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

Percentage Accurate: 99.8% → 99.9%

Time: 11.5s

Alternatives: 12

Speedup: 3.0×

• 29.0% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -7.328498263981314e-21

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

   1. add-sqr-sqrt 33.2%

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

   2. fabs-sqr 33.2%

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

   3. add-sqr-sqrt 99.4%

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

   4. div-inv 99.9%

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

   5. pow1/2 99.9%

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

   6. pow-flip 99.9%

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

   7. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 3: 99.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {x}^{7}\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (sqrt (/ 1.0 PI))
   (fma
    0.6666666666666666
    (pow x 3.0)
    (fma x 2.0 (* 0.047619047619047616 (pow x 7.0)))))))

C

double code(double x) {
	return fabs((sqrt((1.0 / ((double) M_PI))) * fma(0.6666666666666666, pow(x, 3.0), fma(x, 2.0, (0.047619047619047616 * pow(x, 7.0))))));
}

Julia

function code(x)
	return abs(Float64(sqrt(Float64(1.0 / pi)) * fma(0.6666666666666666, (x ^ 3.0), fma(x, 2.0, Float64(0.047619047619047616 * (x ^ 7.0))))))
end

Wolfram

code[x_] := N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision] + N[(x * 2.0 + N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{3} \cdot \left({\left(\left|x\right|\right)}^{3} \cdot \left|x\right|\right), \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{3} \cdot \left(x \cdot x\right), \mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)\right)\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 99.8%

   \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7} + \left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)}\right| \]
6. Step-by-step derivation

   1. +-commutative 99.8%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right| \]

   2. associate-+l+ 99.8%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + \left(2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)}\right| \]

   3. fma-define 99.8%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right| \]

   4. rem-square-sqrt 33.6%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

   5. fabs-sqr 33.6%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

   6. rem-square-sqrt 80.1%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\color{blue}{x}}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

   7. *-commutative 80.1%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\left|x\right| \cdot 2} + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

   8. fma-define 80.1%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\mathsf{fma}\left(\left|x\right|, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right)\right| \]

   9. rem-square-sqrt 33.4%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

   10. fabs-sqr 33.4%

       \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

   11. rem-square-sqrt 80.2%

       \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\color{blue}{x}, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

   12. rem-square-sqrt 33.6%

       \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{7}\right)\right)\right| \]

   13. fabs-sqr 33.6%

       \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{7}\right)\right)\right| \]

   14. rem-square-sqrt 99.8%

       \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\color{blue}{x}}^{7}\right)\right)\right| \]

7. Simplified 99.8%

   \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {x}^{7}\right)\right)}\right| \]

8. Final simplification 99.8%

   \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {x}^{7}\right)\right)\right| \]
9. Add Preprocessing

Alternative 4: 99.0% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 99.7%

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

   1. mul-fabs 99.7%

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

7. Applied egg-rr 99.7%

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

   1. fma-undefine 99.7%

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

9. Applied egg-rr 99.7%

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

10. Final simplification 99.7%

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

Alternative 5: 98.8% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 99.7%

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

6. Taylor expanded in x around inf 99.6%

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

7. Final simplification 99.6%

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

Alternative 6: 89.1% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{{x}^{7}}{\sqrt{\pi} \cdot 21}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (fabs
    (* x (* (sqrt (/ 1.0 PI)) (+ 2.0 (* 0.6666666666666666 (pow x 2.0))))))
   (fabs (/ (pow x 7.0) (* (sqrt PI) 21.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = fabs((x * (sqrt((1.0 / ((double) M_PI))) * (2.0 + (0.6666666666666666 * pow(x, 2.0))))));
	} else {
		tmp = fabs((pow(x, 7.0) / (sqrt(((double) M_PI)) * 21.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = Math.abs((x * (Math.sqrt((1.0 / Math.PI)) * (2.0 + (0.6666666666666666 * Math.pow(x, 2.0))))));
	} else {
		tmp = Math.abs((Math.pow(x, 7.0) / (Math.sqrt(Math.PI) * 21.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.2:
		tmp = math.fabs((x * (math.sqrt((1.0 / math.pi)) * (2.0 + (0.6666666666666666 * math.pow(x, 2.0))))))
	else:
		tmp = math.fabs((math.pow(x, 7.0) / (math.sqrt(math.pi) * 21.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = abs(Float64(x * Float64(sqrt(Float64(1.0 / pi)) * Float64(2.0 + Float64(0.6666666666666666 * (x ^ 2.0))))));
	else
		tmp = abs(Float64((x ^ 7.0) / Float64(sqrt(pi) * 21.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.2)
		tmp = abs((x * (sqrt((1.0 / pi)) * (2.0 + (0.6666666666666666 * (x ^ 2.0))))));
	else
		tmp = abs(((x ^ 7.0) / (sqrt(pi) * 21.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[Abs[N[(x * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(2.0 + N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Power[x, 7.0], $MachinePrecision] / N[(N[Sqrt[Pi], $MachinePrecision] * 21.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.9%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{3} \cdot \left({\left(\left|x\right|\right)}^{3} \cdot \left|x\right|\right), \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{3} \cdot \left(x \cdot x\right), \mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)\right)\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.8%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7} + \left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)}\right| \]
   6. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right| \]

      2. associate-+l+ 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + \left(2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)}\right| \]

      3. fma-define 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right| \]

      4. rem-square-sqrt 33.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      5. fabs-sqr 33.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      6. rem-square-sqrt 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\color{blue}{x}}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      7. *-commutative 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\left|x\right| \cdot 2} + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      8. fma-define 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\mathsf{fma}\left(\left|x\right|, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right)\right| \]

      9. rem-square-sqrt 33.4%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      10. fabs-sqr 33.4%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      11. rem-square-sqrt 80.2%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\color{blue}{x}, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      12. rem-square-sqrt 33.6%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{7}\right)\right)\right| \]

      13. fabs-sqr 33.6%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{7}\right)\right)\right| \]

      14. rem-square-sqrt 99.8%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\color{blue}{x}}^{7}\right)\right)\right| \]

   7. Simplified 99.8%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {x}^{7}\right)\right)}\right| \]

   8. Taylor expanded in x around 0 85.7%

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

      1. associate-*r* 85.7%

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

      2. distribute-rgt-out 85.7%

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

      3. *-commutative 85.7%

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

   10. Simplified 85.7%

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

   if 2.2000000000000002 < x

   1. Initial program 99.9%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{3} \cdot \left({\left(\left|x\right|\right)}^{3} \cdot \left|x\right|\right), \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{3} \cdot \left(x \cdot x\right), \mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)\right)\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.8%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7} + \left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)}\right| \]
   6. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right| \]

      2. associate-+l+ 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + \left(2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)}\right| \]

      3. fma-define 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right| \]

      4. rem-square-sqrt 33.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      5. fabs-sqr 33.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      6. rem-square-sqrt 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\color{blue}{x}}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      7. *-commutative 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\left|x\right| \cdot 2} + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      8. fma-define 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\mathsf{fma}\left(\left|x\right|, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right)\right| \]

      9. rem-square-sqrt 33.4%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      10. fabs-sqr 33.4%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      11. rem-square-sqrt 80.2%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\color{blue}{x}, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      12. rem-square-sqrt 33.6%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{7}\right)\right)\right| \]

      13. fabs-sqr 33.6%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{7}\right)\right)\right| \]

      14. rem-square-sqrt 99.8%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\color{blue}{x}}^{7}\right)\right)\right| \]

   7. Simplified 99.8%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {x}^{7}\right)\right)}\right| \]

   8. Taylor expanded in x around inf 38.2%

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

      1. *-commutative 38.2%

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

      2. associate-*r* 38.1%

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

   10. Simplified 38.1%

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

       1. *-commutative 38.1%

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

       2. sqrt-div 38.1%

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

       3. metadata-eval 38.1%

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

       4. un-div-inv 38.2%

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

   12. Applied egg-rr 38.2%

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

       1. clear-num 38.2%

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

       2. un-div-inv 38.2%

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

       3. div-inv 38.2%

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

       4. metadata-eval 38.2%

          \[\leadsto \left|\frac{{x}^{7}}{\sqrt{\pi} \cdot \color{blue}{21}}\right| \]

   14. Applied egg-rr 38.2%

       \[\leadsto \left|\color{blue}{\frac{{x}^{7}}{\sqrt{\pi} \cdot 21}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{{x}^{7}}{\sqrt{\pi} \cdot 21}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{0.0022675736961451248}{\pi} \cdot {x}^{14}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (fabs (* 2.0 (* x (pow PI -0.5))))
   (fabs (sqrt (* (/ 0.0022675736961451248 PI) (pow x 14.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = fabs((2.0 * (x * pow(((double) M_PI), -0.5))));
	} else {
		tmp = fabs(sqrt(((0.0022675736961451248 / ((double) M_PI)) * pow(x, 14.0))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = Math.abs((2.0 * (x * Math.pow(Math.PI, -0.5))));
	} else {
		tmp = Math.abs(Math.sqrt(((0.0022675736961451248 / Math.PI) * Math.pow(x, 14.0))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = math.fabs((2.0 * (x * math.pow(math.pi, -0.5))))
	else:
		tmp = math.fabs(math.sqrt(((0.0022675736961451248 / math.pi) * math.pow(x, 14.0))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = abs(Float64(2.0 * Float64(x * (pi ^ -0.5))));
	else
		tmp = abs(sqrt(Float64(Float64(0.0022675736961451248 / pi) * (x ^ 14.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = abs((2.0 * (x * (pi ^ -0.5))));
	else
		tmp = abs(sqrt(((0.0022675736961451248 / pi) * (x ^ 14.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[Abs[N[(2.0 * N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[Sqrt[N[(N[(0.0022675736961451248 / Pi), $MachinePrecision] * N[Power[x, 14.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{3} \cdot \left({\left(\left|x\right|\right)}^{3} \cdot \left|x\right|\right), \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{3} \cdot \left(x \cdot x\right), \mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)\right)\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.8%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7} + \left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)}\right| \]
   6. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right| \]

      2. associate-+l+ 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + \left(2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)}\right| \]

      3. fma-define 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right| \]

      4. rem-square-sqrt 33.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      5. fabs-sqr 33.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      6. rem-square-sqrt 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\color{blue}{x}}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      7. *-commutative 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\left|x\right| \cdot 2} + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      8. fma-define 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\mathsf{fma}\left(\left|x\right|, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right)\right| \]

      9. rem-square-sqrt 33.4%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      10. fabs-sqr 33.4%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      11. rem-square-sqrt 80.2%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\color{blue}{x}, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      12. rem-square-sqrt 33.6%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{7}\right)\right)\right| \]

      13. fabs-sqr 33.6%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{7}\right)\right)\right| \]

      14. rem-square-sqrt 99.8%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\color{blue}{x}}^{7}\right)\right)\right| \]

   7. Simplified 99.8%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {x}^{7}\right)\right)}\right| \]

   8. Taylor expanded in x around 0 66.8%

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

      1. *-commutative 66.8%

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

   10. Simplified 66.8%

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

       1. sqrt-div 66.8%

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

       2. metadata-eval 66.8%

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

       3. un-div-inv 66.4%

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

   12. Applied egg-rr 66.4%

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

       1. clear-num 66.4%

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

       2. associate-/r/ 66.8%

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

       3. metadata-eval 66.8%

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

       4. sqrt-div 66.8%

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

       5. inv-pow 66.8%

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

       6. sqrt-pow1 66.8%

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

       7. metadata-eval 66.8%

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

   14. Applied egg-rr 66.8%

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

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{3} \cdot \left({\left(\left|x\right|\right)}^{3} \cdot \left|x\right|\right), \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{3} \cdot \left(x \cdot x\right), \mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)\right)\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.8%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7} + \left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)}\right| \]
   6. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right| \]

      2. associate-+l+ 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + \left(2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)}\right| \]

      3. fma-define 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right| \]

      4. rem-square-sqrt 33.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      5. fabs-sqr 33.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      6. rem-square-sqrt 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\color{blue}{x}}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      7. *-commutative 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\left|x\right| \cdot 2} + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      8. fma-define 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\mathsf{fma}\left(\left|x\right|, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right)\right| \]

      9. rem-square-sqrt 33.4%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      10. fabs-sqr 33.4%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      11. rem-square-sqrt 80.2%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\color{blue}{x}, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      12. rem-square-sqrt 33.6%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{7}\right)\right)\right| \]

      13. fabs-sqr 33.6%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{7}\right)\right)\right| \]

      14. rem-square-sqrt 99.8%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\color{blue}{x}}^{7}\right)\right)\right| \]

   7. Simplified 99.8%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {x}^{7}\right)\right)}\right| \]

   8. Taylor expanded in x around inf 38.2%

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

      1. *-commutative 38.2%

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

      2. associate-*r* 38.1%

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

   10. Simplified 38.1%

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

       1. add-sqr-sqrt 3.3%

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

       2. sqrt-unprod 36.7%

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

       3. swap-sqr 36.7%

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

       4. pow-prod-up 36.7%

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

       5. metadata-eval 36.7%

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

       6. swap-sqr 36.7%

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

       7. add-sqr-sqrt 36.7%

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

       8. metadata-eval 36.7%

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

   12. Applied egg-rr 36.7%

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

       1. *-commutative 36.7%

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

       2. associate-*l/ 36.7%

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

       3. metadata-eval 36.7%

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

   14. Simplified 36.7%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{0.0022675736961451248}{\pi} \cdot {x}^{14}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (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 <= 1.85) {
		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 <= 1.85) {
		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 <= 1.85:
		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 <= 1.85)
		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 <= 1.85)
		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, 1.85], 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 < 1.8500000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{3} \cdot \left({\left(\left|x\right|\right)}^{3} \cdot \left|x\right|\right), \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{3} \cdot \left(x \cdot x\right), \mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)\right)\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.8%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7} + \left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)}\right| \]
   6. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right| \]

      2. associate-+l+ 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + \left(2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)}\right| \]

      3. fma-define 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right| \]

      4. rem-square-sqrt 33.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      5. fabs-sqr 33.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      6. rem-square-sqrt 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\color{blue}{x}}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      7. *-commutative 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\left|x\right| \cdot 2} + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      8. fma-define 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\mathsf{fma}\left(\left|x\right|, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right)\right| \]

      9. rem-square-sqrt 33.4%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      10. fabs-sqr 33.4%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      11. rem-square-sqrt 80.2%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\color{blue}{x}, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      12. rem-square-sqrt 33.6%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{7}\right)\right)\right| \]

      13. fabs-sqr 33.6%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{7}\right)\right)\right| \]

      14. rem-square-sqrt 99.8%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\color{blue}{x}}^{7}\right)\right)\right| \]

   7. Simplified 99.8%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {x}^{7}\right)\right)}\right| \]

   8. Taylor expanded in x around 0 66.8%

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

      1. *-commutative 66.8%

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

   10. Simplified 66.8%

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

       1. sqrt-div 66.8%

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

       2. metadata-eval 66.8%

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

       3. un-div-inv 66.4%

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

   12. Applied egg-rr 66.4%

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

       1. clear-num 66.4%

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

       2. associate-/r/ 66.8%

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

       3. metadata-eval 66.8%

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

       4. sqrt-div 66.8%

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

       5. inv-pow 66.8%

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

       6. sqrt-pow1 66.8%

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

       7. metadata-eval 66.8%

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

   14. Applied egg-rr 66.8%

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

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{3} \cdot \left({\left(\left|x\right|\right)}^{3} \cdot \left|x\right|\right), \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{3} \cdot \left(x \cdot x\right), \mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)\right)\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.8%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7} + \left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)}\right| \]
   6. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right| \]

      2. associate-+l+ 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + \left(2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)}\right| \]

      3. fma-define 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right| \]

      4. rem-square-sqrt 33.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      5. fabs-sqr 33.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      6. rem-square-sqrt 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\color{blue}{x}}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      7. *-commutative 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\left|x\right| \cdot 2} + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      8. fma-define 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\mathsf{fma}\left(\left|x\right|, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right)\right| \]

      9. rem-square-sqrt 33.4%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      10. fabs-sqr 33.4%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      11. rem-square-sqrt 80.2%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\color{blue}{x}, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      12. rem-square-sqrt 33.6%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{7}\right)\right)\right| \]

      13. fabs-sqr 33.6%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{7}\right)\right)\right| \]

      14. rem-square-sqrt 99.8%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\color{blue}{x}}^{7}\right)\right)\right| \]

   7. Simplified 99.8%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {x}^{7}\right)\right)}\right| \]

   8. Taylor expanded in x around inf 38.2%

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

      1. *-commutative 38.2%

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

      2. associate-*r* 38.1%

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

   10. Simplified 38.1%

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

       1. *-commutative 38.1%

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

       2. sqrt-div 38.1%

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

       3. metadata-eval 38.1%

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

       4. un-div-inv 38.2%

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

   12. Applied egg-rr 38.2%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (fabs (* 2.0 (* x (pow PI -0.5))))
   (fabs (/ (* 0.047619047619047616 (pow x 7.0)) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = fabs((2.0 * (x * pow(((double) M_PI), -0.5))));
	} else {
		tmp = fabs(((0.047619047619047616 * pow(x, 7.0)) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = Math.abs((2.0 * (x * Math.pow(Math.PI, -0.5))));
	} else {
		tmp = Math.abs(((0.047619047619047616 * Math.pow(x, 7.0)) / Math.sqrt(Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = math.fabs((2.0 * (x * math.pow(math.pi, -0.5))))
	else:
		tmp = math.fabs(((0.047619047619047616 * math.pow(x, 7.0)) / math.sqrt(math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = abs(Float64(2.0 * Float64(x * (pi ^ -0.5))));
	else
		tmp = abs(Float64(Float64(0.047619047619047616 * (x ^ 7.0)) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = abs((2.0 * (x * (pi ^ -0.5))));
	else
		tmp = abs(((0.047619047619047616 * (x ^ 7.0)) / sqrt(pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[Abs[N[(2.0 * N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{3} \cdot \left({\left(\left|x\right|\right)}^{3} \cdot \left|x\right|\right), \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{3} \cdot \left(x \cdot x\right), \mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)\right)\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.8%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7} + \left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)}\right| \]
   6. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right| \]

      2. associate-+l+ 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + \left(2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)}\right| \]

      3. fma-define 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right| \]

      4. rem-square-sqrt 33.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      5. fabs-sqr 33.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      6. rem-square-sqrt 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\color{blue}{x}}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      7. *-commutative 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\left|x\right| \cdot 2} + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      8. fma-define 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\mathsf{fma}\left(\left|x\right|, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right)\right| \]

      9. rem-square-sqrt 33.4%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      10. fabs-sqr 33.4%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      11. rem-square-sqrt 80.2%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\color{blue}{x}, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      12. rem-square-sqrt 33.6%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{7}\right)\right)\right| \]

      13. fabs-sqr 33.6%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{7}\right)\right)\right| \]

      14. rem-square-sqrt 99.8%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\color{blue}{x}}^{7}\right)\right)\right| \]

   7. Simplified 99.8%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {x}^{7}\right)\right)}\right| \]

   8. Taylor expanded in x around 0 66.8%

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

      1. *-commutative 66.8%

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

   10. Simplified 66.8%

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

       1. sqrt-div 66.8%

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

       2. metadata-eval 66.8%

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

       3. un-div-inv 66.4%

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

   12. Applied egg-rr 66.4%

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

       1. clear-num 66.4%

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

       2. associate-/r/ 66.8%

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

       3. metadata-eval 66.8%

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

       4. sqrt-div 66.8%

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

       5. inv-pow 66.8%

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

       6. sqrt-pow1 66.8%

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

       7. metadata-eval 66.8%

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

   14. Applied egg-rr 66.8%

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

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{3} \cdot \left({\left(\left|x\right|\right)}^{3} \cdot \left|x\right|\right), \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{3} \cdot \left(x \cdot x\right), \mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)\right)\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.8%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7} + \left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)}\right| \]
   6. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right| \]

      2. associate-+l+ 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + \left(2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)}\right| \]

      3. fma-define 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right| \]

      4. rem-square-sqrt 33.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      5. fabs-sqr 33.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      6. rem-square-sqrt 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\color{blue}{x}}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      7. *-commutative 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\left|x\right| \cdot 2} + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      8. fma-define 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\mathsf{fma}\left(\left|x\right|, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right)\right| \]

      9. rem-square-sqrt 33.4%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      10. fabs-sqr 33.4%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      11. rem-square-sqrt 80.2%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\color{blue}{x}, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      12. rem-square-sqrt 33.6%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{7}\right)\right)\right| \]

      13. fabs-sqr 33.6%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{7}\right)\right)\right| \]

      14. rem-square-sqrt 99.8%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\color{blue}{x}}^{7}\right)\right)\right| \]

   7. Simplified 99.8%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {x}^{7}\right)\right)}\right| \]

   8. Taylor expanded in x around inf 38.2%

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

      1. *-commutative 38.2%

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

      2. associate-*r* 38.1%

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

   10. Simplified 38.1%

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

       1. *-commutative 38.1%

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

       2. sqrt-div 38.1%

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

       3. metadata-eval 38.1%

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

       4. un-div-inv 38.2%

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

   12. Applied egg-rr 38.2%

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

       1. associate-*r/ 38.2%

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

       2. *-commutative 38.2%

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

   14. Applied egg-rr 38.2%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{{x}^{7}}{\sqrt{\pi} \cdot 21}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (fabs (* 2.0 (* x (pow PI -0.5))))
   (fabs (/ (pow x 7.0) (* (sqrt PI) 21.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = fabs((2.0 * (x * pow(((double) M_PI), -0.5))));
	} else {
		tmp = fabs((pow(x, 7.0) / (sqrt(((double) M_PI)) * 21.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = Math.abs((2.0 * (x * Math.pow(Math.PI, -0.5))));
	} else {
		tmp = Math.abs((Math.pow(x, 7.0) / (Math.sqrt(Math.PI) * 21.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = math.fabs((2.0 * (x * math.pow(math.pi, -0.5))))
	else:
		tmp = math.fabs((math.pow(x, 7.0) / (math.sqrt(math.pi) * 21.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = abs(Float64(2.0 * Float64(x * (pi ^ -0.5))));
	else
		tmp = abs(Float64((x ^ 7.0) / Float64(sqrt(pi) * 21.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = abs((2.0 * (x * (pi ^ -0.5))));
	else
		tmp = abs(((x ^ 7.0) / (sqrt(pi) * 21.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], 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[(N[Sqrt[Pi], $MachinePrecision] * 21.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{3} \cdot \left({\left(\left|x\right|\right)}^{3} \cdot \left|x\right|\right), \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{3} \cdot \left(x \cdot x\right), \mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)\right)\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.8%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7} + \left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)}\right| \]
   6. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right| \]

      2. associate-+l+ 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + \left(2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)}\right| \]

      3. fma-define 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right| \]

      4. rem-square-sqrt 33.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      5. fabs-sqr 33.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      6. rem-square-sqrt 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\color{blue}{x}}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      7. *-commutative 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\left|x\right| \cdot 2} + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      8. fma-define 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\mathsf{fma}\left(\left|x\right|, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right)\right| \]

      9. rem-square-sqrt 33.4%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      10. fabs-sqr 33.4%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      11. rem-square-sqrt 80.2%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\color{blue}{x}, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      12. rem-square-sqrt 33.6%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{7}\right)\right)\right| \]

      13. fabs-sqr 33.6%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{7}\right)\right)\right| \]

      14. rem-square-sqrt 99.8%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\color{blue}{x}}^{7}\right)\right)\right| \]

   7. Simplified 99.8%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {x}^{7}\right)\right)}\right| \]

   8. Taylor expanded in x around 0 66.8%

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

      1. *-commutative 66.8%

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

   10. Simplified 66.8%

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

       1. sqrt-div 66.8%

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

       2. metadata-eval 66.8%

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

       3. un-div-inv 66.4%

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

   12. Applied egg-rr 66.4%

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

       1. clear-num 66.4%

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

       2. associate-/r/ 66.8%

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

       3. metadata-eval 66.8%

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

       4. sqrt-div 66.8%

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

       5. inv-pow 66.8%

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

       6. sqrt-pow1 66.8%

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

       7. metadata-eval 66.8%

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

   14. Applied egg-rr 66.8%

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

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{3} \cdot \left({\left(\left|x\right|\right)}^{3} \cdot \left|x\right|\right), \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{3} \cdot \left(x \cdot x\right), \mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)\right)\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.8%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7} + \left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)}\right| \]
   6. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right| \]

      2. associate-+l+ 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + \left(2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)}\right| \]

      3. fma-define 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right| \]

      4. rem-square-sqrt 33.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      5. fabs-sqr 33.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      6. rem-square-sqrt 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\color{blue}{x}}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      7. *-commutative 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\left|x\right| \cdot 2} + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

      8. fma-define 80.1%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\mathsf{fma}\left(\left|x\right|, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right)\right| \]

      9. rem-square-sqrt 33.4%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      10. fabs-sqr 33.4%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      11. rem-square-sqrt 80.2%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\color{blue}{x}, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

      12. rem-square-sqrt 33.6%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{7}\right)\right)\right| \]

      13. fabs-sqr 33.6%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{7}\right)\right)\right| \]

      14. rem-square-sqrt 99.8%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\color{blue}{x}}^{7}\right)\right)\right| \]

   7. Simplified 99.8%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {x}^{7}\right)\right)}\right| \]

   8. Taylor expanded in x around inf 38.2%

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

      1. *-commutative 38.2%

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

      2. associate-*r* 38.1%

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

   10. Simplified 38.1%

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

       1. *-commutative 38.1%

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

       2. sqrt-div 38.1%

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

       3. metadata-eval 38.1%

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

       4. un-div-inv 38.2%

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

   12. Applied egg-rr 38.2%

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

       1. clear-num 38.2%

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

       2. un-div-inv 38.2%

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

       3. div-inv 38.2%

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

       4. metadata-eval 38.2%

          \[\leadsto \left|\frac{{x}^{7}}{\sqrt{\pi} \cdot \color{blue}{21}}\right| \]

   14. Applied egg-rr 38.2%

       \[\leadsto \left|\color{blue}{\frac{{x}^{7}}{\sqrt{\pi} \cdot 21}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{{x}^{7}}{\sqrt{\pi} \cdot 21}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.0% 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 99.9%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{3} \cdot \left({\left(\left|x\right|\right)}^{3} \cdot \left|x\right|\right), \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{3} \cdot \left(x \cdot x\right), \mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)\right)\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 99.8%

   \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7} + \left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)}\right| \]
6. Step-by-step derivation

   1. +-commutative 99.8%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right| \]

   2. associate-+l+ 99.8%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + \left(2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)}\right| \]

   3. fma-define 99.8%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right| \]

   4. rem-square-sqrt 33.6%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

   5. fabs-sqr 33.6%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

   6. rem-square-sqrt 80.1%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\color{blue}{x}}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

   7. *-commutative 80.1%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\left|x\right| \cdot 2} + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

   8. fma-define 80.1%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\mathsf{fma}\left(\left|x\right|, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right)\right| \]

   9. rem-square-sqrt 33.4%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

   10. fabs-sqr 33.4%

       \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

   11. rem-square-sqrt 80.2%

       \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\color{blue}{x}, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

   12. rem-square-sqrt 33.6%

       \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{7}\right)\right)\right| \]

   13. fabs-sqr 33.6%

       \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{7}\right)\right)\right| \]

   14. rem-square-sqrt 99.8%

       \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\color{blue}{x}}^{7}\right)\right)\right| \]

7. Simplified 99.8%

   \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {x}^{7}\right)\right)}\right| \]

8. Taylor expanded in x around 0 66.8%

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

   1. *-commutative 66.8%

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

10. Simplified 66.8%

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

    1. sqrt-div 66.8%

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

    2. metadata-eval 66.8%

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

    3. un-div-inv 66.4%

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

12. Applied egg-rr 66.4%

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

    1. clear-num 66.4%

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

    2. associate-/r/ 66.8%

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

    3. metadata-eval 66.8%

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

    4. sqrt-div 66.8%

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

    5. inv-pow 66.8%

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

    6. sqrt-pow1 66.8%

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

    7. metadata-eval 66.8%

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

14. Applied egg-rr 66.8%

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

15. Final simplification 66.8%

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

Alternative 12: 66.6% 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 99.9%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{3} \cdot \left({\left(\left|x\right|\right)}^{3} \cdot \left|x\right|\right), \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{3} \cdot \left(x \cdot x\right), \mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)\right)\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 99.8%

   \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7} + \left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)}\right| \]
6. Step-by-step derivation

   1. +-commutative 99.8%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right| \]

   2. associate-+l+ 99.8%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + \left(2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)}\right| \]

   3. fma-define 99.8%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right| \]

   4. rem-square-sqrt 33.6%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

   5. fabs-sqr 33.6%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

   6. rem-square-sqrt 80.1%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {\color{blue}{x}}^{3}, 2 \cdot \left|x\right| + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

   7. *-commutative 80.1%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\left|x\right| \cdot 2} + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]

   8. fma-define 80.1%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \color{blue}{\mathsf{fma}\left(\left|x\right|, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)}\right)\right| \]

   9. rem-square-sqrt 33.4%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

   10. fabs-sqr 33.4%

       \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

   11. rem-square-sqrt 80.2%

       \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(\color{blue}{x}, 2, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right| \]

   12. rem-square-sqrt 33.6%

       \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{7}\right)\right)\right| \]

   13. fabs-sqr 33.6%

       \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{7}\right)\right)\right| \]

   14. rem-square-sqrt 99.8%

       \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {\color{blue}{x}}^{7}\right)\right)\right| \]

7. Simplified 99.8%

   \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(x, 2, 0.047619047619047616 \cdot {x}^{7}\right)\right)}\right| \]

8. Taylor expanded in x around 0 66.8%

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

   1. *-commutative 66.8%

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

10. Simplified 66.8%

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

    1. sqrt-div 66.8%

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

    2. metadata-eval 66.8%

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

    3. un-div-inv 66.4%

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

12. Applied egg-rr 66.4%

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

13. Final simplification 66.4%

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

Reproduce

herbie shell --seed 2024073 
(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)))))))