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

Percentage Accurate: 99.9% → 99.9%

Time: 37.9s

Alternatives: 11

Speedup: 2.6×

• 28.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% 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, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot \left(x \cdot x\right)\\ t_1 := \left|x\right| \cdot \left(\left|x\right| \cdot t\_0\right)\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + 0.6666666666666666 \cdot t\_0\right) + 0.2 \cdot t\_1\right) + 0.047619047619047616 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot t\_1\right)\right)\right)\right| \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.2 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

Alternative 3: 33.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot {\left(\sqrt{x}\right)}^{2}\right)\right)\right)\right)\right| \end{array} \]

FPCore

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

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * x) + (0.6666666666666666 * pow(x, 3.0))) + (0.2 * ((fabs(x) * (x * x)) * (x * x)))) + (0.047619047619047616 * ((x * x) * ((x * x) * ((x * x) * pow(sqrt(x), 2.0))))))));
}

Java

public static double code(double x) {
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * x) + (0.6666666666666666 * Math.pow(x, 3.0))) + (0.2 * ((Math.abs(x) * (x * x)) * (x * x)))) + (0.047619047619047616 * ((x * x) * ((x * x) * ((x * x) * Math.pow(Math.sqrt(x), 2.0))))))));
}

Python

def code(x):
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * x) + (0.6666666666666666 * math.pow(x, 3.0))) + (0.2 * ((math.fabs(x) * (x * x)) * (x * x)))) + (0.047619047619047616 * ((x * x) * ((x * x) * ((x * x) * math.pow(math.sqrt(x), 2.0))))))))

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * x) + Float64(0.6666666666666666 * (x ^ 3.0))) + Float64(0.2 * Float64(Float64(abs(x) * Float64(x * x)) * Float64(x * x)))) + Float64(0.047619047619047616 * Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(Float64(x * x) * (sqrt(x) ^ 2.0))))))))
end

MATLAB

function tmp = code(x)
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * x) + (0.6666666666666666 * (x ^ 3.0))) + (0.2 * ((abs(x) * (x * x)) * (x * x)))) + (0.047619047619047616 * ((x * x) * ((x * x) * ((x * x) * (sqrt(x) ^ 2.0))))))));
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * x), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[(N[(N[Abs[x], $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[Power[N[Sqrt[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $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 99.8%

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

   1. fma-undefine 99.8%

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

   2. add-sqr-sqrt 30.6%

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

   3. fabs-sqr 30.6%

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

   4. add-sqr-sqrt 99.5%

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

   5. associate-*r* 99.5%

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

   6. add-sqr-sqrt 30.8%

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

   7. fabs-sqr 30.8%

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

   8. add-sqr-sqrt 81.6%

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

   9. associate-*r* 81.6%

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

   10. cube-mult 81.6%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot \color{blue}{{x}^{3}}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

6. Applied egg-rr 81.6%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\color{blue}{\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)} + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]
7. Step-by-step derivation

   1. add-sqr-sqrt 30.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   2. fabs-sqr 30.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   3. pow2 30.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{{\left(\sqrt{x}\right)}^{2}} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

8. Applied egg-rr 30.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{{\left(\sqrt{x}\right)}^{2}} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

9. Final simplification 30.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot {\left(\sqrt{x}\right)}^{2}\right)\right)\right)\right)\right| \]
10. Add Preprocessing

Alternative 4: 99.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(x \cdot x\right) \cdot {x}^{3}\right)\right) + 0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right| \end{array} \]

FPCore

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

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * x) + (0.6666666666666666 * pow(x, 3.0))) + (0.2 * ((x * x) * pow(x, 3.0)))) + (0.047619047619047616 * ((x * x) * ((x * x) * (x * (x * x))))))));
}

Java

public static double code(double x) {
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * x) + (0.6666666666666666 * Math.pow(x, 3.0))) + (0.2 * ((x * x) * Math.pow(x, 3.0)))) + (0.047619047619047616 * ((x * x) * ((x * x) * (x * (x * x))))))));
}

Python

def code(x):
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * x) + (0.6666666666666666 * math.pow(x, 3.0))) + (0.2 * ((x * x) * math.pow(x, 3.0)))) + (0.047619047619047616 * ((x * x) * ((x * x) * (x * (x * x))))))))

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * x) + Float64(0.6666666666666666 * (x ^ 3.0))) + Float64(0.2 * Float64(Float64(x * x) * (x ^ 3.0)))) + Float64(0.047619047619047616 * Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(x * Float64(x * x))))))))
end

MATLAB

function tmp = code(x)
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * x) + (0.6666666666666666 * (x ^ 3.0))) + (0.2 * ((x * x) * (x ^ 3.0)))) + (0.047619047619047616 * ((x * x) * ((x * x) * (x * (x * x))))))));
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * x), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[(N[(x * x), $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $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 99.8%

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

   1. fma-undefine 99.8%

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

   2. add-sqr-sqrt 30.6%

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

   3. fabs-sqr 30.6%

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

   4. add-sqr-sqrt 99.5%

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

   5. associate-*r* 99.5%

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

   6. add-sqr-sqrt 30.8%

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

   7. fabs-sqr 30.8%

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

   8. add-sqr-sqrt 81.6%

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

   9. associate-*r* 81.6%

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

   10. cube-mult 81.6%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot \color{blue}{{x}^{3}}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

6. Applied egg-rr 81.6%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\color{blue}{\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)} + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]
7. Step-by-step derivation

   1. add-sqr-sqrt 30.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   2. fabs-sqr 30.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   3. add-sqr-sqrt 75.9%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{x} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   4. *-un-lft-identity 75.9%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{\left(1 \cdot x\right)} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

8. Applied egg-rr 75.9%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{\left(1 \cdot x\right)} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]
9. Step-by-step derivation

   1. *-lft-identity 75.9%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{x} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

10. Simplified 75.9%

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{x} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

11. Taylor expanded in x around 0 75.9%

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\color{blue}{\left({x}^{2} \cdot \left|x\right|\right)} \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]
12. Step-by-step derivation

    1. unpow2 75.9%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\color{blue}{\left(x \cdot x\right)} \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

    2. rem-square-sqrt 30.8%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left(x \cdot x\right) \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

    3. fabs-sqr 30.8%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left(x \cdot x\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

    4. rem-square-sqrt 99.8%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left(x \cdot x\right) \cdot \color{blue}{x}\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

    5. unpow3 99.8%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\color{blue}{{x}^{3}} \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

13. Simplified 99.8%

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\color{blue}{{x}^{3}} \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

14. Final simplification 99.8%

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(x \cdot x\right) \cdot {x}^{3}\right)\right) + 0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right| \]
15. Add Preprocessing

Alternative 5: 98.9% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

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 99.1%

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

6. Taylor expanded in x around inf 99.0%

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

7. Final simplification 99.0%

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

Alternative 6: 98.9% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(x * N[(N[(2.0 + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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 99.8%

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

5. Taylor expanded in x around 0 99.0%

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

   1. *-commutative 99.0%

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

   2. rem-square-sqrt 30.3%

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

   3. fabs-sqr 30.3%

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

   4. rem-square-sqrt 99.0%

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

7. Simplified 99.0%

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

8. Taylor expanded in x around 0 99.0%

   \[\leadsto \left|\color{blue}{x \cdot \left(0.047619047619047616 \cdot \left(\left({x}^{5} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

10. Simplified 99.0%

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

    1. expm1-log1p-u 99.0%

       \[\leadsto \left|x \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}}\right)\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    2. expm1-undefine 97.9%

       \[\leadsto \left|x \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}}\right)} - 1\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    3. inv-pow 97.9%

       \[\leadsto \left|x \cdot \left(\left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{{\pi}^{-1}}}\right)} - 1\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    4. sqrt-pow1 97.9%

       \[\leadsto \left|x \cdot \left(\left(e^{\mathsf{log1p}\left(\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    5. metadata-eval 97.9%

       \[\leadsto \left|x \cdot \left(\left(e^{\mathsf{log1p}\left({\pi}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

12. Applied egg-rr 97.9%

    \[\leadsto \left|x \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\pi}^{-0.5}\right)} - 1\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]
13. Step-by-step derivation

    1. sub-neg 97.9%

       \[\leadsto \left|x \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\pi}^{-0.5}\right)} + \left(-1\right)\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    2. metadata-eval 97.9%

       \[\leadsto \left|x \cdot \left(\left(e^{\mathsf{log1p}\left({\pi}^{-0.5}\right)} + \color{blue}{-1}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    3. +-commutative 97.9%

       \[\leadsto \left|x \cdot \left(\color{blue}{\left(-1 + e^{\mathsf{log1p}\left({\pi}^{-0.5}\right)}\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    4. log1p-undefine 97.7%

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

    5. rem-exp-log 97.7%

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

    6. associate-+r+ 99.0%

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

    7. metadata-eval 99.0%

       \[\leadsto \left|x \cdot \left(\left(\color{blue}{0} + {\pi}^{-0.5}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    8. exp-to-pow 99.0%

       \[\leadsto \left|x \cdot \left(\left(0 + \color{blue}{e^{\log \pi \cdot -0.5}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    9. metadata-eval 99.0%

       \[\leadsto \left|x \cdot \left(\left(0 + e^{\log \pi \cdot \color{blue}{\left(-0.5\right)}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    10. distribute-rgt-neg-in 99.0%

        \[\leadsto \left|x \cdot \left(\left(0 + e^{\color{blue}{-\log \pi \cdot 0.5}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    11. exp-neg 99.0%

        \[\leadsto \left|x \cdot \left(\left(0 + \color{blue}{\frac{1}{e^{\log \pi \cdot 0.5}}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    12. metadata-eval 99.0%

        \[\leadsto \left|x \cdot \left(\left(0 + \frac{\color{blue}{--1}}{e^{\log \pi \cdot 0.5}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    13. exp-to-pow 99.0%

        \[\leadsto \left|x \cdot \left(\left(0 + \frac{--1}{\color{blue}{{\pi}^{0.5}}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    14. unpow1/2 99.0%

        \[\leadsto \left|x \cdot \left(\left(0 + \frac{--1}{\color{blue}{\sqrt{\pi}}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    15. distribute-neg-frac 99.0%

        \[\leadsto \left|x \cdot \left(\left(0 + \color{blue}{\left(-\frac{-1}{\sqrt{\pi}}\right)}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    16. sub-neg 99.0%

        \[\leadsto \left|x \cdot \left(\color{blue}{\left(0 - \frac{-1}{\sqrt{\pi}}\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    17. neg-sub0 99.0%

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

    18. distribute-neg-frac 99.0%

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

    19. metadata-eval 99.0%

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

    20. unpow1/2 99.0%

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

    21. exp-to-pow 99.0%

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

    22. exp-neg 99.0%

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

    23. distribute-rgt-neg-in 99.0%

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

    24. metadata-eval 99.0%

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

14. Simplified 99.0%

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

15. Final simplification 99.0%

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

Alternative 7: 98.8% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(2 \cdot x + 0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt PI))
   (+
    (* 2.0 x)
    (* 0.047619047619047616 (* (* x x) (* (* x x) (* x (* x x)))))))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((2.0 * x) + (0.047619047619047616 * ((x * x) * ((x * x) * (x * (x * x))))))));
}

Java

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

Python

def code(x):
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((2.0 * x) + (0.047619047619047616 * ((x * x) * ((x * x) * (x * (x * x))))))))

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(2.0 * x) + Float64(0.047619047619047616 * Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(x * Float64(x * x))))))))
end

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * x), $MachinePrecision] + N[(0.047619047619047616 * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $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 99.8%

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

5. Taylor expanded in x around 0 99.0%

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

   1. *-commutative 99.0%

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

   2. rem-square-sqrt 30.3%

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

   3. fabs-sqr 30.3%

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

   4. rem-square-sqrt 99.0%

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

7. Simplified 99.0%

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

   1. add-sqr-sqrt 30.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   2. fabs-sqr 30.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   3. add-sqr-sqrt 75.9%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{x} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   4. *-un-lft-identity 75.9%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{\left(1 \cdot x\right)} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

9. Applied egg-rr 99.0%

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

    1. *-lft-identity 75.9%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{x} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

11. Simplified 99.0%

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

12. Final simplification 99.0%

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

Alternative 8: 34.7% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot \left(x \cdot {x}^{6}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (* x (/ 2.0 (sqrt PI)))
   (* 0.047619047619047616 (* (pow PI -0.5) (* x (pow x 6.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = 0.047619047619047616 * (pow(((double) M_PI), -0.5) * (x * pow(x, 6.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = 0.047619047619047616 * (Math.pow(Math.PI, -0.5) * (x * Math.pow(x, 6.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = 0.047619047619047616 * (math.pow(math.pi, -0.5) * (x * math.pow(x, 6.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(0.047619047619047616 * Float64((pi ^ -0.5) * Float64(x * (x ^ 6.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = 0.047619047619047616 * ((pi ^ -0.5) * (x * (x ^ 6.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * N[Power[x, 6.0], $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 99.8%

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

   5. Taylor expanded in x around 0 70.6%

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

      1. associate-*r* 70.6%

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

      2. rem-square-sqrt 30.3%

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

      3. fabs-sqr 30.3%

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

      4. rem-square-sqrt 70.6%

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

      5. *-commutative 70.6%

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

   7. Simplified 70.6%

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

      1. expm1-log1p-u 99.0%

         \[\leadsto \left|x \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}}\right)\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

      2. expm1-undefine 97.9%

         \[\leadsto \left|x \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}}\right)} - 1\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

      3. inv-pow 97.9%

         \[\leadsto \left|x \cdot \left(\left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{{\pi}^{-1}}}\right)} - 1\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

      4. sqrt-pow1 97.9%

         \[\leadsto \left|x \cdot \left(\left(e^{\mathsf{log1p}\left(\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

      5. metadata-eval 97.9%

         \[\leadsto \left|x \cdot \left(\left(e^{\mathsf{log1p}\left({\pi}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   9. Applied egg-rr 69.5%

      \[\leadsto \left|x \cdot \left(2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\pi}^{-0.5}\right)} - 1\right)}\right)\right| \]
   10. Step-by-step derivation

       1. sub-neg 97.9%

          \[\leadsto \left|x \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\pi}^{-0.5}\right)} + \left(-1\right)\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       2. metadata-eval 97.9%

          \[\leadsto \left|x \cdot \left(\left(e^{\mathsf{log1p}\left({\pi}^{-0.5}\right)} + \color{blue}{-1}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       3. +-commutative 97.9%

          \[\leadsto \left|x \cdot \left(\color{blue}{\left(-1 + e^{\mathsf{log1p}\left({\pi}^{-0.5}\right)}\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       4. log1p-undefine 97.7%

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

       5. rem-exp-log 97.7%

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

       6. associate-+r+ 99.0%

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

       7. metadata-eval 99.0%

          \[\leadsto \left|x \cdot \left(\left(\color{blue}{0} + {\pi}^{-0.5}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       8. exp-to-pow 99.0%

          \[\leadsto \left|x \cdot \left(\left(0 + \color{blue}{e^{\log \pi \cdot -0.5}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       9. metadata-eval 99.0%

          \[\leadsto \left|x \cdot \left(\left(0 + e^{\log \pi \cdot \color{blue}{\left(-0.5\right)}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       10. distribute-rgt-neg-in 99.0%

           \[\leadsto \left|x \cdot \left(\left(0 + e^{\color{blue}{-\log \pi \cdot 0.5}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       11. exp-neg 99.0%

           \[\leadsto \left|x \cdot \left(\left(0 + \color{blue}{\frac{1}{e^{\log \pi \cdot 0.5}}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       12. metadata-eval 99.0%

           \[\leadsto \left|x \cdot \left(\left(0 + \frac{\color{blue}{--1}}{e^{\log \pi \cdot 0.5}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       13. exp-to-pow 99.0%

           \[\leadsto \left|x \cdot \left(\left(0 + \frac{--1}{\color{blue}{{\pi}^{0.5}}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       14. unpow1/2 99.0%

           \[\leadsto \left|x \cdot \left(\left(0 + \frac{--1}{\color{blue}{\sqrt{\pi}}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       15. distribute-neg-frac 99.0%

           \[\leadsto \left|x \cdot \left(\left(0 + \color{blue}{\left(-\frac{-1}{\sqrt{\pi}}\right)}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       16. sub-neg 99.0%

           \[\leadsto \left|x \cdot \left(\color{blue}{\left(0 - \frac{-1}{\sqrt{\pi}}\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       17. neg-sub0 99.0%

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

       18. distribute-neg-frac 99.0%

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

       19. metadata-eval 99.0%

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

       20. unpow1/2 99.0%

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

       21. exp-to-pow 99.0%

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

       22. exp-neg 99.0%

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

       23. distribute-rgt-neg-in 99.0%

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

       24. metadata-eval 99.0%

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

   11. Simplified 70.6%

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

       1. add-sqr-sqrt 30.3%

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

       2. fabs-sqr 30.3%

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

       3. add-sqr-sqrt 32.0%

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

       4. *-commutative 32.0%

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

       5. *-commutative 32.0%

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

       6. metadata-eval 32.0%

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

       7. sqrt-pow2 32.0%

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

       8. inv-pow 32.0%

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

       9. associate-*r* 32.0%

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

       10. *-commutative 32.0%

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

       11. pow1/2 32.0%

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

       12. pow-flip 32.0%

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

       13. metadata-eval 32.0%

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

   13. Applied egg-rr 32.0%

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

       1. expm1-log1p-u 31.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right)\right)} \]

       2. expm1-undefine 5.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right)} - 1} \]

       3. metadata-eval 5.0%

          \[\leadsto e^{\mathsf{log1p}\left({\pi}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot \left(x \cdot 2\right)\right)} - 1 \]

       4. sqrt-pow1 5.0%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{{\pi}^{-1}}} \cdot \left(x \cdot 2\right)\right)} - 1 \]

       5. inv-pow 5.0%

          \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{1}{\pi}}} \cdot \left(x \cdot 2\right)\right)} - 1 \]

       6. *-commutative 5.0%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(x \cdot 2\right) \cdot \sqrt{\frac{1}{\pi}}}\right)} - 1 \]

       7. associate-*l* 5.0%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{\pi}}\right)}\right)} - 1 \]

       8. sqrt-div 5.0%

          \[\leadsto e^{\mathsf{log1p}\left(x \cdot \left(2 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)\right)} - 1 \]

       9. metadata-eval 5.0%

          \[\leadsto e^{\mathsf{log1p}\left(x \cdot \left(2 \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)\right)} - 1 \]

       10. un-div-inv 5.0%

           \[\leadsto e^{\mathsf{log1p}\left(x \cdot \color{blue}{\frac{2}{\sqrt{\pi}}}\right)} - 1 \]

   15. Applied egg-rr 5.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2}{\sqrt{\pi}}\right)} - 1} \]
   16. Step-by-step derivation

       1. sub-neg 5.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2}{\sqrt{\pi}}\right)} + \left(-1\right)} \]

       2. metadata-eval 5.0%

          \[\leadsto e^{\mathsf{log1p}\left(x \cdot \frac{2}{\sqrt{\pi}}\right)} + \color{blue}{-1} \]

       3. +-commutative 5.0%

          \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(x \cdot \frac{2}{\sqrt{\pi}}\right)}} \]

       4. log1p-undefine 5.0%

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

       5. rem-exp-log 5.1%

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

       6. associate-+r+ 32.0%

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

       7. metadata-eval 32.0%

          \[\leadsto \color{blue}{0} + x \cdot \frac{2}{\sqrt{\pi}} \]

       8. metadata-eval 32.0%

          \[\leadsto \color{blue}{2 \cdot 0} + x \cdot \frac{2}{\sqrt{\pi}} \]

       9. associate-*r/ 31.8%

          \[\leadsto 2 \cdot 0 + \color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}} \]

       10. *-commutative 31.8%

           \[\leadsto 2 \cdot 0 + \frac{\color{blue}{2 \cdot x}}{\sqrt{\pi}} \]

       11. associate-/l* 31.8%

           \[\leadsto 2 \cdot 0 + \color{blue}{2 \cdot \frac{x}{\sqrt{\pi}}} \]

       12. distribute-lft-in 31.8%

           \[\leadsto \color{blue}{2 \cdot \left(0 + \frac{x}{\sqrt{\pi}}\right)} \]

       13. +-lft-identity 31.8%

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

       14. associate-/l* 31.8%

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

       15. *-commutative 31.8%

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

       16. associate-*r/ 32.0%

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

   17. Simplified 32.0%

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

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.8%

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

      1. fma-undefine 99.8%

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

      2. add-sqr-sqrt 30.6%

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

      3. fabs-sqr 30.6%

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

      4. add-sqr-sqrt 99.5%

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

      5. associate-*r* 99.5%

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

      6. add-sqr-sqrt 30.8%

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

      7. fabs-sqr 30.8%

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

      8. add-sqr-sqrt 81.6%

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

      9. associate-*r* 81.6%

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

      10. cube-mult 81.6%

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot \color{blue}{{x}^{3}}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   6. Applied egg-rr 81.6%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\color{blue}{\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)} + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   7. Taylor expanded in x around inf 34.1%

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

   9. Simplified 34.1%

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

   11. Applied egg-rr 3.9%

       \[\leadsto \color{blue}{\left({\pi}^{-0.5} \cdot \left({x}^{6} \cdot x\right)\right) \cdot 0.047619047619047616} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot \left(x \cdot {x}^{6}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.7% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 70.6%

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

      1. associate-*r* 70.6%

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

      2. rem-square-sqrt 30.3%

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

      3. fabs-sqr 30.3%

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

      4. rem-square-sqrt 70.6%

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

      5. *-commutative 70.6%

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

   7. Simplified 70.6%

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

      1. expm1-log1p-u 99.0%

         \[\leadsto \left|x \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}}\right)\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

      2. expm1-undefine 97.9%

         \[\leadsto \left|x \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}}\right)} - 1\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

      3. inv-pow 97.9%

         \[\leadsto \left|x \cdot \left(\left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{{\pi}^{-1}}}\right)} - 1\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

      4. sqrt-pow1 97.9%

         \[\leadsto \left|x \cdot \left(\left(e^{\mathsf{log1p}\left(\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

      5. metadata-eval 97.9%

         \[\leadsto \left|x \cdot \left(\left(e^{\mathsf{log1p}\left({\pi}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   9. Applied egg-rr 69.5%

      \[\leadsto \left|x \cdot \left(2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\pi}^{-0.5}\right)} - 1\right)}\right)\right| \]
   10. Step-by-step derivation

       1. sub-neg 97.9%

          \[\leadsto \left|x \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\pi}^{-0.5}\right)} + \left(-1\right)\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       2. metadata-eval 97.9%

          \[\leadsto \left|x \cdot \left(\left(e^{\mathsf{log1p}\left({\pi}^{-0.5}\right)} + \color{blue}{-1}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       3. +-commutative 97.9%

          \[\leadsto \left|x \cdot \left(\color{blue}{\left(-1 + e^{\mathsf{log1p}\left({\pi}^{-0.5}\right)}\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       4. log1p-undefine 97.7%

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

       5. rem-exp-log 97.7%

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

       6. associate-+r+ 99.0%

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

       7. metadata-eval 99.0%

          \[\leadsto \left|x \cdot \left(\left(\color{blue}{0} + {\pi}^{-0.5}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       8. exp-to-pow 99.0%

          \[\leadsto \left|x \cdot \left(\left(0 + \color{blue}{e^{\log \pi \cdot -0.5}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       9. metadata-eval 99.0%

          \[\leadsto \left|x \cdot \left(\left(0 + e^{\log \pi \cdot \color{blue}{\left(-0.5\right)}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       10. distribute-rgt-neg-in 99.0%

           \[\leadsto \left|x \cdot \left(\left(0 + e^{\color{blue}{-\log \pi \cdot 0.5}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       11. exp-neg 99.0%

           \[\leadsto \left|x \cdot \left(\left(0 + \color{blue}{\frac{1}{e^{\log \pi \cdot 0.5}}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       12. metadata-eval 99.0%

           \[\leadsto \left|x \cdot \left(\left(0 + \frac{\color{blue}{--1}}{e^{\log \pi \cdot 0.5}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       13. exp-to-pow 99.0%

           \[\leadsto \left|x \cdot \left(\left(0 + \frac{--1}{\color{blue}{{\pi}^{0.5}}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       14. unpow1/2 99.0%

           \[\leadsto \left|x \cdot \left(\left(0 + \frac{--1}{\color{blue}{\sqrt{\pi}}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       15. distribute-neg-frac 99.0%

           \[\leadsto \left|x \cdot \left(\left(0 + \color{blue}{\left(-\frac{-1}{\sqrt{\pi}}\right)}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       16. sub-neg 99.0%

           \[\leadsto \left|x \cdot \left(\color{blue}{\left(0 - \frac{-1}{\sqrt{\pi}}\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       17. neg-sub0 99.0%

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

       18. distribute-neg-frac 99.0%

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

       19. metadata-eval 99.0%

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

       20. unpow1/2 99.0%

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

       21. exp-to-pow 99.0%

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

       22. exp-neg 99.0%

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

       23. distribute-rgt-neg-in 99.0%

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

       24. metadata-eval 99.0%

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

   11. Simplified 70.6%

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

       1. add-sqr-sqrt 30.3%

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

       2. fabs-sqr 30.3%

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

       3. add-sqr-sqrt 32.0%

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

       4. *-commutative 32.0%

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

       5. *-commutative 32.0%

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

       6. metadata-eval 32.0%

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

       7. sqrt-pow2 32.0%

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

       8. inv-pow 32.0%

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

       9. associate-*r* 32.0%

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

       10. *-commutative 32.0%

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

       11. pow1/2 32.0%

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

       12. pow-flip 32.0%

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

       13. metadata-eval 32.0%

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

   13. Applied egg-rr 32.0%

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

       1. expm1-log1p-u 31.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right)\right)} \]

       2. expm1-undefine 5.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right)} - 1} \]

       3. metadata-eval 5.0%

          \[\leadsto e^{\mathsf{log1p}\left({\pi}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot \left(x \cdot 2\right)\right)} - 1 \]

       4. sqrt-pow1 5.0%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{{\pi}^{-1}}} \cdot \left(x \cdot 2\right)\right)} - 1 \]

       5. inv-pow 5.0%

          \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{1}{\pi}}} \cdot \left(x \cdot 2\right)\right)} - 1 \]

       6. *-commutative 5.0%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(x \cdot 2\right) \cdot \sqrt{\frac{1}{\pi}}}\right)} - 1 \]

       7. associate-*l* 5.0%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{\pi}}\right)}\right)} - 1 \]

       8. sqrt-div 5.0%

          \[\leadsto e^{\mathsf{log1p}\left(x \cdot \left(2 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)\right)} - 1 \]

       9. metadata-eval 5.0%

          \[\leadsto e^{\mathsf{log1p}\left(x \cdot \left(2 \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)\right)} - 1 \]

       10. un-div-inv 5.0%

           \[\leadsto e^{\mathsf{log1p}\left(x \cdot \color{blue}{\frac{2}{\sqrt{\pi}}}\right)} - 1 \]

   15. Applied egg-rr 5.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2}{\sqrt{\pi}}\right)} - 1} \]
   16. Step-by-step derivation

       1. sub-neg 5.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2}{\sqrt{\pi}}\right)} + \left(-1\right)} \]

       2. metadata-eval 5.0%

          \[\leadsto e^{\mathsf{log1p}\left(x \cdot \frac{2}{\sqrt{\pi}}\right)} + \color{blue}{-1} \]

       3. +-commutative 5.0%

          \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(x \cdot \frac{2}{\sqrt{\pi}}\right)}} \]

       4. log1p-undefine 5.0%

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

       5. rem-exp-log 5.1%

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

       6. associate-+r+ 32.0%

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

       7. metadata-eval 32.0%

          \[\leadsto \color{blue}{0} + x \cdot \frac{2}{\sqrt{\pi}} \]

       8. metadata-eval 32.0%

          \[\leadsto \color{blue}{2 \cdot 0} + x \cdot \frac{2}{\sqrt{\pi}} \]

       9. associate-*r/ 31.8%

          \[\leadsto 2 \cdot 0 + \color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}} \]

       10. *-commutative 31.8%

           \[\leadsto 2 \cdot 0 + \frac{\color{blue}{2 \cdot x}}{\sqrt{\pi}} \]

       11. associate-/l* 31.8%

           \[\leadsto 2 \cdot 0 + \color{blue}{2 \cdot \frac{x}{\sqrt{\pi}}} \]

       12. distribute-lft-in 31.8%

           \[\leadsto \color{blue}{2 \cdot \left(0 + \frac{x}{\sqrt{\pi}}\right)} \]

       13. +-lft-identity 31.8%

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

       14. associate-/l* 31.8%

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

       15. *-commutative 31.8%

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

       16. associate-*r/ 32.0%

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

   17. Simplified 32.0%

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

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.8%

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

      1. fma-undefine 99.8%

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

      2. add-sqr-sqrt 30.6%

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

      3. fabs-sqr 30.6%

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

      4. add-sqr-sqrt 99.5%

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

      5. associate-*r* 99.5%

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

      6. add-sqr-sqrt 30.8%

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

      7. fabs-sqr 30.8%

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

      8. add-sqr-sqrt 81.6%

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

      9. associate-*r* 81.6%

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

      10. cube-mult 81.6%

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot \color{blue}{{x}^{3}}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   6. Applied egg-rr 81.6%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\color{blue}{\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)} + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   7. Taylor expanded in x around inf 34.1%

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

   9. Simplified 34.1%

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

   11. Applied egg-rr 3.9%

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

   12. Taylor expanded in x around 0 3.9%

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

4. Final simplification 32.0%

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

Alternative 10: 34.7% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.0022675736961451248 \cdot \frac{{x}^{14}}{\pi}}\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.0022675736961451248 * N[(N[Power[x, 14.0], $MachinePrecision] / 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 99.8%

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

   5. Taylor expanded in x around 0 70.6%

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

      1. associate-*r* 70.6%

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

      2. rem-square-sqrt 30.3%

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

      3. fabs-sqr 30.3%

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

      4. rem-square-sqrt 70.6%

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

      5. *-commutative 70.6%

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

   7. Simplified 70.6%

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

      1. expm1-log1p-u 99.0%

         \[\leadsto \left|x \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}}\right)\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

      2. expm1-undefine 97.9%

         \[\leadsto \left|x \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}}\right)} - 1\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

      3. inv-pow 97.9%

         \[\leadsto \left|x \cdot \left(\left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{{\pi}^{-1}}}\right)} - 1\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

      4. sqrt-pow1 97.9%

         \[\leadsto \left|x \cdot \left(\left(e^{\mathsf{log1p}\left(\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

      5. metadata-eval 97.9%

         \[\leadsto \left|x \cdot \left(\left(e^{\mathsf{log1p}\left({\pi}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   9. Applied egg-rr 69.5%

      \[\leadsto \left|x \cdot \left(2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\pi}^{-0.5}\right)} - 1\right)}\right)\right| \]
   10. Step-by-step derivation

       1. sub-neg 97.9%

          \[\leadsto \left|x \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\pi}^{-0.5}\right)} + \left(-1\right)\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       2. metadata-eval 97.9%

          \[\leadsto \left|x \cdot \left(\left(e^{\mathsf{log1p}\left({\pi}^{-0.5}\right)} + \color{blue}{-1}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       3. +-commutative 97.9%

          \[\leadsto \left|x \cdot \left(\color{blue}{\left(-1 + e^{\mathsf{log1p}\left({\pi}^{-0.5}\right)}\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       4. log1p-undefine 97.7%

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

       5. rem-exp-log 97.7%

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

       6. associate-+r+ 99.0%

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

       7. metadata-eval 99.0%

          \[\leadsto \left|x \cdot \left(\left(\color{blue}{0} + {\pi}^{-0.5}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       8. exp-to-pow 99.0%

          \[\leadsto \left|x \cdot \left(\left(0 + \color{blue}{e^{\log \pi \cdot -0.5}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       9. metadata-eval 99.0%

          \[\leadsto \left|x \cdot \left(\left(0 + e^{\log \pi \cdot \color{blue}{\left(-0.5\right)}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       10. distribute-rgt-neg-in 99.0%

           \[\leadsto \left|x \cdot \left(\left(0 + e^{\color{blue}{-\log \pi \cdot 0.5}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       11. exp-neg 99.0%

           \[\leadsto \left|x \cdot \left(\left(0 + \color{blue}{\frac{1}{e^{\log \pi \cdot 0.5}}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       12. metadata-eval 99.0%

           \[\leadsto \left|x \cdot \left(\left(0 + \frac{\color{blue}{--1}}{e^{\log \pi \cdot 0.5}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       13. exp-to-pow 99.0%

           \[\leadsto \left|x \cdot \left(\left(0 + \frac{--1}{\color{blue}{{\pi}^{0.5}}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       14. unpow1/2 99.0%

           \[\leadsto \left|x \cdot \left(\left(0 + \frac{--1}{\color{blue}{\sqrt{\pi}}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       15. distribute-neg-frac 99.0%

           \[\leadsto \left|x \cdot \left(\left(0 + \color{blue}{\left(-\frac{-1}{\sqrt{\pi}}\right)}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       16. sub-neg 99.0%

           \[\leadsto \left|x \cdot \left(\color{blue}{\left(0 - \frac{-1}{\sqrt{\pi}}\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

       17. neg-sub0 99.0%

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

       18. distribute-neg-frac 99.0%

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

       19. metadata-eval 99.0%

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

       20. unpow1/2 99.0%

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

       21. exp-to-pow 99.0%

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

       22. exp-neg 99.0%

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

       23. distribute-rgt-neg-in 99.0%

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

       24. metadata-eval 99.0%

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

   11. Simplified 70.6%

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

       1. add-sqr-sqrt 30.3%

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

       2. fabs-sqr 30.3%

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

       3. add-sqr-sqrt 32.0%

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

       4. *-commutative 32.0%

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

       5. *-commutative 32.0%

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

       6. metadata-eval 32.0%

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

       7. sqrt-pow2 32.0%

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

       8. inv-pow 32.0%

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

       9. associate-*r* 32.0%

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

       10. *-commutative 32.0%

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

       11. pow1/2 32.0%

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

       12. pow-flip 32.0%

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

       13. metadata-eval 32.0%

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

   13. Applied egg-rr 32.0%

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

       1. expm1-log1p-u 31.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right)\right)} \]

       2. expm1-undefine 5.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right)} - 1} \]

       3. metadata-eval 5.0%

          \[\leadsto e^{\mathsf{log1p}\left({\pi}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot \left(x \cdot 2\right)\right)} - 1 \]

       4. sqrt-pow1 5.0%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{{\pi}^{-1}}} \cdot \left(x \cdot 2\right)\right)} - 1 \]

       5. inv-pow 5.0%

          \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{1}{\pi}}} \cdot \left(x \cdot 2\right)\right)} - 1 \]

       6. *-commutative 5.0%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(x \cdot 2\right) \cdot \sqrt{\frac{1}{\pi}}}\right)} - 1 \]

       7. associate-*l* 5.0%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{\pi}}\right)}\right)} - 1 \]

       8. sqrt-div 5.0%

          \[\leadsto e^{\mathsf{log1p}\left(x \cdot \left(2 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)\right)} - 1 \]

       9. metadata-eval 5.0%

          \[\leadsto e^{\mathsf{log1p}\left(x \cdot \left(2 \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)\right)} - 1 \]

       10. un-div-inv 5.0%

           \[\leadsto e^{\mathsf{log1p}\left(x \cdot \color{blue}{\frac{2}{\sqrt{\pi}}}\right)} - 1 \]

   15. Applied egg-rr 5.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2}{\sqrt{\pi}}\right)} - 1} \]
   16. Step-by-step derivation

       1. sub-neg 5.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2}{\sqrt{\pi}}\right)} + \left(-1\right)} \]

       2. metadata-eval 5.0%

          \[\leadsto e^{\mathsf{log1p}\left(x \cdot \frac{2}{\sqrt{\pi}}\right)} + \color{blue}{-1} \]

       3. +-commutative 5.0%

          \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(x \cdot \frac{2}{\sqrt{\pi}}\right)}} \]

       4. log1p-undefine 5.0%

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

       5. rem-exp-log 5.1%

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

       6. associate-+r+ 32.0%

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

       7. metadata-eval 32.0%

          \[\leadsto \color{blue}{0} + x \cdot \frac{2}{\sqrt{\pi}} \]

       8. metadata-eval 32.0%

          \[\leadsto \color{blue}{2 \cdot 0} + x \cdot \frac{2}{\sqrt{\pi}} \]

       9. associate-*r/ 31.8%

          \[\leadsto 2 \cdot 0 + \color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}} \]

       10. *-commutative 31.8%

           \[\leadsto 2 \cdot 0 + \frac{\color{blue}{2 \cdot x}}{\sqrt{\pi}} \]

       11. associate-/l* 31.8%

           \[\leadsto 2 \cdot 0 + \color{blue}{2 \cdot \frac{x}{\sqrt{\pi}}} \]

       12. distribute-lft-in 31.8%

           \[\leadsto \color{blue}{2 \cdot \left(0 + \frac{x}{\sqrt{\pi}}\right)} \]

       13. +-lft-identity 31.8%

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

       14. associate-/l* 31.8%

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

       15. *-commutative 31.8%

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

       16. associate-*r/ 32.0%

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

   17. Simplified 32.0%

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

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.8%

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

      1. fma-undefine 99.8%

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

      2. add-sqr-sqrt 30.6%

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

      3. fabs-sqr 30.6%

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

      4. add-sqr-sqrt 99.5%

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

      5. associate-*r* 99.5%

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

      6. add-sqr-sqrt 30.8%

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

      7. fabs-sqr 30.8%

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

      8. add-sqr-sqrt 81.6%

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

      9. associate-*r* 81.6%

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

      10. cube-mult 81.6%

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot \color{blue}{{x}^{3}}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   6. Applied egg-rr 81.6%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\color{blue}{\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)} + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   7. Taylor expanded in x around inf 34.1%

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

   9. Simplified 34.1%

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

   11. Applied egg-rr 3.9%

       \[\leadsto \color{blue}{\left({\pi}^{-0.5} \cdot \left({x}^{6} \cdot x\right)\right) \cdot 0.047619047619047616} \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 3.7%

          \[\leadsto \color{blue}{\sqrt{\left({\pi}^{-0.5} \cdot \left({x}^{6} \cdot x\right)\right) \cdot 0.047619047619047616} \cdot \sqrt{\left({\pi}^{-0.5} \cdot \left({x}^{6} \cdot x\right)\right) \cdot 0.047619047619047616}} \]

       2. sqrt-unprod 32.2%

          \[\leadsto \color{blue}{\sqrt{\left(\left({\pi}^{-0.5} \cdot \left({x}^{6} \cdot x\right)\right) \cdot 0.047619047619047616\right) \cdot \left(\left({\pi}^{-0.5} \cdot \left({x}^{6} \cdot x\right)\right) \cdot 0.047619047619047616\right)}} \]

       3. swap-sqr 32.2%

          \[\leadsto \sqrt{\color{blue}{\left(\left({\pi}^{-0.5} \cdot \left({x}^{6} \cdot x\right)\right) \cdot \left({\pi}^{-0.5} \cdot \left({x}^{6} \cdot x\right)\right)\right) \cdot \left(0.047619047619047616 \cdot 0.047619047619047616\right)}} \]

   13. Applied egg-rr 32.3%

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

       1. *-commutative 32.3%

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

       2. associate-*l/ 32.3%

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

       3. *-lft-identity 32.3%

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

   15. Simplified 32.3%

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

Alternative 11: 34.7% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 70.6%

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

   1. associate-*r* 70.6%

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

   2. rem-square-sqrt 30.3%

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

   3. fabs-sqr 30.3%

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

   4. rem-square-sqrt 70.6%

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

   5. *-commutative 70.6%

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

7. Simplified 70.6%

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

   1. expm1-log1p-u 99.0%

      \[\leadsto \left|x \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}}\right)\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   2. expm1-undefine 97.9%

      \[\leadsto \left|x \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}}\right)} - 1\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   3. inv-pow 97.9%

      \[\leadsto \left|x \cdot \left(\left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{{\pi}^{-1}}}\right)} - 1\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   4. sqrt-pow1 97.9%

      \[\leadsto \left|x \cdot \left(\left(e^{\mathsf{log1p}\left(\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   5. metadata-eval 97.9%

      \[\leadsto \left|x \cdot \left(\left(e^{\mathsf{log1p}\left({\pi}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

9. Applied egg-rr 69.5%

   \[\leadsto \left|x \cdot \left(2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\pi}^{-0.5}\right)} - 1\right)}\right)\right| \]
10. Step-by-step derivation

    1. sub-neg 97.9%

       \[\leadsto \left|x \cdot \left(\color{blue}{\left(e^{\mathsf{log1p}\left({\pi}^{-0.5}\right)} + \left(-1\right)\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    2. metadata-eval 97.9%

       \[\leadsto \left|x \cdot \left(\left(e^{\mathsf{log1p}\left({\pi}^{-0.5}\right)} + \color{blue}{-1}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    3. +-commutative 97.9%

       \[\leadsto \left|x \cdot \left(\color{blue}{\left(-1 + e^{\mathsf{log1p}\left({\pi}^{-0.5}\right)}\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    4. log1p-undefine 97.7%

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

    5. rem-exp-log 97.7%

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

    6. associate-+r+ 99.0%

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

    7. metadata-eval 99.0%

       \[\leadsto \left|x \cdot \left(\left(\color{blue}{0} + {\pi}^{-0.5}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    8. exp-to-pow 99.0%

       \[\leadsto \left|x \cdot \left(\left(0 + \color{blue}{e^{\log \pi \cdot -0.5}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    9. metadata-eval 99.0%

       \[\leadsto \left|x \cdot \left(\left(0 + e^{\log \pi \cdot \color{blue}{\left(-0.5\right)}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    10. distribute-rgt-neg-in 99.0%

        \[\leadsto \left|x \cdot \left(\left(0 + e^{\color{blue}{-\log \pi \cdot 0.5}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    11. exp-neg 99.0%

        \[\leadsto \left|x \cdot \left(\left(0 + \color{blue}{\frac{1}{e^{\log \pi \cdot 0.5}}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    12. metadata-eval 99.0%

        \[\leadsto \left|x \cdot \left(\left(0 + \frac{\color{blue}{--1}}{e^{\log \pi \cdot 0.5}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    13. exp-to-pow 99.0%

        \[\leadsto \left|x \cdot \left(\left(0 + \frac{--1}{\color{blue}{{\pi}^{0.5}}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    14. unpow1/2 99.0%

        \[\leadsto \left|x \cdot \left(\left(0 + \frac{--1}{\color{blue}{\sqrt{\pi}}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    15. distribute-neg-frac 99.0%

        \[\leadsto \left|x \cdot \left(\left(0 + \color{blue}{\left(-\frac{-1}{\sqrt{\pi}}\right)}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    16. sub-neg 99.0%

        \[\leadsto \left|x \cdot \left(\color{blue}{\left(0 - \frac{-1}{\sqrt{\pi}}\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    17. neg-sub0 99.0%

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

    18. distribute-neg-frac 99.0%

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

    19. metadata-eval 99.0%

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

    20. unpow1/2 99.0%

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

    21. exp-to-pow 99.0%

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

    22. exp-neg 99.0%

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

    23. distribute-rgt-neg-in 99.0%

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

    24. metadata-eval 99.0%

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

11. Simplified 70.6%

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

    1. add-sqr-sqrt 30.3%

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

    2. fabs-sqr 30.3%

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

    3. add-sqr-sqrt 32.0%

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

    4. *-commutative 32.0%

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

    5. *-commutative 32.0%

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

    6. metadata-eval 32.0%

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

    7. sqrt-pow2 32.0%

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

    8. inv-pow 32.0%

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

    9. associate-*r* 32.0%

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

    10. *-commutative 32.0%

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

    11. pow1/2 32.0%

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

    12. pow-flip 32.0%

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

    13. metadata-eval 32.0%

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

13. Applied egg-rr 32.0%

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

    1. expm1-log1p-u 31.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right)\right)} \]

    2. expm1-undefine 5.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right)} - 1} \]

    3. metadata-eval 5.0%

       \[\leadsto e^{\mathsf{log1p}\left({\pi}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot \left(x \cdot 2\right)\right)} - 1 \]

    4. sqrt-pow1 5.0%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{{\pi}^{-1}}} \cdot \left(x \cdot 2\right)\right)} - 1 \]

    5. inv-pow 5.0%

       \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{1}{\pi}}} \cdot \left(x \cdot 2\right)\right)} - 1 \]

    6. *-commutative 5.0%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(x \cdot 2\right) \cdot \sqrt{\frac{1}{\pi}}}\right)} - 1 \]

    7. associate-*l* 5.0%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{\pi}}\right)}\right)} - 1 \]

    8. sqrt-div 5.0%

       \[\leadsto e^{\mathsf{log1p}\left(x \cdot \left(2 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)\right)} - 1 \]

    9. metadata-eval 5.0%

       \[\leadsto e^{\mathsf{log1p}\left(x \cdot \left(2 \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)\right)} - 1 \]

    10. un-div-inv 5.0%

        \[\leadsto e^{\mathsf{log1p}\left(x \cdot \color{blue}{\frac{2}{\sqrt{\pi}}}\right)} - 1 \]

15. Applied egg-rr 5.0%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2}{\sqrt{\pi}}\right)} - 1} \]
16. Step-by-step derivation

    1. sub-neg 5.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2}{\sqrt{\pi}}\right)} + \left(-1\right)} \]

    2. metadata-eval 5.0%

       \[\leadsto e^{\mathsf{log1p}\left(x \cdot \frac{2}{\sqrt{\pi}}\right)} + \color{blue}{-1} \]

    3. +-commutative 5.0%

       \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(x \cdot \frac{2}{\sqrt{\pi}}\right)}} \]

    4. log1p-undefine 5.0%

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

    5. rem-exp-log 5.1%

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

    6. associate-+r+ 32.0%

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

    7. metadata-eval 32.0%

       \[\leadsto \color{blue}{0} + x \cdot \frac{2}{\sqrt{\pi}} \]

    8. metadata-eval 32.0%

       \[\leadsto \color{blue}{2 \cdot 0} + x \cdot \frac{2}{\sqrt{\pi}} \]

    9. associate-*r/ 31.8%

       \[\leadsto 2 \cdot 0 + \color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}} \]

    10. *-commutative 31.8%

        \[\leadsto 2 \cdot 0 + \frac{\color{blue}{2 \cdot x}}{\sqrt{\pi}} \]

    11. associate-/l* 31.8%

        \[\leadsto 2 \cdot 0 + \color{blue}{2 \cdot \frac{x}{\sqrt{\pi}}} \]

    12. distribute-lft-in 31.8%

        \[\leadsto \color{blue}{2 \cdot \left(0 + \frac{x}{\sqrt{\pi}}\right)} \]

    13. +-lft-identity 31.8%

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

    14. associate-/l* 31.8%

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

    15. *-commutative 31.8%

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

    16. associate-*r/ 32.0%

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

17. Simplified 32.0%

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

Reproduce

herbie shell --seed 2024116 
(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)))))))