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

Percentage Accurate: 99.8% → 99.8%

Time: 8.6s

Alternatives: 12

Speedup: 1.5×

• 27.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = Float64(abs(x) * Float64(x * x))
	t_1 = Float64(Float64(x * x) * t_0)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(fma(2.0, abs(x), Float64(0.6666666666666666 * t_0)) + Float64(0.2 * t_1)) + Float64(0.047619047619047616 * Float64(Float64(x * 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[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision] + N[(0.6666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.2 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[(N[(x * x), $MachinePrecision] * t$95$1), $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. Final simplification 99.8%

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

Alternative 3: 99.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

   1. expm1-log1p-u 99.2%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left|x\right|}{\sqrt{\pi}}\right)\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   2. expm1-udef 37.7%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\left|x\right|}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   3. add-sqr-sqrt 3.4%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   4. fabs-sqr 3.4%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   5. add-sqr-sqrt 6.3%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{x}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

5. Applied egg-rr 6.3%

   \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]
6. Step-by-step derivation

   1. expm1-def 67.8%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}}\right)\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   2. expm1-log1p 99.4%

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

7. Simplified 99.4%

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

   1. metadata-eval 99.4%

      \[\leadsto \left|\frac{x}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(\color{blue}{\frac{1}{5}}, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   2. fma-udef 99.4%

      \[\leadsto \left|\frac{x}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \color{blue}{\left(\frac{1}{5} \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)}\right)\right| \]

   3. metadata-eval 99.4%

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

9. Applied egg-rr 99.4%

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

10. Final simplification 99.4%

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

Alternative 4: 93.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(0.2 \cdot {x}^{4} + \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (fabs
    (*
     (* x (pow PI -0.5))
     (+ (* 0.2 (pow x 4.0)) (+ 2.0 (* 0.6666666666666666 (* x x))))))
   (fabs
    (*
     (sqrt (/ 1.0 PI))
     (+ (* 0.2 (pow x 5.0)) (* 0.047619047619047616 (pow x 7.0)))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = fabs(((x * pow(((double) M_PI), -0.5)) * ((0.2 * pow(x, 4.0)) + (2.0 + (0.6666666666666666 * (x * x))))));
	} else {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * ((0.2 * pow(x, 5.0)) + (0.047619047619047616 * pow(x, 7.0)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = Math.abs(((x * Math.pow(Math.PI, -0.5)) * ((0.2 * Math.pow(x, 4.0)) + (2.0 + (0.6666666666666666 * (x * x))))));
	} else {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * ((0.2 * Math.pow(x, 5.0)) + (0.047619047619047616 * Math.pow(x, 7.0)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.2:
		tmp = math.fabs(((x * math.pow(math.pi, -0.5)) * ((0.2 * math.pow(x, 4.0)) + (2.0 + (0.6666666666666666 * (x * x))))))
	else:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * ((0.2 * math.pow(x, 5.0)) + (0.047619047619047616 * math.pow(x, 7.0)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = abs(Float64(Float64(x * (pi ^ -0.5)) * Float64(Float64(0.2 * (x ^ 4.0)) + Float64(2.0 + Float64(0.6666666666666666 * Float64(x * x))))));
	else
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(0.2 * (x ^ 5.0)) + Float64(0.047619047619047616 * (x ^ 7.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.2)
		tmp = abs(((x * (pi ^ -0.5)) * ((0.2 * (x ^ 4.0)) + (2.0 + (0.6666666666666666 * (x * x))))));
	else
		tmp = abs((sqrt((1.0 / pi)) * ((0.2 * (x ^ 5.0)) + (0.047619047619047616 * (x ^ 7.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[Abs[N[(N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

      1. expm1-log1p-u 99.2%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left|x\right|}{\sqrt{\pi}}\right)\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

      2. expm1-udef 37.7%

         \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\left|x\right|}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

      3. add-sqr-sqrt 3.4%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

      4. fabs-sqr 3.4%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

      5. add-sqr-sqrt 6.3%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{x}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   5. Applied egg-rr 6.3%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]
   6. Step-by-step derivation

      1. expm1-def 67.8%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}}\right)\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

      2. expm1-log1p 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in x around 0 93.0%

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

      1. div-inv 93.4%

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

      2. pow1/2 93.4%

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

      3. pow-flip 93.4%

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

      4. metadata-eval 93.4%

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

   10. Applied egg-rr 93.4%

       \[\leadsto \left|\color{blue}{\left(x \cdot {\pi}^{-0.5}\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + 0.2 \cdot {x}^{4}\right)\right| \]
   11. Step-by-step derivation

       1. fma-udef 93.4%

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

   12. Applied egg-rr 93.4%

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

   if 2.2000000000000002 < x

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around inf 35.6%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left(\left({x}^{4} \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}\right) + 0.2 \cdot \left(\left({x}^{2} \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

   6. Simplified 35.6%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(0.2 \cdot {x}^{4} + \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \end{array} \]

Alternative 5: 93.7% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7:\\ \;\;\;\;\left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(0.2 \cdot {x}^{4} + \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.7)
   (fabs
    (*
     (* x (pow PI -0.5))
     (+ (* 0.2 (pow x 4.0)) (+ 2.0 (* 0.6666666666666666 (* x x))))))
   (fabs (* (sqrt (/ 1.0 PI)) (* 0.047619047619047616 (pow x 7.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.7) {
		tmp = fabs(((x * pow(((double) M_PI), -0.5)) * ((0.2 * pow(x, 4.0)) + (2.0 + (0.6666666666666666 * (x * x))))));
	} else {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * (0.047619047619047616 * pow(x, 7.0))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.7) {
		tmp = Math.abs(((x * Math.pow(Math.PI, -0.5)) * ((0.2 * Math.pow(x, 4.0)) + (2.0 + (0.6666666666666666 * (x * x))))));
	} else {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * (0.047619047619047616 * Math.pow(x, 7.0))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.7:
		tmp = math.fabs(((x * math.pow(math.pi, -0.5)) * ((0.2 * math.pow(x, 4.0)) + (2.0 + (0.6666666666666666 * (x * x))))))
	else:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * (0.047619047619047616 * math.pow(x, 7.0))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.7)
		tmp = abs(Float64(Float64(x * (pi ^ -0.5)) * Float64(Float64(0.2 * (x ^ 4.0)) + Float64(2.0 + Float64(0.6666666666666666 * Float64(x * x))))));
	else
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(0.047619047619047616 * (x ^ 7.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.7)
		tmp = abs(((x * (pi ^ -0.5)) * ((0.2 * (x ^ 4.0)) + (2.0 + (0.6666666666666666 * (x * x))))));
	else
		tmp = abs((sqrt((1.0 / pi)) * (0.047619047619047616 * (x ^ 7.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.7], N[Abs[N[(N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7000000000000002

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

      1. expm1-log1p-u 99.2%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left|x\right|}{\sqrt{\pi}}\right)\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

      2. expm1-udef 37.7%

         \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\left|x\right|}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

      3. add-sqr-sqrt 3.4%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

      4. fabs-sqr 3.4%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

      5. add-sqr-sqrt 6.3%

         \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{x}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   5. Applied egg-rr 6.3%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]
   6. Step-by-step derivation

      1. expm1-def 67.8%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}}\right)\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

      2. expm1-log1p 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in x around 0 93.0%

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

      1. div-inv 93.4%

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

      2. pow1/2 93.4%

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

      3. pow-flip 93.4%

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

      4. metadata-eval 93.4%

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

   10. Applied egg-rr 93.4%

       \[\leadsto \left|\color{blue}{\left(x \cdot {\pi}^{-0.5}\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + 0.2 \cdot {x}^{4}\right)\right| \]
   11. Step-by-step derivation

       1. fma-udef 93.4%

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

   12. Applied egg-rr 93.4%

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

   if 2.7000000000000002 < x

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around inf 35.4%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left(\left({x}^{4} \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   5. Step-by-step derivation

      1. *-commutative 35.4%

         \[\leadsto \left|\color{blue}{\left(\left({x}^{4} \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}\right) \cdot 0.047619047619047616}\right| \]

      2. *-commutative 35.4%

         \[\leadsto \left|\color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left({x}^{4} \cdot {\left(\left|x\right|\right)}^{3}\right)\right)} \cdot 0.047619047619047616\right| \]

      3. associate-*l* 35.4%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left({x}^{4} \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot 0.047619047619047616\right)}\right| \]

      4. unpow3 35.4%

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

      5. sqr-abs 35.4%

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

      6. associate-*r* 35.4%

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

      7. metadata-eval 35.4%

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

      8. pow-sqr 35.4%

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

      9. unpow2 35.4%

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

      10. unpow2 35.4%

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

      11. unpow3 35.3%

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

      12. cube-prod 35.4%

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

      13. pow-sqr 35.4%

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

      14. metadata-eval 35.4%

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

      15. rem-square-sqrt 2.2%

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

      16. fabs-sqr 2.2%

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

      17. rem-square-sqrt 35.4%

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

   6. Simplified 35.4%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7:\\ \;\;\;\;\left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(0.2 \cdot {x}^{4} + \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \end{array} \]

Alternative 6: 89.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (fabs (* (pow PI -0.5) (+ (* 0.6666666666666666 (pow x 3.0)) (* 2.0 x))))
   (fabs (* (sqrt (/ 1.0 PI)) (* 0.047619047619047616 (pow x 7.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = fabs((pow(((double) M_PI), -0.5) * ((0.6666666666666666 * pow(x, 3.0)) + (2.0 * x))));
	} else {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * (0.047619047619047616 * pow(x, 7.0))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = Math.abs((Math.pow(Math.PI, -0.5) * ((0.6666666666666666 * Math.pow(x, 3.0)) + (2.0 * x))));
	} else {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * (0.047619047619047616 * Math.pow(x, 7.0))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.2:
		tmp = math.fabs((math.pow(math.pi, -0.5) * ((0.6666666666666666 * math.pow(x, 3.0)) + (2.0 * x))))
	else:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * (0.047619047619047616 * math.pow(x, 7.0))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = abs(Float64((pi ^ -0.5) * Float64(Float64(0.6666666666666666 * (x ^ 3.0)) + Float64(2.0 * x))));
	else
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(0.047619047619047616 * (x ^ 7.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.2)
		tmp = abs(((pi ^ -0.5) * ((0.6666666666666666 * (x ^ 3.0)) + (2.0 * x))));
	else
		tmp = abs((sqrt((1.0 / pi)) * (0.047619047619047616 * (x ^ 7.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 89.8%

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

      1. fma-def 89.8%

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

      2. rem-square-sqrt 37.3%

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

      3. fabs-sqr 37.3%

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

      4. rem-square-sqrt 89.6%

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

      5. *-commutative 89.6%

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

      6. rem-square-sqrt 37.0%

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

      7. fabs-sqr 37.0%

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

      8. rem-square-sqrt 89.8%

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

   6. Simplified 89.8%

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

      1. fma-udef 89.8%

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

   8. Applied egg-rr 89.8%

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

      1. distribute-lft-in 89.8%

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

      2. inv-pow 89.8%

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

      3. sqrt-pow1 89.8%

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

      4. metadata-eval 89.8%

         \[\leadsto \left|{\pi}^{\color{blue}{-0.5}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2\right)\right| \]

      5. *-commutative 89.8%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right) + \color{blue}{\left(x \cdot 2\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

      6. inv-pow 89.8%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right) + \left(x \cdot 2\right) \cdot \sqrt{\color{blue}{{\pi}^{-1}}}\right| \]

      7. sqrt-pow1 89.8%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right) + \left(x \cdot 2\right) \cdot \color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}\right| \]

      8. metadata-eval 89.8%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right) + \left(x \cdot 2\right) \cdot {\pi}^{\color{blue}{-0.5}}\right| \]

   10. Applied egg-rr 89.8%

       \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right) + \left(x \cdot 2\right) \cdot {\pi}^{-0.5}}\right| \]
   11. Step-by-step derivation

       1. *-commutative 89.8%

          \[\leadsto \left|{\pi}^{-0.5} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right) + \color{blue}{{\pi}^{-0.5} \cdot \left(x \cdot 2\right)}\right| \]

       2. distribute-lft-in 89.8%

          \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(0.6666666666666666 \cdot {x}^{3} + x \cdot 2\right)}\right| \]

       3. *-commutative 89.8%

          \[\leadsto \left|{\pi}^{-0.5} \cdot \left(0.6666666666666666 \cdot {x}^{3} + \color{blue}{2 \cdot x}\right)\right| \]

   12. Simplified 89.8%

       \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right)}\right| \]

   if 2.2000000000000002 < x

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around inf 35.4%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left(\left({x}^{4} \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   5. Step-by-step derivation

      1. *-commutative 35.4%

         \[\leadsto \left|\color{blue}{\left(\left({x}^{4} \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}\right) \cdot 0.047619047619047616}\right| \]

      2. *-commutative 35.4%

         \[\leadsto \left|\color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left({x}^{4} \cdot {\left(\left|x\right|\right)}^{3}\right)\right)} \cdot 0.047619047619047616\right| \]

      3. associate-*l* 35.4%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left({x}^{4} \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot 0.047619047619047616\right)}\right| \]

      4. unpow3 35.4%

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

      5. sqr-abs 35.4%

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

      6. associate-*r* 35.4%

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

      7. metadata-eval 35.4%

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

      8. pow-sqr 35.4%

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

      9. unpow2 35.4%

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

      10. unpow2 35.4%

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

      11. unpow3 35.3%

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

      12. cube-prod 35.4%

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

      13. pow-sqr 35.4%

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

      14. metadata-eval 35.4%

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

      15. rem-square-sqrt 2.2%

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

      16. fabs-sqr 2.2%

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

      17. rem-square-sqrt 35.4%

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

   6. Simplified 35.4%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \end{array} \]

Alternative 7: 68.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(2 \cdot x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.9)
   (fabs (* (pow PI -0.5) (* 2.0 x)))
   (fabs (* (sqrt (/ 1.0 PI)) (* 0.047619047619047616 (pow x 7.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = fabs((pow(((double) M_PI), -0.5) * (2.0 * x)));
	} else {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * (0.047619047619047616 * pow(x, 7.0))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = Math.abs((Math.pow(Math.PI, -0.5) * (2.0 * x)));
	} else {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * (0.047619047619047616 * Math.pow(x, 7.0))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.9:
		tmp = math.fabs((math.pow(math.pi, -0.5) * (2.0 * x)))
	else:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * (0.047619047619047616 * math.pow(x, 7.0))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.9)
		tmp = abs(Float64((pi ^ -0.5) * Float64(2.0 * x)));
	else
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(0.047619047619047616 * (x ^ 7.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.9)
		tmp = abs(((pi ^ -0.5) * (2.0 * x)));
	else
		tmp = abs((sqrt((1.0 / pi)) * (0.047619047619047616 * (x ^ 7.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.9], N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(2.0 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 89.8%

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

      1. fma-def 89.8%

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

      2. rem-square-sqrt 37.3%

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

      3. fabs-sqr 37.3%

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

      4. rem-square-sqrt 89.6%

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

      5. *-commutative 89.6%

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

      6. rem-square-sqrt 37.0%

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

      7. fabs-sqr 37.0%

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

      8. rem-square-sqrt 89.8%

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

   6. Simplified 89.8%

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

   7. Taylor expanded in x around 0 69.2%

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

      1. associate-*r* 69.2%

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

      2. *-commutative 69.2%

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

   9. Simplified 69.2%

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

       1. expm1-log1p-u 67.4%

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

       2. expm1-udef 5.8%

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

       3. inv-pow 5.8%

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

       4. sqrt-pow1 5.8%

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

       5. metadata-eval 5.8%

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

   11. Applied egg-rr 5.8%

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

       1. expm1-def 67.4%

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

       2. expm1-log1p 69.2%

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

       3. *-commutative 69.2%

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

   13. Simplified 69.2%

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

   if 1.8999999999999999 < x

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around inf 35.4%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left(\left({x}^{4} \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   5. Step-by-step derivation

      1. *-commutative 35.4%

         \[\leadsto \left|\color{blue}{\left(\left({x}^{4} \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}\right) \cdot 0.047619047619047616}\right| \]

      2. *-commutative 35.4%

         \[\leadsto \left|\color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left({x}^{4} \cdot {\left(\left|x\right|\right)}^{3}\right)\right)} \cdot 0.047619047619047616\right| \]

      3. associate-*l* 35.4%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left({x}^{4} \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot 0.047619047619047616\right)}\right| \]

      4. unpow3 35.4%

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

      5. sqr-abs 35.4%

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

      6. associate-*r* 35.4%

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

      7. metadata-eval 35.4%

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

      8. pow-sqr 35.4%

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

      9. unpow2 35.4%

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

      10. unpow2 35.4%

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

      11. unpow3 35.3%

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

      12. cube-prod 35.4%

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

      13. pow-sqr 35.4%

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

      14. metadata-eval 35.4%

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

      15. rem-square-sqrt 2.2%

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

      16. fabs-sqr 2.2%

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

      17. rem-square-sqrt 35.4%

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

   6. Simplified 35.4%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(2 \cdot x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \end{array} \]

Alternative 8: 68.3% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.9)
		tmp = abs(Float64((pi ^ -0.5) * Float64(2.0 * x)));
	else
		tmp = abs(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 <= 1.9)
		tmp = abs(((pi ^ -0.5) * (2.0 * x)));
	else
		tmp = abs((0.047619047619047616 * ((pi ^ -0.5) * (x ^ 7.0))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 89.8%

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

      1. fma-def 89.8%

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

      2. rem-square-sqrt 37.3%

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

      3. fabs-sqr 37.3%

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

      4. rem-square-sqrt 89.6%

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

      5. *-commutative 89.6%

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

      6. rem-square-sqrt 37.0%

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

      7. fabs-sqr 37.0%

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

      8. rem-square-sqrt 89.8%

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

   6. Simplified 89.8%

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

   7. Taylor expanded in x around 0 69.2%

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

      1. associate-*r* 69.2%

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

      2. *-commutative 69.2%

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

   9. Simplified 69.2%

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

       1. expm1-log1p-u 67.4%

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

       2. expm1-udef 5.8%

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

       3. inv-pow 5.8%

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

       4. sqrt-pow1 5.8%

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

       5. metadata-eval 5.8%

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

   11. Applied egg-rr 5.8%

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

       1. expm1-def 67.4%

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

       2. expm1-log1p 69.2%

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

       3. *-commutative 69.2%

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

   13. Simplified 69.2%

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

   if 1.8999999999999999 < x

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around inf 35.4%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left(\left({x}^{4} \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   5. Step-by-step derivation

      1. *-commutative 35.4%

         \[\leadsto \left|\color{blue}{\left(\left({x}^{4} \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}\right) \cdot 0.047619047619047616}\right| \]

      2. *-commutative 35.4%

         \[\leadsto \left|\color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left({x}^{4} \cdot {\left(\left|x\right|\right)}^{3}\right)\right)} \cdot 0.047619047619047616\right| \]

      3. associate-*l* 35.4%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left({x}^{4} \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot 0.047619047619047616\right)}\right| \]

      4. unpow3 35.4%

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

      5. sqr-abs 35.4%

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

      6. associate-*r* 35.4%

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

      7. metadata-eval 35.4%

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

      8. pow-sqr 35.4%

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

      9. unpow2 35.4%

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

      10. unpow2 35.4%

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

      11. unpow3 35.3%

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

      12. cube-prod 35.4%

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

      13. pow-sqr 35.4%

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

      14. metadata-eval 35.4%

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

      15. rem-square-sqrt 2.2%

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

      16. fabs-sqr 2.2%

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

      17. rem-square-sqrt 35.4%

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

   6. Simplified 35.4%

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

      1. expm1-log1p-u 3.9%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right)\right)}\right| \]

      2. expm1-udef 3.7%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right)} - 1}\right| \]

      3. *-commutative 3.7%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}}\right)} - 1\right| \]

      4. inv-pow 3.7%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\color{blue}{{\pi}^{-1}}}\right)} - 1\right| \]

      5. sqrt-pow1 3.7%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right| \]

      6. metadata-eval 3.7%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot {\pi}^{\color{blue}{-0.5}}\right)} - 1\right| \]

   8. Applied egg-rr 3.7%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot {\pi}^{-0.5}\right)} - 1}\right| \]
   9. Step-by-step derivation

      1. expm1-def 3.9%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot {\pi}^{-0.5}\right)\right)}\right| \]

      2. expm1-log1p 35.4%

         \[\leadsto \left|\color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot {\pi}^{-0.5}}\right| \]

      3. associate-*l* 35.4%

         \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)}\right| \]

   10. Simplified 35.4%

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

4. Final simplification 69.2%

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

Alternative 9: 68.3% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(2 \cdot x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi} \cdot \left(\left(x \cdot x\right) \cdot 4\right)}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e-20)
   (fabs (* (pow PI -0.5) (* 2.0 x)))
   (fabs (sqrt (* (/ 1.0 PI) (* (* x x) 4.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 5e-20) {
		tmp = fabs((pow(((double) M_PI), -0.5) * (2.0 * x)));
	} else {
		tmp = fabs(sqrt(((1.0 / ((double) M_PI)) * ((x * x) * 4.0))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5e-20) {
		tmp = Math.abs((Math.pow(Math.PI, -0.5) * (2.0 * x)));
	} else {
		tmp = Math.abs(Math.sqrt(((1.0 / Math.PI) * ((x * x) * 4.0))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5e-20:
		tmp = math.fabs((math.pow(math.pi, -0.5) * (2.0 * x)))
	else:
		tmp = math.fabs(math.sqrt(((1.0 / math.pi) * ((x * x) * 4.0))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e-20)
		tmp = abs(Float64((pi ^ -0.5) * Float64(2.0 * x)));
	else
		tmp = abs(sqrt(Float64(Float64(1.0 / pi) * Float64(Float64(x * x) * 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5e-20)
		tmp = abs(((pi ^ -0.5) * (2.0 * x)));
	else
		tmp = abs(sqrt(((1.0 / pi) * ((x * x) * 4.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5e-20], N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(2.0 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[Sqrt[N[(N[(1.0 / Pi), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.9999999999999999e-20

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 89.8%

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

      1. fma-def 89.8%

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

      2. rem-square-sqrt 36.0%

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

      3. fabs-sqr 36.0%

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

      4. rem-square-sqrt 89.5%

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

      5. *-commutative 89.5%

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

      6. rem-square-sqrt 35.7%

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

      7. fabs-sqr 35.7%

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

      8. rem-square-sqrt 89.8%

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

   6. Simplified 89.8%

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

   7. Taylor expanded in x around 0 69.0%

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

      1. associate-*r* 69.0%

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

      2. *-commutative 69.0%

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

   9. Simplified 69.0%

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

       1. expm1-log1p-u 67.2%

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

       2. expm1-udef 4.8%

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

       3. inv-pow 4.8%

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

       4. sqrt-pow1 4.8%

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

       5. metadata-eval 4.8%

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

   11. Applied egg-rr 4.8%

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

       1. expm1-def 67.2%

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

       2. expm1-log1p 69.0%

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

       3. *-commutative 69.0%

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

   13. Simplified 69.0%

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

   if 4.9999999999999999e-20 < x

   1. Initial program 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 92.0%

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

      1. fma-def 92.0%

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

      2. rem-square-sqrt 92.0%

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

      3. fabs-sqr 92.0%

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

      4. rem-square-sqrt 92.0%

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

      5. *-commutative 92.0%

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

      6. rem-square-sqrt 92.0%

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

      7. fabs-sqr 92.0%

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

      8. rem-square-sqrt 92.0%

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

   6. Simplified 92.0%

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

   7. Taylor expanded in x around 0 76.8%

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

      1. associate-*r* 76.8%

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

      2. *-commutative 76.8%

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

   9. Simplified 76.8%

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

       1. expm1-log1p-u 76.8%

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

       2. expm1-udef 47.4%

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

       3. inv-pow 47.4%

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

       4. sqrt-pow1 47.4%

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

       5. metadata-eval 47.4%

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

   11. Applied egg-rr 47.4%

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

       1. expm1-def 76.8%

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

       2. expm1-log1p 76.8%

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

       3. *-commutative 76.8%

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

   13. Simplified 76.8%

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

       1. *-commutative 76.8%

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

       2. add-sqr-sqrt 77.1%

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

       3. sqrt-unprod 76.8%

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

       4. *-commutative 76.8%

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

       5. associate-*l* 76.8%

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

       6. *-commutative 76.8%

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

       7. associate-*l* 76.8%

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

       8. swap-sqr 76.8%

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

       9. metadata-eval 76.8%

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

       10. swap-sqr 76.6%

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

       11. pow-prod-up 77.3%

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

       12. metadata-eval 77.3%

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

   15. Applied egg-rr 77.3%

       \[\leadsto \left|\color{blue}{\sqrt{4 \cdot \left(\left(x \cdot x\right) \cdot {\pi}^{-1}\right)}}\right| \]
   16. Step-by-step derivation

       1. associate-*r* 77.3%

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

       2. unpow-1 77.3%

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

   17. Simplified 77.3%

       \[\leadsto \left|\color{blue}{\sqrt{\left(4 \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{\pi}}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(2 \cdot x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi} \cdot \left(\left(x \cdot x\right) \cdot 4\right)}\right|\\ \end{array} \]

Alternative 10: 68.3% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(2 \cdot x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{4 \cdot \left(x \cdot \frac{x}{\pi}\right)}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e-21)
   (fabs (* (pow PI -0.5) (* 2.0 x)))
   (fabs (sqrt (* 4.0 (* x (/ x PI)))))))

C

double code(double x) {
	double tmp;
	if (x <= 2e-21) {
		tmp = fabs((pow(((double) M_PI), -0.5) * (2.0 * x)));
	} else {
		tmp = fabs(sqrt((4.0 * (x * (x / ((double) M_PI))))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2e-21) {
		tmp = Math.abs((Math.pow(Math.PI, -0.5) * (2.0 * x)));
	} else {
		tmp = Math.abs(Math.sqrt((4.0 * (x * (x / Math.PI)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2e-21:
		tmp = math.fabs((math.pow(math.pi, -0.5) * (2.0 * x)))
	else:
		tmp = math.fabs(math.sqrt((4.0 * (x * (x / math.pi)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e-21)
		tmp = abs(Float64((pi ^ -0.5) * Float64(2.0 * x)));
	else
		tmp = abs(sqrt(Float64(4.0 * Float64(x * Float64(x / pi)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2e-21)
		tmp = abs(((pi ^ -0.5) * (2.0 * x)));
	else
		tmp = abs(sqrt((4.0 * (x * (x / pi)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2e-21], N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(2.0 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[Sqrt[N[(4.0 * N[(x * N[(x / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999999999982e-21

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 89.8%

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

      1. fma-def 89.8%

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

      2. rem-square-sqrt 36.0%

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

      3. fabs-sqr 36.0%

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

      4. rem-square-sqrt 89.5%

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

      5. *-commutative 89.5%

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

      6. rem-square-sqrt 35.7%

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

      7. fabs-sqr 35.7%

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

      8. rem-square-sqrt 89.8%

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

   6. Simplified 89.8%

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

   7. Taylor expanded in x around 0 69.0%

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

      1. associate-*r* 69.0%

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

      2. *-commutative 69.0%

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

   9. Simplified 69.0%

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

       1. expm1-log1p-u 67.2%

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

       2. expm1-udef 4.8%

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

       3. inv-pow 4.8%

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

       4. sqrt-pow1 4.8%

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

       5. metadata-eval 4.8%

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

   11. Applied egg-rr 4.8%

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

       1. expm1-def 67.2%

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

       2. expm1-log1p 69.0%

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

       3. *-commutative 69.0%

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

   13. Simplified 69.0%

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

   if 1.99999999999999982e-21 < x

   1. Initial program 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 92.0%

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

      1. fma-def 92.0%

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

      2. rem-square-sqrt 92.0%

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

      3. fabs-sqr 92.0%

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

      4. rem-square-sqrt 92.0%

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

      5. *-commutative 92.0%

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

      6. rem-square-sqrt 92.0%

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

      7. fabs-sqr 92.0%

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

      8. rem-square-sqrt 92.0%

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

   6. Simplified 92.0%

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

   7. Taylor expanded in x around 0 76.8%

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

      1. associate-*r* 76.8%

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

      2. *-commutative 76.8%

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

   9. Simplified 76.8%

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

       1. expm1-log1p-u 76.8%

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

       2. expm1-udef 47.4%

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

       3. inv-pow 47.4%

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

       4. sqrt-pow1 47.4%

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

       5. metadata-eval 47.4%

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

   11. Applied egg-rr 47.4%

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

       1. expm1-def 76.8%

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

       2. expm1-log1p 76.8%

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

       3. *-commutative 76.8%

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

   13. Simplified 76.8%

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

       1. *-commutative 76.8%

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

       2. add-sqr-sqrt 77.1%

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

       3. sqrt-unprod 76.8%

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

       4. *-commutative 76.8%

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

       5. associate-*l* 76.8%

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

       6. *-commutative 76.8%

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

       7. associate-*l* 76.8%

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

       8. swap-sqr 76.8%

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

       9. metadata-eval 76.8%

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

       10. swap-sqr 76.6%

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

       11. pow-prod-up 77.3%

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

       12. metadata-eval 77.3%

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

   15. Applied egg-rr 77.3%

       \[\leadsto \left|\color{blue}{\sqrt{4 \cdot \left(\left(x \cdot x\right) \cdot {\pi}^{-1}\right)}}\right| \]
   16. Step-by-step derivation

       1. *-commutative 77.3%

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

       2. associate-*l* 77.3%

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

       3. unpow-1 77.3%

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

       4. associate-*r/ 76.8%

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

       5. *-rgt-identity 76.8%

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

   17. Simplified 76.8%

       \[\leadsto \left|\color{blue}{\sqrt{\left(x \cdot \frac{x}{\pi}\right) \cdot 4}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(2 \cdot x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{4 \cdot \left(x \cdot \frac{x}{\pi}\right)}\right|\\ \end{array} \]

Alternative 11: 68.3% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(2 \cdot x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{4 \cdot \frac{x \cdot x}{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e-20)
   (fabs (* (pow PI -0.5) (* 2.0 x)))
   (fabs (sqrt (* 4.0 (/ (* x x) PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 5e-20) {
		tmp = fabs((pow(((double) M_PI), -0.5) * (2.0 * x)));
	} else {
		tmp = fabs(sqrt((4.0 * ((x * x) / ((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5e-20) {
		tmp = Math.abs((Math.pow(Math.PI, -0.5) * (2.0 * x)));
	} else {
		tmp = Math.abs(Math.sqrt((4.0 * ((x * x) / Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5e-20:
		tmp = math.fabs((math.pow(math.pi, -0.5) * (2.0 * x)))
	else:
		tmp = math.fabs(math.sqrt((4.0 * ((x * x) / math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e-20)
		tmp = abs(Float64((pi ^ -0.5) * Float64(2.0 * x)));
	else
		tmp = abs(sqrt(Float64(4.0 * Float64(Float64(x * x) / pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5e-20)
		tmp = abs(((pi ^ -0.5) * (2.0 * x)));
	else
		tmp = abs(sqrt((4.0 * ((x * x) / pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5e-20], N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(2.0 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[Sqrt[N[(4.0 * N[(N[(x * x), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.9999999999999999e-20

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 89.8%

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

      1. fma-def 89.8%

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

      2. rem-square-sqrt 36.0%

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

      3. fabs-sqr 36.0%

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

      4. rem-square-sqrt 89.5%

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

      5. *-commutative 89.5%

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

      6. rem-square-sqrt 35.7%

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

      7. fabs-sqr 35.7%

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

      8. rem-square-sqrt 89.8%

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

   6. Simplified 89.8%

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

   7. Taylor expanded in x around 0 69.0%

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

      1. associate-*r* 69.0%

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

      2. *-commutative 69.0%

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

   9. Simplified 69.0%

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

       1. expm1-log1p-u 67.2%

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

       2. expm1-udef 4.8%

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

       3. inv-pow 4.8%

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

       4. sqrt-pow1 4.8%

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

       5. metadata-eval 4.8%

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

   11. Applied egg-rr 4.8%

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

       1. expm1-def 67.2%

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

       2. expm1-log1p 69.0%

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

       3. *-commutative 69.0%

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

   13. Simplified 69.0%

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

   if 4.9999999999999999e-20 < x

   1. Initial program 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 92.0%

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

      1. fma-def 92.0%

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

      2. rem-square-sqrt 92.0%

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

      3. fabs-sqr 92.0%

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

      4. rem-square-sqrt 92.0%

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

      5. *-commutative 92.0%

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

      6. rem-square-sqrt 92.0%

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

      7. fabs-sqr 92.0%

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

      8. rem-square-sqrt 92.0%

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

   6. Simplified 92.0%

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

   7. Taylor expanded in x around 0 76.8%

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

      1. associate-*r* 76.8%

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

      2. *-commutative 76.8%

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

   9. Simplified 76.8%

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

       1. expm1-log1p-u 76.8%

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

       2. expm1-udef 47.4%

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

       3. inv-pow 47.4%

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

       4. sqrt-pow1 47.4%

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

       5. metadata-eval 47.4%

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

   11. Applied egg-rr 47.4%

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

       1. expm1-def 76.8%

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

       2. expm1-log1p 76.8%

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

       3. *-commutative 76.8%

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

   13. Simplified 76.8%

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

       1. *-commutative 76.8%

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

       2. add-sqr-sqrt 77.1%

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

       3. sqrt-unprod 76.8%

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

       4. *-commutative 76.8%

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

       5. associate-*l* 76.8%

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

       6. *-commutative 76.8%

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

       7. associate-*l* 76.8%

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

       8. swap-sqr 76.8%

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

       9. metadata-eval 76.8%

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

       10. swap-sqr 76.6%

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

       11. pow-prod-up 77.3%

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

       12. metadata-eval 77.3%

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

   15. Applied egg-rr 77.3%

       \[\leadsto \left|\color{blue}{\sqrt{4 \cdot \left(\left(x \cdot x\right) \cdot {\pi}^{-1}\right)}}\right| \]
   16. Step-by-step derivation

       1. *-commutative 77.3%

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

       2. unpow-1 77.3%

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

       3. associate-*r/ 77.3%

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

       4. *-rgt-identity 77.3%

          \[\leadsto \left|\sqrt{\frac{\color{blue}{x \cdot x}}{\pi} \cdot 4}\right| \]

   17. Simplified 77.3%

       \[\leadsto \left|\color{blue}{\sqrt{\frac{x \cdot x}{\pi} \cdot 4}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(2 \cdot x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{4 \cdot \frac{x \cdot x}{\pi}}\right|\\ \end{array} \]

Alternative 12: 68.3% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around 0 89.8%

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

   1. fma-def 89.8%

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

   2. rem-square-sqrt 37.3%

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

   3. fabs-sqr 37.3%

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

   4. rem-square-sqrt 89.6%

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

   5. *-commutative 89.6%

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

   6. rem-square-sqrt 37.0%

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

   7. fabs-sqr 37.0%

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

   8. rem-square-sqrt 89.8%

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

6. Simplified 89.8%

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

7. Taylor expanded in x around 0 69.2%

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

   1. associate-*r* 69.2%

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

   2. *-commutative 69.2%

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

9. Simplified 69.2%

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

    1. expm1-log1p-u 67.4%

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

    2. expm1-udef 5.8%

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

    3. inv-pow 5.8%

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

    4. sqrt-pow1 5.8%

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

    5. metadata-eval 5.8%

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

11. Applied egg-rr 5.8%

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

    1. expm1-def 67.4%

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

    2. expm1-log1p 69.2%

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

    3. *-commutative 69.2%

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

13. Simplified 69.2%

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

14. Final simplification 69.2%

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

Reproduce

herbie shell --seed 2023293 
(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)))))))