-sqr 35.2%

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

   5. sqr-pow 67.6%

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

   6. unpow1 67.6%

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

7. Simplified 67.6%

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

   1. sqrt-div 67.6%

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

   2. metadata-eval 67.6%

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

   3. un-div-inv 67.1%

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

9. Applied egg-rr 67.1%

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

    1. div-inv 67.6%

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

    2. pow1/2 67.6%

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

    3. pow-flip 67.6%

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

    4. metadata-eval 67.6%

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

11. Applied egg-rr 67.6%

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

12. Final simplification 67.6%

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

Alternative 7: 68.5% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(2.0 * N[(x / 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%

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

5. Taylor expanded in x around 0 67.6%

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

   1. *-commutative 67.6%

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

   2. unpow1 67.6%

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

   3. sqr-pow 35.2%

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

   4. fabs-sqr 35.2%

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

   5. sqr-pow 67.6%

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

   6. unpow1 67.6%

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

7. Simplified 67.6%

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

   1. sqrt-div 67.6%

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

   2. metadata-eval 67.6%

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

   3. un-div-inv 67.1%

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

9. Applied egg-rr 67.1%

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

10. Final simplification 67.1%

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

Reproduce

herbie shell --seed 2024062 
(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)))))))