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

Percentage Accurate: 99.8% → 99.8%

Time: 10.7s

Alternatives: 14

Speedup: 2.4×

• 29.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (* x x))) (t_1 (* (* 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(fma(2.0, abs(x), Float64(0.6666666666666666 * t_0)) + Float64(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[(2.0 * N[Abs[x], $MachinePrecision] + N[(0.6666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(0.2 * t$95$1), $MachinePrecision] + N[(0.047619047619047616 * N[(N[(x * x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 3: 99.9% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

   1. div-inv 99.8%

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

   2. pow1/2 99.8%

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

   3. pow-flip 99.8%

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

   4. metadata-eval 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 4: 99.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

   1. fma-udef 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 5: 99.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \left|\left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \left(0.047619047619047616 \cdot {x}^{6} + 0.2 \cdot {x}^{4}\right)\right) \cdot \frac{1}{\frac{\sqrt{\pi}}{x}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (+
    (fma 0.6666666666666666 (* x x) 2.0)
    (+ (* 0.047619047619047616 (pow x 6.0)) (* 0.2 (pow x 4.0))))
   (/ 1.0 (/ (sqrt PI) x)))))

C

double code(double x) {
	return fabs(((fma(0.6666666666666666, (x * x), 2.0) + ((0.047619047619047616 * pow(x, 6.0)) + (0.2 * pow(x, 4.0)))) * (1.0 / (sqrt(((double) M_PI)) / x))));
}

Julia

function code(x)
	return abs(Float64(Float64(fma(0.6666666666666666, Float64(x * x), 2.0) + Float64(Float64(0.047619047619047616 * (x ^ 6.0)) + Float64(0.2 * (x ^ 4.0)))) * Float64(1.0 / Float64(sqrt(pi) / x))))
end

Wolfram

code[x_] := N[Abs[N[(N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] + N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[Sqrt[Pi], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

   1. div-inv 99.8%

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

   2. pow1/2 99.8%

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

   3. pow-flip 99.8%

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

   4. metadata-eval 99.8%

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

5. Applied egg-rr 99.8%

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

   1. add-sqr-sqrt 30.6%

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

   2. fabs-sqr 30.6%

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

   3. add-sqr-sqrt 99.8%

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

   4. metadata-eval 99.8%

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

   5. sqrt-pow1 99.8%

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

   6. inv-pow 99.8%

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

   7. *-commutative 99.8%

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

   8. sqrt-div 99.8%

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

   9. metadata-eval 99.8%

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

   10. associate-/r/ 99.4%

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

7. Applied egg-rr 99.4%

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

   1. fma-udef 99.4%

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

9. Applied egg-rr 99.4%

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

10. Final simplification 99.4%

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

Alternative 6: 99.2% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

   1. div-inv 99.8%

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

   2. pow1/2 99.8%

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

   3. pow-flip 99.8%

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

   4. metadata-eval 99.8%

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

5. Applied egg-rr 99.8%

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

6. Taylor expanded in x around inf 98.7%

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

7. Final simplification 98.7%

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

Alternative 7: 98.7% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around inf 98.3%

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

5. Final simplification 98.3%

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

Alternative 8: 67.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.2:\\ \;\;\;\;\frac{x}{\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(\left(\left|x\right| \cdot {x}^{6}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.2)
   (/ x (* (sqrt PI) (fma (pow x 2.0) -0.16666666666666666 0.5)))
   (fabs
    (* 0.047619047619047616 (* (* (fabs x) (pow x 6.0)) (sqrt (/ 1.0 PI)))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.2) {
		tmp = x / (sqrt(((double) M_PI)) * fma(pow(x, 2.0), -0.16666666666666666, 0.5));
	} else {
		tmp = fabs((0.047619047619047616 * ((fabs(x) * pow(x, 6.0)) * sqrt((1.0 / ((double) M_PI))))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.2)
		tmp = Float64(x / Float64(sqrt(pi) * fma((x ^ 2.0), -0.16666666666666666, 0.5)));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64(Float64(abs(x) * (x ^ 6.0)) * sqrt(Float64(1.0 / pi)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.2], N[(x / N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[Power[x, 2.0], $MachinePrecision] * -0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[(N[Abs[x], $MachinePrecision] * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.20000000000000001

   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%

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

   4. Taylor expanded in x around 0 98.7%

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

      1. *-commutative 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in x around 0 98.5%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{-0.16666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\pi}\right) + 0.5 \cdot \sqrt{\pi}}\right|} \]
   8. Step-by-step derivation

      1. +-commutative 98.5%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{0.5 \cdot \sqrt{\pi} + -0.16666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\pi}\right)}\right|} \]

      2. associate-*r* 98.5%

         \[\leadsto \frac{\left|x\right|}{\left|0.5 \cdot \sqrt{\pi} + \color{blue}{\left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot \sqrt{\pi}}\right|} \]

      3. distribute-rgt-out 98.5%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot \left(0.5 + -0.16666666666666666 \cdot {x}^{2}\right)}\right|} \]

      4. *-commutative 98.5%

         \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \left(0.5 + \color{blue}{{x}^{2} \cdot -0.16666666666666666}\right)\right|} \]

   9. Simplified 98.5%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot \left(0.5 + {x}^{2} \cdot -0.16666666666666666\right)}\right|} \]

   10. Taylor expanded in x around 0 98.5%

       \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \left(0.5 + -0.16666666666666666 \cdot {x}^{2}\right)\right|}} \]
   11. Step-by-step derivation

       1. +-commutative 98.5%

          \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \color{blue}{\left(-0.16666666666666666 \cdot {x}^{2} + 0.5\right)}\right|} \]

       2. *-commutative 98.5%

          \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \left(\color{blue}{{x}^{2} \cdot -0.16666666666666666} + 0.5\right)\right|} \]

       3. fma-udef 98.5%

          \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}\right|} \]

       4. rem-square-sqrt 98.9%

          \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)} \cdot \sqrt{\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}}\right|} \]

       5. fabs-sqr 98.9%

          \[\leadsto \frac{\left|x\right|}{\color{blue}{\sqrt{\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)} \cdot \sqrt{\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}}} \]

       6. rem-square-sqrt 98.5%

          \[\leadsto \frac{\left|x\right|}{\color{blue}{\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]

       7. unpow1 98.5%

          \[\leadsto \frac{\left|\color{blue}{{x}^{1}}\right|}{\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)} \]

       8. sqr-pow 45.5%

          \[\leadsto \frac{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}{\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)} \]

       9. fabs-sqr 45.5%

          \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}}{\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)} \]

       10. sqr-pow 47.7%

           \[\leadsto \frac{\color{blue}{{x}^{1}}}{\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)} \]

       11. unpow1 47.7%

           \[\leadsto \frac{\color{blue}{x}}{\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)} \]

   12. Simplified 47.7%

       \[\leadsto \color{blue}{\frac{x}{\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]

   if 0.20000000000000001 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. div-inv 99.9%

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

      2. pow1/2 99.9%

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

      3. pow-flip 99.9%

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

      4. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 98.8%

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

   7. Taylor expanded in x around inf 98.0%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.2:\\ \;\;\;\;\frac{x}{\sqrt{\pi} \cdot \mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(\left(\left|x\right| \cdot {x}^{6}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right|\\ \end{array} \]

Alternative 9: 98.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\frac{\sqrt{\pi}}{x}} \cdot \left(0.047619047619047616 \cdot {x}^{6} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (/ (sqrt PI) x))
   (+
    (* 0.047619047619047616 (pow x 6.0))
    (fma 0.6666666666666666 (* x x) 2.0)))))

C

double code(double x) {
	return fabs(((1.0 / (sqrt(((double) M_PI)) / x)) * ((0.047619047619047616 * pow(x, 6.0)) + fma(0.6666666666666666, (x * x), 2.0))));
}

Julia

function code(x)
	return abs(Float64(Float64(1.0 / Float64(sqrt(pi) / x)) * Float64(Float64(0.047619047619047616 * (x ^ 6.0)) + fma(0.6666666666666666, Float64(x * x), 2.0))))
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[(N[Sqrt[Pi], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

   1. div-inv 99.8%

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

   2. pow1/2 99.8%

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

   3. pow-flip 99.8%

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

   4. metadata-eval 99.8%

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

5. Applied egg-rr 99.8%

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

   1. add-sqr-sqrt 30.6%

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

   2. fabs-sqr 30.6%

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

   3. add-sqr-sqrt 99.8%

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

   4. metadata-eval 99.8%

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

   5. sqrt-pow1 99.8%

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

   6. inv-pow 99.8%

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

   7. *-commutative 99.8%

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

   8. sqrt-div 99.8%

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

   9. metadata-eval 99.8%

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

   10. associate-/r/ 99.4%

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

7. Applied egg-rr 99.4%

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

8. Taylor expanded in x around inf 98.3%

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

9. Final simplification 98.3%

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

Alternative 10: 98.4% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around inf 98.3%

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

5. Taylor expanded in x around 0 97.9%

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

6. Final simplification 97.9%

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

Alternative 11: 33.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{\sqrt{\pi \cdot 0.25}}\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log1p (expm1 (/ x (sqrt (* PI 0.25))))))

C

double code(double x) {
	return log1p(expm1((x / sqrt((((double) M_PI) * 0.25)))));
}

Java

public static double code(double x) {
	return Math.log1p(Math.expm1((x / Math.sqrt((Math.PI * 0.25)))));
}

Python

def code(x):
	return math.log1p(math.expm1((x / math.sqrt((math.pi * 0.25)))))

Julia

function code(x)
	return log1p(expm1(Float64(x / sqrt(Float64(pi * 0.25)))))
end

Wolfram

code[x_] := N[Log[1 + N[(Exp[N[(x / N[Sqrt[N[(Pi * 0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around 0 93.5%

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

   1. *-commutative 93.5%

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

6. Simplified 93.5%

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

7. Taylor expanded in x around 0 66.9%

   \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{0.5 \cdot \sqrt{\pi}}\right|} \]
8. Step-by-step derivation

   1. expm1-log1p-u 66.9%

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

   2. expm1-udef 6.7%

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

   3. div-fabs 6.7%

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

   4. *-commutative 6.7%

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

9. Applied egg-rr 6.7%

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

    1. expm1-def 66.9%

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

    2. expm1-log1p 66.9%

       \[\leadsto \color{blue}{\left|\frac{x}{\sqrt{\pi} \cdot 0.5}\right|} \]

    3. fabs-div 66.9%

       \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\sqrt{\pi} \cdot 0.5\right|}} \]

    4. unpow1 66.9%

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

    5. sqr-pow 30.0%

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

    6. fabs-sqr 30.0%

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

    7. sqr-pow 31.5%

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

    8. unpow1 31.5%

       \[\leadsto \frac{\color{blue}{x}}{\left|\sqrt{\pi} \cdot 0.5\right|} \]

    9. rem-square-sqrt 31.7%

       \[\leadsto \frac{x}{\left|\color{blue}{\sqrt{\sqrt{\pi} \cdot 0.5} \cdot \sqrt{\sqrt{\pi} \cdot 0.5}}\right|} \]

    10. fabs-sqr 31.7%

        \[\leadsto \frac{x}{\color{blue}{\sqrt{\sqrt{\pi} \cdot 0.5} \cdot \sqrt{\sqrt{\pi} \cdot 0.5}}} \]

    11. rem-square-sqrt 31.5%

        \[\leadsto \frac{x}{\color{blue}{\sqrt{\pi} \cdot 0.5}} \]

11. Simplified 31.5%

    \[\leadsto \color{blue}{\frac{x}{\sqrt{\pi} \cdot 0.5}} \]

13. Applied egg-rr 31.4%

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{\sqrt{\pi \cdot 0.25}}\right)\right)} \]

14. Final simplification 31.4%

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{\sqrt{\pi \cdot 0.25}}\right)\right) \]

Alternative 12: 34.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.8)
   (* x (/ 2.0 (sqrt PI)))
   (* 0.2 (* (sqrt (/ 1.0 PI)) (pow x 5.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = 0.2 * (sqrt((1.0 / ((double) M_PI))) * pow(x, 5.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = 0.2 * (Math.sqrt((1.0 / Math.PI)) * Math.pow(x, 5.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.8:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = 0.2 * (math.sqrt((1.0 / math.pi)) * math.pow(x, 5.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.8)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(0.2 * Float64(sqrt(Float64(1.0 / pi)) * (x ^ 5.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.8)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = 0.2 * (sqrt((1.0 / pi)) * (x ^ 5.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.8], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.2 * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.80000000000000004

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 93.5%

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

      1. *-commutative 93.5%

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

   6. Simplified 93.5%

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

   7. Taylor expanded in x around 0 66.9%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{0.5 \cdot \sqrt{\pi}}\right|} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 66.9%

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

      2. expm1-udef 6.7%

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

      3. div-fabs 6.7%

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

      4. *-commutative 6.7%

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

   9. Applied egg-rr 6.7%

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

       1. expm1-def 66.9%

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

       2. expm1-log1p 66.9%

          \[\leadsto \color{blue}{\left|\frac{x}{\sqrt{\pi} \cdot 0.5}\right|} \]

       3. fabs-div 66.9%

          \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\sqrt{\pi} \cdot 0.5\right|}} \]

       4. unpow1 66.9%

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

       5. sqr-pow 30.0%

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

       6. fabs-sqr 30.0%

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

       7. sqr-pow 31.5%

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

       8. unpow1 31.5%

          \[\leadsto \frac{\color{blue}{x}}{\left|\sqrt{\pi} \cdot 0.5\right|} \]

       9. rem-square-sqrt 31.7%

          \[\leadsto \frac{x}{\left|\color{blue}{\sqrt{\sqrt{\pi} \cdot 0.5} \cdot \sqrt{\sqrt{\pi} \cdot 0.5}}\right|} \]

       10. fabs-sqr 31.7%

           \[\leadsto \frac{x}{\color{blue}{\sqrt{\sqrt{\pi} \cdot 0.5} \cdot \sqrt{\sqrt{\pi} \cdot 0.5}}} \]

       11. rem-square-sqrt 31.5%

           \[\leadsto \frac{x}{\color{blue}{\sqrt{\pi} \cdot 0.5}} \]

   11. Simplified 31.5%

       \[\leadsto \color{blue}{\frac{x}{\sqrt{\pi} \cdot 0.5}} \]
   12. Step-by-step derivation

       1. div-inv 31.7%

          \[\leadsto \color{blue}{x \cdot \frac{1}{\sqrt{\pi} \cdot 0.5}} \]

       2. *-commutative 31.7%

          \[\leadsto \color{blue}{\frac{1}{\sqrt{\pi} \cdot 0.5} \cdot x} \]

       3. *-commutative 31.7%

          \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \sqrt{\pi}}} \cdot x \]

       4. associate-/r* 31.7%

          \[\leadsto \color{blue}{\frac{\frac{1}{0.5}}{\sqrt{\pi}}} \cdot x \]

       5. metadata-eval 31.7%

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

   13. Applied egg-rr 31.7%

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

   if 1.80000000000000004 < x

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 93.5%

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

      1. *-commutative 93.5%

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

   6. Simplified 93.5%

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

      1. div-inv 93.9%

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

      2. add-sqr-sqrt 30.4%

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

      3. fabs-sqr 30.4%

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

      4. add-sqr-sqrt 32.0%

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

      5. *-commutative 32.0%

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

   8. Applied egg-rr 32.0%

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

   9. Taylor expanded in x around inf 3.6%

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

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.8:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{5}\right)\\ \end{array} \]

Alternative 13: 34.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-32}:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{2} \cdot \frac{4}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e-32) (* x (/ 2.0 (sqrt PI))) (sqrt (* (pow x 2.0) (/ 4.0 PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 5e-32) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = sqrt((pow(x, 2.0) * (4.0 / ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5e-32) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.sqrt((Math.pow(x, 2.0) * (4.0 / Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5e-32:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.sqrt((math.pow(x, 2.0) * (4.0 / math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e-32)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = sqrt(Float64((x ^ 2.0) * Float64(4.0 / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5e-32)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = sqrt(((x ^ 2.0) * (4.0 / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5e-32], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[x, 2.0], $MachinePrecision] * N[(4.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e-32

   1. Initial program 99.9%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 93.4%

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

      1. *-commutative 93.4%

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

   6. Simplified 93.4%

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

   7. Taylor expanded in x around 0 66.1%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{0.5 \cdot \sqrt{\pi}}\right|} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 66.1%

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

      2. expm1-udef 6.2%

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

      3. div-fabs 6.2%

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

      4. *-commutative 6.2%

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

   9. Applied egg-rr 6.2%

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

       1. expm1-def 66.1%

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

       2. expm1-log1p 66.1%

          \[\leadsto \color{blue}{\left|\frac{x}{\sqrt{\pi} \cdot 0.5}\right|} \]

       3. fabs-div 66.1%

          \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\sqrt{\pi} \cdot 0.5\right|}} \]

       4. unpow1 66.1%

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

       5. sqr-pow 27.7%

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

       6. fabs-sqr 27.7%

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

       7. sqr-pow 29.4%

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

       8. unpow1 29.4%

          \[\leadsto \frac{\color{blue}{x}}{\left|\sqrt{\pi} \cdot 0.5\right|} \]

       9. rem-square-sqrt 29.5%

          \[\leadsto \frac{x}{\left|\color{blue}{\sqrt{\sqrt{\pi} \cdot 0.5} \cdot \sqrt{\sqrt{\pi} \cdot 0.5}}\right|} \]

       10. fabs-sqr 29.5%

           \[\leadsto \frac{x}{\color{blue}{\sqrt{\sqrt{\pi} \cdot 0.5} \cdot \sqrt{\sqrt{\pi} \cdot 0.5}}} \]

       11. rem-square-sqrt 29.4%

           \[\leadsto \frac{x}{\color{blue}{\sqrt{\pi} \cdot 0.5}} \]

   11. Simplified 29.4%

       \[\leadsto \color{blue}{\frac{x}{\sqrt{\pi} \cdot 0.5}} \]
   12. Step-by-step derivation

       1. div-inv 29.6%

          \[\leadsto \color{blue}{x \cdot \frac{1}{\sqrt{\pi} \cdot 0.5}} \]

       2. *-commutative 29.6%

          \[\leadsto \color{blue}{\frac{1}{\sqrt{\pi} \cdot 0.5} \cdot x} \]

       3. *-commutative 29.6%

          \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \sqrt{\pi}}} \cdot x \]

       4. associate-/r* 29.6%

          \[\leadsto \color{blue}{\frac{\frac{1}{0.5}}{\sqrt{\pi}}} \cdot x \]

       5. metadata-eval 29.6%

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

   13. Applied egg-rr 29.6%

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

   if 5e-32 < x

   1. Initial program 99.7%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 94.7%

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

      1. *-commutative 94.7%

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

   6. Simplified 94.7%

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

   7. Taylor expanded in x around 0 85.1%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{0.5 \cdot \sqrt{\pi}}\right|} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 85.1%

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

      2. expm1-udef 18.9%

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

      3. div-fabs 18.9%

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

      4. *-commutative 18.9%

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

   9. Applied egg-rr 18.9%

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

       1. expm1-def 85.1%

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

       2. expm1-log1p 85.1%

          \[\leadsto \color{blue}{\left|\frac{x}{\sqrt{\pi} \cdot 0.5}\right|} \]

       3. fabs-div 85.1%

          \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\sqrt{\pi} \cdot 0.5\right|}} \]

       4. unpow1 85.1%

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

       5. sqr-pow 84.8%

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

       6. fabs-sqr 84.8%

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

       7. sqr-pow 85.1%

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

       8. unpow1 85.1%

          \[\leadsto \frac{\color{blue}{x}}{\left|\sqrt{\pi} \cdot 0.5\right|} \]

       9. rem-square-sqrt 84.9%

          \[\leadsto \frac{x}{\left|\color{blue}{\sqrt{\sqrt{\pi} \cdot 0.5} \cdot \sqrt{\sqrt{\pi} \cdot 0.5}}\right|} \]

       10. fabs-sqr 84.9%

           \[\leadsto \frac{x}{\color{blue}{\sqrt{\sqrt{\pi} \cdot 0.5} \cdot \sqrt{\sqrt{\pi} \cdot 0.5}}} \]

       11. rem-square-sqrt 85.1%

           \[\leadsto \frac{x}{\color{blue}{\sqrt{\pi} \cdot 0.5}} \]

   11. Simplified 85.1%

       \[\leadsto \color{blue}{\frac{x}{\sqrt{\pi} \cdot 0.5}} \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 84.9%

          \[\leadsto \color{blue}{\sqrt{\frac{x}{\sqrt{\pi} \cdot 0.5}} \cdot \sqrt{\frac{x}{\sqrt{\pi} \cdot 0.5}}} \]

       2. sqrt-unprod 85.1%

          \[\leadsto \color{blue}{\sqrt{\frac{x}{\sqrt{\pi} \cdot 0.5} \cdot \frac{x}{\sqrt{\pi} \cdot 0.5}}} \]

       3. div-inv 85.2%

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

       4. div-inv 85.2%

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

       5. swap-sqr 84.9%

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

       6. unpow2 84.9%

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

       7. *-commutative 84.9%

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

       8. associate-/r* 84.9%

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

       9. metadata-eval 84.9%

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

       10. *-commutative 84.9%

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

       11. associate-/r* 84.9%

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

       12. metadata-eval 84.9%

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

       13. frac-times 84.5%

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

       14. metadata-eval 84.5%

           \[\leadsto \sqrt{{x}^{2} \cdot \frac{\color{blue}{4}}{\sqrt{\pi} \cdot \sqrt{\pi}}} \]

       15. add-sqr-sqrt 85.1%

           \[\leadsto \sqrt{{x}^{2} \cdot \frac{4}{\color{blue}{\pi}}} \]

   13. Applied egg-rr 85.1%

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

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-32}:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{2} \cdot \frac{4}{\pi}}\\ \end{array} \]

Alternative 14: 34.1% accurate, 9.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around 0 93.5%

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

   1. *-commutative 93.5%

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

6. Simplified 93.5%

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

7. Taylor expanded in x around 0 66.9%

   \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{0.5 \cdot \sqrt{\pi}}\right|} \]
8. Step-by-step derivation

   1. expm1-log1p-u 66.9%

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

   2. expm1-udef 6.7%

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

   3. div-fabs 6.7%

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

   4. *-commutative 6.7%

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

9. Applied egg-rr 6.7%

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

    1. expm1-def 66.9%

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

    2. expm1-log1p 66.9%

       \[\leadsto \color{blue}{\left|\frac{x}{\sqrt{\pi} \cdot 0.5}\right|} \]

    3. fabs-div 66.9%

       \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\sqrt{\pi} \cdot 0.5\right|}} \]

    4. unpow1 66.9%

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

    5. sqr-pow 30.0%

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

    6. fabs-sqr 30.0%

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

    7. sqr-pow 31.5%

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

    8. unpow1 31.5%

       \[\leadsto \frac{\color{blue}{x}}{\left|\sqrt{\pi} \cdot 0.5\right|} \]

    9. rem-square-sqrt 31.7%

       \[\leadsto \frac{x}{\left|\color{blue}{\sqrt{\sqrt{\pi} \cdot 0.5} \cdot \sqrt{\sqrt{\pi} \cdot 0.5}}\right|} \]

    10. fabs-sqr 31.7%

        \[\leadsto \frac{x}{\color{blue}{\sqrt{\sqrt{\pi} \cdot 0.5} \cdot \sqrt{\sqrt{\pi} \cdot 0.5}}} \]

    11. rem-square-sqrt 31.5%

        \[\leadsto \frac{x}{\color{blue}{\sqrt{\pi} \cdot 0.5}} \]

11. Simplified 31.5%

    \[\leadsto \color{blue}{\frac{x}{\sqrt{\pi} \cdot 0.5}} \]
12. Step-by-step derivation

    1. div-inv 31.7%

       \[\leadsto \color{blue}{x \cdot \frac{1}{\sqrt{\pi} \cdot 0.5}} \]

    2. *-commutative 31.7%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{\pi} \cdot 0.5} \cdot x} \]

    3. *-commutative 31.7%

       \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \sqrt{\pi}}} \cdot x \]

    4. associate-/r* 31.7%

       \[\leadsto \color{blue}{\frac{\frac{1}{0.5}}{\sqrt{\pi}}} \cdot x \]

    5. metadata-eval 31.7%

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

13. Applied egg-rr 31.7%

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

14. Final simplification 31.7%

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

Reproduce

herbie shell --seed 2023322 
(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)))))))