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

Percentage Accurate: 99.8% → 99.8%

Time: 17.2s

Alternatives: 12

Speedup: 1.5×

• 28.3% 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 99.5%

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

2. Final simplification 99.5%

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

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 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 3: 99.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return abs(Float64(Float64(x / sqrt(pi)) * Float64(Float64(2.0 + Float64(0.6666666666666666 * Float64(x * x))) + fma(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[(2.0 + N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[Power[x, 4.0], $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.5%

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

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. *-un-lft-identity 99.4%

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

5. Applied egg-rr 99.4%

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

   1. *-lft-identity 99.4%

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

   2. unpow1 99.4%

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

   3. sqr-pow 33.2%

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

   4. fabs-sqr 33.2%

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

   5. sqr-pow 99.4%

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

   6. unpow1 99.4%

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

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%

      \[\leadsto \left|\frac{x}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{2}{3}}, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   2. fma-udef 99.4%

      \[\leadsto \left|\frac{x}{\sqrt{\pi}} \cdot \left(\color{blue}{\left(\frac{2}{3} \cdot \left(x \cdot x\right) + 2\right)} + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   3. metadata-eval 99.4%

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

9. Applied egg-rr 99.4%

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

10. Final simplification 99.4%

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

Alternative 4: 99.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.01:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.01)
   (fabs (* (pow PI -0.5) (* x (+ 2.0 (* 0.6666666666666666 (* x x))))))
   (fabs
    (/
     (+ (* 0.2 (pow x 5.0)) (* 0.047619047619047616 (pow x 7.0)))
     (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.01) {
		tmp = fabs((pow(((double) M_PI), -0.5) * (x * (2.0 + (0.6666666666666666 * (x * x))))));
	} else {
		tmp = fabs((((0.2 * pow(x, 5.0)) + (0.047619047619047616 * pow(x, 7.0))) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.01) {
		tmp = Math.abs((Math.pow(Math.PI, -0.5) * (x * (2.0 + (0.6666666666666666 * (x * x))))));
	} else {
		tmp = Math.abs((((0.2 * Math.pow(x, 5.0)) + (0.047619047619047616 * Math.pow(x, 7.0))) / Math.sqrt(Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.01:
		tmp = math.fabs((math.pow(math.pi, -0.5) * (x * (2.0 + (0.6666666666666666 * (x * x))))))
	else:
		tmp = math.fabs((((0.2 * math.pow(x, 5.0)) + (0.047619047619047616 * math.pow(x, 7.0))) / math.sqrt(math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.01)
		tmp = abs(Float64((pi ^ -0.5) * Float64(x * Float64(2.0 + Float64(0.6666666666666666 * Float64(x * x))))));
	else
		tmp = abs(Float64(Float64(Float64(0.2 * (x ^ 5.0)) + Float64(0.047619047619047616 * (x ^ 7.0))) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.01)
		tmp = abs(((pi ^ -0.5) * (x * (2.0 + (0.6666666666666666 * (x * x))))));
	else
		tmp = abs((((0.2 * (x ^ 5.0)) + (0.047619047619047616 * (x ^ 7.0))) / sqrt(pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.01], N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * N[(2.0 + N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0100000000000000002

   1. Initial program 99.8%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 98.9%

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

      1. +-commutative 98.9%

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

      2. *-commutative 98.9%

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

      3. cube-mult 98.9%

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

      4. sqr-abs 98.9%

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

      5. unpow2 98.9%

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

      6. associate-*l* 98.9%

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

      7. unpow2 98.9%

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

      8. associate-*l* 98.9%

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

      9. *-commutative 98.9%

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

      10. *-commutative 98.9%

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

      11. distribute-lft-in 98.9%

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

      12. unpow1 98.9%

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

      13. sqr-pow 50.8%

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

      14. fabs-sqr 50.8%

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

      15. sqr-pow 98.9%

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

      16. unpow1 98.9%

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

      17. *-commutative 98.9%

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

      18. fma-def 98.9%

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

   6. Simplified 98.9%

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

      1. div-inv 99.5%

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

      2. metadata-eval 99.5%

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

      3. sqrt-div 99.5%

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

      4. associate-*l* 99.5%

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

      5. inv-pow 99.5%

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

      6. sqrt-pow1 99.5%

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

      7. metadata-eval 99.5%

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

   8. Applied egg-rr 99.5%

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

      1. associate-*r* 99.5%

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

      2. *-commutative 99.5%

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

   10. Simplified 99.5%

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

       1. fma-udef 99.5%

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

       2. associate-*r* 99.5%

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

       3. *-commutative 99.5%

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

   12. Applied egg-rr 99.5%

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

   if 0.0100000000000000002 < (fabs.f64 x)

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in x around inf 98.9%

      \[\leadsto \left|\frac{\color{blue}{0.047619047619047616 \cdot \left({\left(\left|x\right|\right)}^{3} \cdot {x}^{4}\right) + 0.2 \cdot \left({\left(\left|x\right|\right)}^{3} \cdot {x}^{2}\right)}}{\sqrt{\pi}}\right| \]
   5. Step-by-step derivation

      1. fma-def 98.9%

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

   6. Simplified 99.0%

      \[\leadsto \left|\frac{\color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)}}{\sqrt{\pi}}\right| \]

   7. Taylor expanded in x around 0 99.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.01:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 5: 89.0% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.5%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around 0 88.7%

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

      1. +-commutative 88.7%

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

      2. *-commutative 88.7%

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

      3. cube-mult 88.7%

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

      4. sqr-abs 88.7%

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

      5. unpow2 88.7%

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

      6. associate-*l* 88.7%

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

      7. unpow2 88.7%

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

      8. associate-*l* 88.7%

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

      9. *-commutative 88.7%

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

      10. *-commutative 88.7%

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

      11. distribute-lft-in 88.7%

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

      12. unpow1 88.7%

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

      13. sqr-pow 33.2%

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

      14. fabs-sqr 33.2%

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

      15. sqr-pow 88.7%

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

      16. unpow1 88.7%

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

      17. *-commutative 88.7%

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

      18. fma-def 88.7%

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

   6. Simplified 88.7%

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

      1. div-inv 89.1%

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

      2. metadata-eval 89.1%

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

      3. sqrt-div 89.1%

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

      4. associate-*l* 89.1%

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

      5. inv-pow 89.1%

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

      6. sqrt-pow1 89.1%

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

      7. metadata-eval 89.1%

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

   8. Applied egg-rr 89.1%

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

      1. associate-*r* 89.1%

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

      2. *-commutative 89.1%

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

   10. Simplified 89.1%

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

       1. fma-udef 89.1%

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

       2. associate-*r* 89.1%

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

       3. *-commutative 89.1%

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

   12. Applied egg-rr 89.1%

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

   if 2.2000000000000002 < x

   1. Initial program 99.5%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around inf 37.6%

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

      1. associate-*r* 37.6%

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

      2. *-commutative 37.6%

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

      3. *-commutative 37.6%

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

      4. unpow1 37.6%

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

      5. sqr-pow 1.9%

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

      6. fabs-sqr 1.9%

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

      7. sqr-pow 37.6%

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

      8. unpow1 37.6%

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

      9. *-commutative 37.6%

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

      10. *-commutative 37.6%

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

      11. cube-mult 37.6%

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

      12. sqr-abs 37.6%

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

      13. unpow2 37.6%

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

      14. *-commutative 37.6%

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

   6. Simplified 37.7%

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

      1. expm1-log1p-u 3.6%

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

      2. expm1-udef 3.4%

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

      3. associate-*r* 3.4%

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

      4. *-commutative 3.4%

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

      5. sqrt-div 3.4%

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

      6. metadata-eval 3.4%

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

      7. associate-*l/ 3.4%

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

      8. metadata-eval 3.4%

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

   8. Applied egg-rr 3.4%

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

      1. expm1-def 3.6%

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

      2. expm1-log1p 37.7%

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

   10. Simplified 37.7%

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

4. Final simplification 89.1%

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

Alternative 6: 89.0% accurate, 4.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = fabs((pow(((double) M_PI), -0.5) * (x * (2.0 + (0.6666666666666666 * (x * x))))));
	} else {
		tmp = fabs((0.047619047619047616 / (sqrt(((double) M_PI)) / 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) * (x * (2.0 + (0.6666666666666666 * (x * x))))));
	} else {
		tmp = Math.abs((0.047619047619047616 / (Math.sqrt(Math.PI) / 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) * (x * (2.0 + (0.6666666666666666 * (x * x))))))
	else:
		tmp = math.fabs((0.047619047619047616 / (math.sqrt(math.pi) / 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(x * Float64(2.0 + Float64(0.6666666666666666 * Float64(x * x))))));
	else
		tmp = abs(Float64(0.047619047619047616 / Float64(sqrt(pi) / (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) * (x * (2.0 + (0.6666666666666666 * (x * x))))));
	else
		tmp = abs((0.047619047619047616 / (sqrt(pi) / (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[(x * N[(2.0 + N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 / N[(N[Sqrt[Pi], $MachinePrecision] / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.5%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around 0 88.7%

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

      1. +-commutative 88.7%

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

      2. *-commutative 88.7%

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

      3. cube-mult 88.7%

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

      4. sqr-abs 88.7%

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

      5. unpow2 88.7%

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

      6. associate-*l* 88.7%

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

      7. unpow2 88.7%

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

      8. associate-*l* 88.7%

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

      9. *-commutative 88.7%

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

      10. *-commutative 88.7%

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

      11. distribute-lft-in 88.7%

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

      12. unpow1 88.7%

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

      13. sqr-pow 33.2%

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

      14. fabs-sqr 33.2%

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

      15. sqr-pow 88.7%

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

      16. unpow1 88.7%

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

      17. *-commutative 88.7%

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

      18. fma-def 88.7%

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

   6. Simplified 88.7%

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

      1. div-inv 89.1%

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

      2. metadata-eval 89.1%

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

      3. sqrt-div 89.1%

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

      4. associate-*l* 89.1%

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

      5. inv-pow 89.1%

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

      6. sqrt-pow1 89.1%

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

      7. metadata-eval 89.1%

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

   8. Applied egg-rr 89.1%

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

      1. associate-*r* 89.1%

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

      2. *-commutative 89.1%

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

   10. Simplified 89.1%

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

       1. fma-udef 89.1%

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

       2. associate-*r* 89.1%

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

       3. *-commutative 89.1%

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

   12. Applied egg-rr 89.1%

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

   if 2.2000000000000002 < x

   1. Initial program 99.5%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around inf 38.1%

      \[\leadsto \left|\frac{\color{blue}{0.047619047619047616 \cdot \left({\left(\left|x\right|\right)}^{3} \cdot {x}^{4}\right) + 0.2 \cdot \left({\left(\left|x\right|\right)}^{3} \cdot {x}^{2}\right)}}{\sqrt{\pi}}\right| \]
   5. Step-by-step derivation

      1. fma-def 38.1%

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

   6. Simplified 38.1%

      \[\leadsto \left|\frac{\color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)}}{\sqrt{\pi}}\right| \]

   7. Taylor expanded in x around inf 37.7%

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

      1. expm1-log1p-u 3.6%

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

      2. expm1-udef 3.4%

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

      3. *-un-lft-identity 3.4%

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

      4. times-frac 3.4%

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

      5. metadata-eval 3.4%

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

   9. Applied egg-rr 3.4%

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

       1. expm1-def 3.6%

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

       2. expm1-log1p 37.6%

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

       3. associate-*r/ 37.7%

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

       4. associate-/l* 37.7%

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

   11. Simplified 37.7%

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

4. Final simplification 89.1%

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

Alternative 7: 89.0% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.5%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around 0 88.7%

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

      1. +-commutative 88.7%

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

      2. *-commutative 88.7%

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

      3. cube-mult 88.7%

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

      4. sqr-abs 88.7%

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

      5. unpow2 88.7%

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

      6. associate-*l* 88.7%

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

      7. unpow2 88.7%

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

      8. associate-*l* 88.7%

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

      9. *-commutative 88.7%

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

      10. *-commutative 88.7%

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

      11. distribute-lft-in 88.7%

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

      12. unpow1 88.7%

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

      13. sqr-pow 33.2%

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

      14. fabs-sqr 33.2%

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

      15. sqr-pow 88.7%

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

      16. unpow1 88.7%

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

      17. *-commutative 88.7%

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

      18. fma-def 88.7%

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

   6. Simplified 88.7%

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

      1. div-inv 89.1%

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

      2. metadata-eval 89.1%

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

      3. sqrt-div 89.1%

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

      4. associate-*l* 89.1%

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

      5. inv-pow 89.1%

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

      6. sqrt-pow1 89.1%

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

      7. metadata-eval 89.1%

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

   8. Applied egg-rr 89.1%

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

      1. associate-*r* 89.1%

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

      2. *-commutative 89.1%

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

   10. Simplified 89.1%

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

       1. fma-udef 89.1%

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

       2. associate-*r* 89.1%

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

       3. *-commutative 89.1%

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

   12. Applied egg-rr 89.1%

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

   if 2.2000000000000002 < x

   1. Initial program 99.5%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around inf 38.1%

      \[\leadsto \left|\frac{\color{blue}{0.047619047619047616 \cdot \left({\left(\left|x\right|\right)}^{3} \cdot {x}^{4}\right) + 0.2 \cdot \left({\left(\left|x\right|\right)}^{3} \cdot {x}^{2}\right)}}{\sqrt{\pi}}\right| \]
   5. Step-by-step derivation

      1. fma-def 38.1%

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

   6. Simplified 38.1%

      \[\leadsto \left|\frac{\color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)}}{\sqrt{\pi}}\right| \]

   7. Taylor expanded in x around inf 37.7%

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

4. Final simplification 89.1%

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

Alternative 8: 89.0% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in x around 0 88.7%

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

   1. +-commutative 88.7%

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

   2. *-commutative 88.7%

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

   3. cube-mult 88.7%

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

   4. sqr-abs 88.7%

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

   5. unpow2 88.7%

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

   6. associate-*l* 88.7%

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

   7. unpow2 88.7%

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

   8. associate-*l* 88.7%

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

   9. *-commutative 88.7%

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

   10. *-commutative 88.7%

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

   11. distribute-lft-in 88.7%

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

   12. unpow1 88.7%

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

   13. sqr-pow 33.2%

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

   14. fabs-sqr 33.2%

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

   15. sqr-pow 88.7%

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

   16. unpow1 88.7%

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

   17. *-commutative 88.7%

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

   18. fma-def 88.7%

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

6. Simplified 88.7%

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

   1. div-inv 89.1%

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

   2. metadata-eval 89.1%

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

   3. sqrt-div 89.1%

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

   4. associate-*l* 89.1%

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

   5. inv-pow 89.1%

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

   6. sqrt-pow1 89.1%

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

   7. metadata-eval 89.1%

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

8. Applied egg-rr 89.1%

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

   1. associate-*r* 89.1%

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

   2. *-commutative 89.1%

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

10. Simplified 89.1%

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

    1. fma-udef 89.1%

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

    2. associate-*r* 89.1%

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

    3. *-commutative 89.1%

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

12. Applied egg-rr 89.1%

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

13. Final simplification 89.1%

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

Alternative 9: 67.8% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-41}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(2 \cdot x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{\left(x \cdot x\right) \cdot 4}{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e-41)
   (fabs (* (pow PI -0.5) (* 2.0 x)))
   (fabs (sqrt (/ (* (* x x) 4.0) PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 5e-41) {
		tmp = fabs((pow(((double) M_PI), -0.5) * (2.0 * x)));
	} else {
		tmp = fabs(sqrt((((x * x) * 4.0) / ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5e-41) {
		tmp = Math.abs((Math.pow(Math.PI, -0.5) * (2.0 * x)));
	} else {
		tmp = Math.abs(Math.sqrt((((x * x) * 4.0) / Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5e-41:
		tmp = math.fabs((math.pow(math.pi, -0.5) * (2.0 * x)))
	else:
		tmp = math.fabs(math.sqrt((((x * x) * 4.0) / math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e-41)
		tmp = abs(Float64((pi ^ -0.5) * Float64(2.0 * x)));
	else
		tmp = abs(sqrt(Float64(Float64(Float64(x * x) * 4.0) / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5e-41)
		tmp = abs(((pi ^ -0.5) * (2.0 * x)));
	else
		tmp = abs(sqrt((((x * x) * 4.0) / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5e-41], N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(2.0 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[Sqrt[N[(N[(N[(x * x), $MachinePrecision] * 4.0), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.9999999999999996e-41

   1. Initial program 99.5%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around 0 88.3%

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

      1. +-commutative 88.3%

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

      2. *-commutative 88.3%

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

      3. cube-mult 88.3%

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

      4. sqr-abs 88.3%

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

      5. unpow2 88.3%

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

      6. associate-*l* 88.3%

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

      7. unpow2 88.3%

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

      8. associate-*l* 88.3%

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

      9. *-commutative 88.3%

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

      10. *-commutative 88.3%

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

      11. distribute-lft-in 88.3%

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

      12. unpow1 88.3%

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

      13. sqr-pow 30.3%

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

      14. fabs-sqr 30.3%

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

      15. sqr-pow 88.3%

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

      16. unpow1 88.3%

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

      17. *-commutative 88.3%

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

      18. fma-def 88.3%

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

   6. Simplified 88.3%

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

   7. Taylor expanded in x around 0 64.9%

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

      1. *-commutative 64.9%

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

   9. Simplified 64.9%

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

       1. div-inv 65.3%

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

       2. pow1/2 65.3%

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

       3. pow-flip 65.3%

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

       4. metadata-eval 65.3%

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

   11. Applied egg-rr 65.3%

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

   if 4.9999999999999996e-41 < x

   1. Initial program 99.6%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 98.1%

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

      1. +-commutative 98.1%

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

      2. *-commutative 98.1%

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

      3. cube-mult 98.1%

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

      4. sqr-abs 98.1%

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

      5. unpow2 98.1%

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

      6. associate-*l* 98.1%

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

      7. unpow2 98.1%

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

      8. associate-*l* 98.1%

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

      9. *-commutative 98.1%

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

      10. *-commutative 98.1%

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

      11. distribute-lft-in 98.1%

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

      12. unpow1 98.1%

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

      13. sqr-pow 97.4%

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

      14. fabs-sqr 97.4%

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

      15. sqr-pow 98.1%

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

      16. unpow1 98.1%

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

      17. *-commutative 98.1%

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

      18. fma-def 98.1%

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

   6. Simplified 98.1%

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

   7. Taylor expanded in x around 0 92.2%

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

      1. *-commutative 92.2%

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

   9. Simplified 92.2%

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

       1. add-sqr-sqrt 91.9%

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

       2. sqrt-unprod 92.2%

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

       3. frac-times 92.2%

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

       4. pow2 92.2%

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

       5. add-sqr-sqrt 92.8%

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

   11. Applied egg-rr 92.8%

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

       1. unpow2 92.8%

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

       2. swap-sqr 92.8%

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

       3. metadata-eval 92.8%

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

   13. Simplified 92.8%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-41}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(2 \cdot x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{\left(x \cdot x\right) \cdot 4}{\pi}}\right|\\ \end{array} \]

Alternative 10: 67.8% 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 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in x around 0 88.7%

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

   1. +-commutative 88.7%

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

   2. *-commutative 88.7%

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

   3. cube-mult 88.7%

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

   4. sqr-abs 88.7%

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

   5. unpow2 88.7%

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

   6. associate-*l* 88.7%

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

   7. unpow2 88.7%

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

   8. associate-*l* 88.7%

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

   9. *-commutative 88.7%

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

   10. *-commutative 88.7%

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

   11. distribute-lft-in 88.7%

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

   12. unpow1 88.7%

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

   13. sqr-pow 33.2%

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

   14. fabs-sqr 33.2%

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

   15. sqr-pow 88.7%

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

   16. unpow1 88.7%

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

   17. *-commutative 88.7%

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

   18. fma-def 88.7%

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

6. Simplified 88.7%

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

7. Taylor expanded in x around 0 66.1%

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

   1. *-commutative 66.1%

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

9. Simplified 66.1%

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

    1. div-inv 66.5%

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

    2. pow1/2 66.5%

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

    3. pow-flip 66.5%

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

    4. metadata-eval 66.5%

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

11. Applied egg-rr 66.5%

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

12. Final simplification 66.5%

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

Alternative 11: 67.4% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in x around 0 88.7%

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

   1. +-commutative 88.7%

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

   2. *-commutative 88.7%

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

   3. cube-mult 88.7%

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

   4. sqr-abs 88.7%

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

   5. unpow2 88.7%

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

   6. associate-*l* 88.7%

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

   7. unpow2 88.7%

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

   8. associate-*l* 88.7%

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

   9. *-commutative 88.7%

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

   10. *-commutative 88.7%

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

   11. distribute-lft-in 88.7%

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

   12. unpow1 88.7%

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

   13. sqr-pow 33.2%

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

   14. fabs-sqr 33.2%

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

   15. sqr-pow 88.7%

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

   16. unpow1 88.7%

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

   17. *-commutative 88.7%

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

   18. fma-def 88.7%

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

6. Simplified 88.7%

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

7. Taylor expanded in x around 0 66.1%

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

   1. *-commutative 66.1%

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

9. Simplified 66.1%

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

10. Final simplification 66.1%

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

Alternative 12: 4.1% accurate, 1948.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in x around 0 88.7%

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

   1. +-commutative 88.7%

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

   2. *-commutative 88.7%

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

   3. cube-mult 88.7%

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

   4. sqr-abs 88.7%

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

   5. unpow2 88.7%

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

   6. associate-*l* 88.7%

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

   7. unpow2 88.7%

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

   8. associate-*l* 88.7%

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

   9. *-commutative 88.7%

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

   10. *-commutative 88.7%

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

   11. distribute-lft-in 88.7%

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

   12. unpow1 88.7%

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

   13. sqr-pow 33.2%

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

   14. fabs-sqr 33.2%

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

   15. sqr-pow 88.7%

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

   16. unpow1 88.7%

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

   17. *-commutative 88.7%

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

   18. fma-def 88.7%

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

6. Simplified 88.7%

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

   1. expm1-log1p-u 64.5%

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

   2. expm1-udef 5.4%

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

   3. *-commutative 5.4%

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

   4. associate-/l* 5.4%

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

8. Applied egg-rr 5.4%

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

9. Taylor expanded in x around 0 4.0%

   \[\leadsto \left|\color{blue}{1} - 1\right| \]

10. Final simplification 4.0%

    \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023274 
(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)))))))