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

Percentage Accurate: 99.8% → 99.8%

Time: 10.3s

Alternatives: 17

Speedup: 3.7×

• 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 99.8%

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

2. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. *-rgt-identity 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 3: 99.9% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return Math.abs(((x * Math.sqrt((1.0 / Math.PI))) * ((2.0 + (0.6666666666666666 * (x * x))) + ((0.2 * Math.pow(x, 4.0)) + (0.047619047619047616 * Math.pow(x, 6.0))))));
}

Python

def code(x):
	return math.fabs(((x * math.sqrt((1.0 / math.pi))) * ((2.0 + (0.6666666666666666 * (x * x))) + ((0.2 * math.pow(x, 4.0)) + (0.047619047619047616 * math.pow(x, 6.0))))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = abs(((x * sqrt((1.0 / pi))) * ((2.0 + (0.6666666666666666 * (x * x))) + ((0.2 * (x ^ 4.0)) + (0.047619047619047616 * (x ^ 6.0))))));
end

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. associate-*l/ 99.4%

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

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. Taylor expanded in x around 0 99.8%

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

   1. unpow1 99.8%

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

   2. sqr-pow 31.4%

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

   3. fabs-sqr 31.4%

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

   4. sqr-pow 99.8%

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

   5. unpow1 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in x around 0 99.8%

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

   1. metadata-eval 92.1%

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

   2. fma-udef 92.1%

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

   3. metadata-eval 92.1%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 4: 99.4% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return Math.abs((((2.0 + (0.6666666666666666 * (x * x))) + ((0.2 * Math.pow(x, 4.0)) + (0.047619047619047616 * Math.pow(x, 6.0)))) * (x / Math.sqrt(Math.PI))));
}

Python

def code(x):
	return math.fabs((((2.0 + (0.6666666666666666 * (x * x))) + ((0.2 * math.pow(x, 4.0)) + (0.047619047619047616 * math.pow(x, 6.0)))) * (x / math.sqrt(math.pi))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = abs((((2.0 + (0.6666666666666666 * (x * x))) + ((0.2 * (x ^ 4.0)) + (0.047619047619047616 * (x ^ 6.0)))) * (x / sqrt(pi))));
end

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. associate-*l/ 99.4%

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

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. Taylor expanded in x around 0 99.8%

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

   1. unpow1 99.8%

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

   2. sqr-pow 31.4%

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

   3. fabs-sqr 31.4%

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

   4. sqr-pow 99.8%

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

   5. unpow1 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in x around 0 99.8%

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

   1. metadata-eval 92.1%

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

   2. fma-udef 92.1%

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

   3. metadata-eval 92.1%

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

9. Applied egg-rr 99.8%

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

    1. expm1-log1p-u 64.3%

       \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \cdot \left(\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    2. expm1-udef 6.4%

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

    3. sqrt-div 6.4%

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

    4. metadata-eval 6.4%

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

    5. un-div-inv 6.4%

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

11. Applied egg-rr 6.4%

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

    1. expm1-def 63.9%

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

    2. expm1-log1p 99.4%

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

13. Simplified 99.4%

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

14. Final simplification 99.4%

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

Alternative 5: 99.2% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. associate-*l/ 99.4%

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

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. Taylor expanded in x around 0 99.8%

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

   1. unpow1 99.8%

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

   2. sqr-pow 31.4%

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

   3. fabs-sqr 31.4%

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

   4. sqr-pow 99.8%

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

   5. unpow1 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in x around inf 98.4%

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

8. Final simplification 98.4%

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

Alternative 6: 99.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;\left|t_0 \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(x \cdot t_0\right) \cdot \left(\left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {x}^{4}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (if (<= x -2.2)
     (fabs
      (* t_0 (+ (* 0.2 (pow x 5.0)) (* 0.047619047619047616 (pow x 7.0)))))
     (fabs
      (*
       (* x t_0)
       (+ (+ 2.0 (* 0.6666666666666666 (* x x))) (* 0.2 (pow x 4.0))))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double tmp;
	if (x <= -2.2) {
		tmp = fabs((t_0 * ((0.2 * pow(x, 5.0)) + (0.047619047619047616 * pow(x, 7.0)))));
	} else {
		tmp = fabs(((x * t_0) * ((2.0 + (0.6666666666666666 * (x * x))) + (0.2 * pow(x, 4.0)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	double tmp;
	if (x <= -2.2) {
		tmp = Math.abs((t_0 * ((0.2 * Math.pow(x, 5.0)) + (0.047619047619047616 * Math.pow(x, 7.0)))));
	} else {
		tmp = Math.abs(((x * t_0) * ((2.0 + (0.6666666666666666 * (x * x))) + (0.2 * Math.pow(x, 4.0)))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / math.pi))
	tmp = 0
	if x <= -2.2:
		tmp = math.fabs((t_0 * ((0.2 * math.pow(x, 5.0)) + (0.047619047619047616 * math.pow(x, 7.0)))))
	else:
		tmp = math.fabs(((x * t_0) * ((2.0 + (0.6666666666666666 * (x * x))) + (0.2 * math.pow(x, 4.0)))))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / pi))
	tmp = 0.0
	if (x <= -2.2)
		tmp = abs(Float64(t_0 * Float64(Float64(0.2 * (x ^ 5.0)) + Float64(0.047619047619047616 * (x ^ 7.0)))));
	else
		tmp = abs(Float64(Float64(x * t_0) * Float64(Float64(2.0 + Float64(0.6666666666666666 * Float64(x * x))) + Float64(0.2 * (x ^ 4.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 / pi));
	tmp = 0.0;
	if (x <= -2.2)
		tmp = abs((t_0 * ((0.2 * (x ^ 5.0)) + (0.047619047619047616 * (x ^ 7.0)))));
	else
		tmp = abs(((x * t_0) * ((2.0 + (0.6666666666666666 * (x * x))) + (0.2 * (x ^ 4.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -2.2], N[Abs[N[(t$95$0 * N[(N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x * t$95$0), $MachinePrecision] * N[(N[(2.0 + N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2000000000000002

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Step-by-step derivation

      1. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 99.4%

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

      1. associate-*r* 99.4%

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

      2. associate-*r* 99.4%

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

   6. Simplified 99.5%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)}\right| \]

   7. Taylor expanded in x around 0 99.5%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)}\right| \]

   if -2.2000000000000002 < x

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Step-by-step derivation

      1. associate-*l/ 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around 0 99.8%

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

      1. unpow1 99.8%

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

      2. sqr-pow 48.7%

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

      3. fabs-sqr 48.7%

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

      4. sqr-pow 99.8%

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

      5. unpow1 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in x around 0 98.8%

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

      1. metadata-eval 98.8%

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

      2. fma-udef 98.8%

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

      3. metadata-eval 98.8%

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

   9. Applied egg-rr 98.8%

      \[\leadsto \left|\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\color{blue}{\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right)} + 0.2 \cdot {x}^{4}\right)\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

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

Alternative 7: 98.4% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. associate-+l+ 99.8%

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

   2. +-commutative 99.8%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

4. Taylor expanded in x around inf 97.3%

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

5. Final simplification 97.3%

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

Alternative 8: 99.3% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -2.7000000000000002

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Step-by-step derivation

      1. associate-+l+ 99.8%

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

      2. +-commutative 99.8%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around inf 98.3%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-udef 0.0%

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

      3. sqrt-div 0.0%

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

      4. metadata-eval 0.0%

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

      5. un-div-inv 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. expm1-def 0.0%

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

      2. expm1-log1p 98.3%

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

      3. associate-*r/ 98.4%

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

      4. associate-/l* 98.4%

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

      5. associate-/r/ 98.4%

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

   8. Simplified 98.4%

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

   if -2.7000000000000002 < x

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Step-by-step derivation

      1. associate-*l/ 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around 0 99.8%

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

      1. unpow1 99.8%

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

      2. sqr-pow 48.7%

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

      3. fabs-sqr 48.7%

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

      4. sqr-pow 99.8%

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

      5. unpow1 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in x around 0 98.8%

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

      1. metadata-eval 98.8%

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

      2. fma-udef 98.8%

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

      3. metadata-eval 98.8%

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

   9. Applied egg-rr 98.8%

      \[\leadsto \left|\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\color{blue}{\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right)} + 0.2 \cdot {x}^{4}\right)\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

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

Alternative 9: 97.1% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;\left|\sqrt{\frac{{x}^{14}}{\pi} \cdot 0.0022675736961451248}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.6666666666666666\right)\right) + 2 \cdot x\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.2)
   (fabs (sqrt (* (/ (pow x 14.0) PI) 0.0022675736961451248)))
   (fabs
    (* (sqrt (/ 1.0 PI)) (+ (* x (* x (* x 0.6666666666666666))) (* 2.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -2.2) {
		tmp = fabs(sqrt(((pow(x, 14.0) / ((double) M_PI)) * 0.0022675736961451248)));
	} else {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * ((x * (x * (x * 0.6666666666666666))) + (2.0 * x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.2) {
		tmp = Math.abs(Math.sqrt(((Math.pow(x, 14.0) / Math.PI) * 0.0022675736961451248)));
	} else {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * ((x * (x * (x * 0.6666666666666666))) + (2.0 * x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.2:
		tmp = math.fabs(math.sqrt(((math.pow(x, 14.0) / math.pi) * 0.0022675736961451248)))
	else:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * ((x * (x * (x * 0.6666666666666666))) + (2.0 * x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.2)
		tmp = abs(sqrt(Float64(Float64((x ^ 14.0) / pi) * 0.0022675736961451248)));
	else
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(x * Float64(x * Float64(x * 0.6666666666666666))) + Float64(2.0 * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.2)
		tmp = abs(sqrt((((x ^ 14.0) / pi) * 0.0022675736961451248)));
	else
		tmp = abs((sqrt((1.0 / pi)) * ((x * (x * (x * 0.6666666666666666))) + (2.0 * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.2], N[Abs[N[Sqrt[N[(N[(N[Power[x, 14.0], $MachinePrecision] / Pi), $MachinePrecision] * 0.0022675736961451248), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(x * N[(x * N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2000000000000002

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Step-by-step derivation

      1. associate-+l+ 99.8%

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

      2. +-commutative 99.8%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around inf 98.3%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 92.1%

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

      3. *-commutative 92.1%

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

      4. *-commutative 92.1%

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

      5. swap-sqr 92.1%

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

      6. *-commutative 92.1%

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

      7. *-commutative 92.1%

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

      8. swap-sqr 92.1%

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

      9. add-sqr-sqrt 92.1%

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

      10. pow-prod-up 92.1%

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

      11. metadata-eval 92.1%

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

      12. metadata-eval 92.2%

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

   6. Applied egg-rr 92.2%

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

      1. metadata-eval 92.2%

         \[\leadsto \left|\sqrt{\left(\frac{1}{\pi} \cdot {x}^{\color{blue}{\left(2 \cdot 7\right)}}\right) \cdot 0.0022675736961451248}\right| \]

      2. pow-sqr 92.2%

         \[\leadsto \left|\sqrt{\left(\frac{1}{\pi} \cdot \color{blue}{\left({x}^{7} \cdot {x}^{7}\right)}\right) \cdot 0.0022675736961451248}\right| \]

      3. associate-*l/ 92.2%

         \[\leadsto \left|\sqrt{\color{blue}{\frac{1 \cdot \left({x}^{7} \cdot {x}^{7}\right)}{\pi}} \cdot 0.0022675736961451248}\right| \]

      4. *-lft-identity 92.2%

         \[\leadsto \left|\sqrt{\frac{\color{blue}{{x}^{7} \cdot {x}^{7}}}{\pi} \cdot 0.0022675736961451248}\right| \]

      5. pow-sqr 92.2%

         \[\leadsto \left|\sqrt{\frac{\color{blue}{{x}^{\left(2 \cdot 7\right)}}}{\pi} \cdot 0.0022675736961451248}\right| \]

      6. metadata-eval 92.2%

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

   8. Simplified 92.2%

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

   if -2.2000000000000002 < x

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Step-by-step derivation

      1. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 98.5%

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

      1. associate-*r* 98.5%

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

      2. associate-*r* 98.5%

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

      3. distribute-rgt-out 98.5%

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

      4. *-commutative 98.5%

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

      5. associate-*r* 98.5%

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

      6. distribute-rgt-in 98.5%

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

      7. +-commutative 98.5%

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

      8. fma-def 98.5%

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

      9. unpow2 98.5%

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

      10. unpow1 98.5%

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

      11. sqr-pow 48.1%

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

      12. fabs-sqr 48.1%

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

      13. sqr-pow 98.5%

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

      14. unpow1 98.5%

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

      15. fma-def 98.5%

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

      16. associate-*r* 98.5%

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

      17. fma-def 98.5%

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

   6. Simplified 98.5%

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

      1. fma-udef 98.5%

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

      2. distribute-rgt-in 98.5%

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

      3. *-commutative 98.5%

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

   8. Applied egg-rr 98.5%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(\left(\left(x \cdot 0.6666666666666666\right) \cdot x\right) \cdot x + 2 \cdot x\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;\left|\sqrt{\frac{{x}^{14}}{\pi} \cdot 0.0022675736961451248}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.6666666666666666\right)\right) + 2 \cdot x\right)\right|\\ \end{array} \]

Alternative 10: 99.2% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -2.2000000000000002

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Step-by-step derivation

      1. associate-+l+ 99.8%

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

      2. +-commutative 99.8%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around inf 98.3%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-udef 0.0%

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

      3. sqrt-div 0.0%

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

      4. metadata-eval 0.0%

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

      5. un-div-inv 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. expm1-def 0.0%

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

      2. expm1-log1p 98.3%

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

   8. Simplified 98.3%

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

   if -2.2000000000000002 < x

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Step-by-step derivation

      1. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 98.5%

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

      1. associate-*r* 98.5%

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

      2. associate-*r* 98.5%

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

      3. distribute-rgt-out 98.5%

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

      4. *-commutative 98.5%

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

      5. associate-*r* 98.5%

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

      6. distribute-rgt-in 98.5%

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

      7. +-commutative 98.5%

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

      8. fma-def 98.5%

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

      9. unpow2 98.5%

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

      10. unpow1 98.5%

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

      11. sqr-pow 48.1%

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

      12. fabs-sqr 48.1%

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

      13. sqr-pow 98.5%

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

      14. unpow1 98.5%

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

      15. fma-def 98.5%

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

      16. associate-*r* 98.5%

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

      17. fma-def 98.5%

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

   6. Simplified 98.5%

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

      1. fma-udef 98.5%

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

      2. distribute-rgt-in 98.5%

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

      3. *-commutative 98.5%

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

   8. Applied egg-rr 98.5%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(\left(\left(x \cdot 0.6666666666666666\right) \cdot x\right) \cdot x + 2 \cdot x\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

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

Alternative 11: 99.2% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -2.2000000000000002

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Step-by-step derivation

      1. associate-+l+ 99.8%

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

      2. +-commutative 99.8%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around inf 98.3%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-udef 0.0%

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

      3. sqrt-div 0.0%

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

      4. metadata-eval 0.0%

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

      5. un-div-inv 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. expm1-def 0.0%

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

      2. expm1-log1p 98.3%

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

      3. associate-*r/ 98.4%

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

      4. associate-/l* 98.4%

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

      5. associate-/r/ 98.4%

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

   8. Simplified 98.4%

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

   if -2.2000000000000002 < x

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Step-by-step derivation

      1. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 98.5%

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

      1. associate-*r* 98.5%

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

      2. associate-*r* 98.5%

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

      3. distribute-rgt-out 98.5%

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

      4. *-commutative 98.5%

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

      5. associate-*r* 98.5%

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

      6. distribute-rgt-in 98.5%

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

      7. +-commutative 98.5%

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

      8. fma-def 98.5%

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

      9. unpow2 98.5%

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

      10. unpow1 98.5%

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

      11. sqr-pow 48.1%

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

      12. fabs-sqr 48.1%

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

      13. sqr-pow 98.5%

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

      14. unpow1 98.5%

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

      15. fma-def 98.5%

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

      16. associate-*r* 98.5%

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

      17. fma-def 98.5%

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

   6. Simplified 98.5%

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

      1. fma-udef 98.5%

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

      2. distribute-rgt-in 98.5%

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

      3. *-commutative 98.5%

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

   8. Applied egg-rr 98.5%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(\left(\left(x \cdot 0.6666666666666666\right) \cdot x\right) \cdot x + 2 \cdot x\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

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

Alternative 12: 91.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot 0.6666666666666666\right)\\ \mathbf{if}\;x \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\left|\sqrt{\frac{\left(x \cdot x\right) \cdot 4}{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \frac{t_0 \cdot t_0 - 4}{t_0 - 2}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x 0.6666666666666666))))
   (if (<= x -2e+154)
     (fabs (sqrt (/ (* (* x x) 4.0) PI)))
     (fabs (* (sqrt (/ 1.0 PI)) (* x (/ (- (* t_0 t_0) 4.0) (- t_0 2.0))))))))

C

double code(double x) {
	double t_0 = x * (x * 0.6666666666666666);
	double tmp;
	if (x <= -2e+154) {
		tmp = fabs(sqrt((((x * x) * 4.0) / ((double) M_PI))));
	} else {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * (x * (((t_0 * t_0) - 4.0) / (t_0 - 2.0)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = x * (x * 0.6666666666666666);
	double tmp;
	if (x <= -2e+154) {
		tmp = Math.abs(Math.sqrt((((x * x) * 4.0) / Math.PI)));
	} else {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * (x * (((t_0 * t_0) - 4.0) / (t_0 - 2.0)))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * 0.6666666666666666)
	tmp = 0
	if x <= -2e+154:
		tmp = math.fabs(math.sqrt((((x * x) * 4.0) / math.pi)))
	else:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * (x * (((t_0 * t_0) - 4.0) / (t_0 - 2.0)))))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * 0.6666666666666666))
	tmp = 0.0
	if (x <= -2e+154)
		tmp = abs(sqrt(Float64(Float64(Float64(x * x) * 4.0) / pi)));
	else
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(x * Float64(Float64(Float64(t_0 * t_0) - 4.0) / Float64(t_0 - 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * 0.6666666666666666);
	tmp = 0.0;
	if (x <= -2e+154)
		tmp = abs(sqrt((((x * x) * 4.0) / pi)));
	else
		tmp = abs((sqrt((1.0 / pi)) * (x * (((t_0 * t_0) - 4.0) / (t_0 - 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2e+154], N[Abs[N[Sqrt[N[(N[(N[(x * x), $MachinePrecision] * 4.0), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x * N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - 4.0), $MachinePrecision] / N[(t$95$0 - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.00000000000000007e154

   1. Initial program 100.0%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Step-by-step derivation

      1. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 7.1%

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

      1. associate-*r* 7.1%

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

      2. *-commutative 7.1%

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

      3. associate-*l* 7.1%

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

      4. unpow1 7.1%

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

      5. sqr-pow 0.0%

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

      6. fabs-sqr 0.0%

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

      7. sqr-pow 7.1%

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

      8. unpow1 7.1%

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

   6. Simplified 7.1%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-udef 0.0%

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

      3. associate-*r* 0.0%

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

      4. sqrt-div 0.0%

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

      5. metadata-eval 0.0%

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

      6. un-div-inv 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. expm1-def 0.0%

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

      2. expm1-log1p 7.1%

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

   10. Simplified 7.1%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 100.0%

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

       3. *-commutative 100.0%

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

       4. *-commutative 100.0%

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

       5. frac-times 100.0%

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

       6. swap-sqr 100.0%

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

       7. metadata-eval 100.0%

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

       8. add-sqr-sqrt 100.0%

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

   12. Applied egg-rr 100.0%

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

   if -2.00000000000000007e154 < x

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Step-by-step derivation

      1. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 84.0%

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

      1. associate-*r* 84.0%

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

      2. associate-*r* 84.0%

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

      3. distribute-rgt-out 84.0%

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

      4. *-commutative 84.0%

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

      5. associate-*r* 84.0%

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

      6. distribute-rgt-in 84.0%

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

      7. +-commutative 84.0%

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

      8. fma-def 84.0%

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

      9. unpow2 84.0%

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

      10. unpow1 84.0%

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

      11. sqr-pow 37.1%

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

      12. fabs-sqr 37.1%

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

      13. sqr-pow 84.0%

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

      14. unpow1 84.0%

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

      15. fma-def 84.0%

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

      16. associate-*r* 84.0%

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

      17. fma-def 84.0%

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

   6. Simplified 84.0%

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

      1. fma-udef 84.0%

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

      2. flip-+ 86.6%

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

      3. *-commutative 86.6%

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

      4. *-commutative 86.6%

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

      5. metadata-eval 86.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \frac{\left(\left(x \cdot 0.6666666666666666\right) \cdot x\right) \cdot \left(\left(x \cdot 0.6666666666666666\right) \cdot x\right) - \color{blue}{4}}{\left(0.6666666666666666 \cdot x\right) \cdot x - 2}\right)\right| \]

      6. *-commutative 86.6%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \frac{\left(\left(x \cdot 0.6666666666666666\right) \cdot x\right) \cdot \left(\left(x \cdot 0.6666666666666666\right) \cdot x\right) - 4}{\color{blue}{\left(x \cdot 0.6666666666666666\right)} \cdot x - 2}\right)\right| \]

   8. Applied egg-rr 86.6%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \color{blue}{\frac{\left(\left(x \cdot 0.6666666666666666\right) \cdot x\right) \cdot \left(\left(x \cdot 0.6666666666666666\right) \cdot x\right) - 4}{\left(x \cdot 0.6666666666666666\right) \cdot x - 2}}\right)\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\left|\sqrt{\frac{\left(x \cdot x\right) \cdot 4}{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \frac{\left(x \cdot \left(x \cdot 0.6666666666666666\right)\right) \cdot \left(x \cdot \left(x \cdot 0.6666666666666666\right)\right) - 4}{x \cdot \left(x \cdot 0.6666666666666666\right) - 2}\right)\right|\\ \end{array} \]

Alternative 13: 89.3% accurate, 6.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	return fabs((sqrt((1.0 / ((double) M_PI))) * ((x * (x * (x * 0.6666666666666666))) + (2.0 * x))));
}

Java

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

Python

def code(x):
	return math.fabs((math.sqrt((1.0 / math.pi)) * ((x * (x * (x * 0.6666666666666666))) + (2.0 * x))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. *-rgt-identity 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around 0 86.6%

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

   1. associate-*r* 86.6%

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

   2. associate-*r* 86.6%

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

   3. distribute-rgt-out 86.6%

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

   4. *-commutative 86.6%

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

   5. associate-*r* 86.6%

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

   6. distribute-rgt-in 86.6%

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

   7. +-commutative 86.6%

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

   8. fma-def 86.6%

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

   9. unpow2 86.6%

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

   10. unpow1 86.6%

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

   11. sqr-pow 31.0%

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

   12. fabs-sqr 31.0%

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

   13. sqr-pow 86.6%

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

   14. unpow1 86.6%

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

   15. fma-def 86.6%

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

   16. associate-*r* 86.6%

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

   17. fma-def 86.6%

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

6. Simplified 86.6%

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

   1. fma-udef 86.6%

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

   2. distribute-rgt-in 86.6%

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

   3. *-commutative 86.6%

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

8. Applied egg-rr 86.6%

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

9. Final simplification 86.6%

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

Alternative 14: 88.9% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.7)
   (fabs (* (sqrt (/ 1.0 PI)) (* x (* 0.6666666666666666 (* x x)))))
   (fabs (* x (/ 2.0 (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.7) {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * (x * (0.6666666666666666 * (x * x)))));
	} else {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.7) {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * (x * (0.6666666666666666 * (x * x)))));
	} else {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.7:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * (x * (0.6666666666666666 * (x * x)))))
	else:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.7)
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(x * Float64(0.6666666666666666 * Float64(x * x)))));
	else
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.7)
		tmp = abs((sqrt((1.0 / pi)) * (x * (0.6666666666666666 * (x * x)))));
	else
		tmp = abs((x * (2.0 / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.7], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x * N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.69999999999999996

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Step-by-step derivation

      1. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 65.1%

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

      1. associate-*r* 65.1%

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

      2. associate-*r* 65.1%

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

      3. distribute-rgt-out 65.1%

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

      4. *-commutative 65.1%

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

      5. associate-*r* 65.1%

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

      6. distribute-rgt-in 65.1%

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

      7. +-commutative 65.1%

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

      8. fma-def 65.1%

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

      9. unpow2 65.1%

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

      10. unpow1 65.1%

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

      11. sqr-pow 0.0%

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

      12. fabs-sqr 0.0%

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

      13. sqr-pow 65.1%

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

      14. unpow1 65.1%

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

      15. fma-def 65.1%

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

      16. associate-*r* 65.1%

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

      17. fma-def 65.1%

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

   6. Simplified 65.1%

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

   7. Taylor expanded in x around inf 65.1%

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

      1. unpow2 65.1%

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

   9. Simplified 65.1%

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

   if -1.69999999999999996 < x

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Step-by-step derivation

      1. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 97.3%

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

      1. associate-*r* 97.3%

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

      2. *-commutative 97.3%

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

      3. associate-*l* 97.3%

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

      4. unpow1 97.3%

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

      5. sqr-pow 47.8%

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

      6. fabs-sqr 47.8%

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

      7. sqr-pow 97.3%

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

      8. unpow1 97.3%

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

   6. Simplified 97.3%

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

      1. expm1-log1p-u 97.3%

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

      2. expm1-udef 8.0%

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

      3. associate-*r* 8.0%

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

      4. sqrt-div 8.0%

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

      5. metadata-eval 8.0%

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

      6. un-div-inv 8.0%

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

   8. Applied egg-rr 8.0%

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

      1. expm1-def 96.7%

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

      2. expm1-log1p 96.7%

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

   10. Simplified 96.7%

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

       1. expm1-log1p-u 96.7%

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

       2. expm1-udef 8.0%

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

       3. *-commutative 8.0%

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

       4. *-un-lft-identity 8.0%

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

       5. times-frac 8.0%

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

       6. metadata-eval 8.0%

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

   12. Applied egg-rr 8.0%

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

       1. expm1-def 96.7%

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

       2. expm1-log1p 96.7%

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

       3. associate-*r/ 96.7%

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

       4. *-commutative 96.7%

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

       5. associate-*r/ 97.3%

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

   14. Simplified 97.3%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 15: 89.3% accurate, 6.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	return fabs((sqrt((1.0 / ((double) M_PI))) * (x * (2.0 + (x * (x * 0.6666666666666666))))));
}

Java

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

Python

def code(x):
	return math.fabs((math.sqrt((1.0 / math.pi)) * (x * (2.0 + (x * (x * 0.6666666666666666))))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. *-rgt-identity 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around 0 86.6%

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

   1. associate-*r* 86.6%

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

   2. associate-*r* 86.6%

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

   3. distribute-rgt-out 86.6%

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

   4. *-commutative 86.6%

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

   5. associate-*r* 86.6%

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

   6. distribute-rgt-in 86.6%

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

   7. +-commutative 86.6%

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

   8. fma-def 86.6%

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

   9. unpow2 86.6%

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

   10. unpow1 86.6%

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

   11. sqr-pow 31.0%

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

   12. fabs-sqr 31.0%

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

   13. sqr-pow 86.6%

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

   14. unpow1 86.6%

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

   15. fma-def 86.6%

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

   16. associate-*r* 86.6%

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

   17. fma-def 86.6%

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

6. Simplified 86.6%

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

   1. fma-udef 86.6%

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

   2. *-commutative 86.6%

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

8. Applied egg-rr 86.6%

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

9. Final simplification 86.6%

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

Alternative 16: 83.3% accurate, 6.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-5)
   (fabs (sqrt (/ (* (* x x) 4.0) PI)))
   (fabs (* x (/ 2.0 (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= -2e-5) {
		tmp = fabs(sqrt((((x * x) * 4.0) / ((double) M_PI))));
	} else {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -2e-5) {
		tmp = Math.abs(Math.sqrt((((x * x) * 4.0) / Math.PI)));
	} else {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2e-5:
		tmp = math.fabs(math.sqrt((((x * x) * 4.0) / math.pi)))
	else:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-5)
		tmp = abs(sqrt(Float64(Float64(Float64(x * x) * 4.0) / pi)));
	else
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2e-5)
		tmp = abs(sqrt((((x * x) * 4.0) / pi)));
	else
		tmp = abs((x * (2.0 / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2e-5], N[Abs[N[Sqrt[N[(N[(N[(x * x), $MachinePrecision] * 4.0), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.00000000000000016e-5

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Step-by-step derivation

      1. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 8.2%

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

      1. associate-*r* 8.2%

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

      2. *-commutative 8.2%

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

      3. associate-*l* 8.2%

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

      4. unpow1 8.2%

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

      5. sqr-pow 0.0%

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

      6. fabs-sqr 0.0%

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

      7. sqr-pow 8.2%

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

      8. unpow1 8.2%

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

   6. Simplified 8.2%

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

      1. expm1-log1p-u 2.6%

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

      2. expm1-udef 2.6%

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

      3. associate-*r* 2.6%

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

      4. sqrt-div 2.6%

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

      5. metadata-eval 2.6%

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

      6. un-div-inv 2.6%

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

   8. Applied egg-rr 2.6%

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

      1. expm1-def 2.6%

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

      2. expm1-log1p 8.2%

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

   10. Simplified 8.2%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 48.8%

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

       3. *-commutative 48.8%

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

       4. *-commutative 48.8%

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

       5. frac-times 48.8%

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

       6. swap-sqr 48.8%

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

       7. metadata-eval 48.8%

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

       8. add-sqr-sqrt 48.8%

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

   12. Applied egg-rr 48.8%

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

   if -2.00000000000000016e-5 < x

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Step-by-step derivation

      1. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 98.8%

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

      1. associate-*r* 98.8%

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

      2. *-commutative 98.8%

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

      3. associate-*l* 98.8%

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

      4. unpow1 98.8%

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

      5. sqr-pow 49.2%

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

      6. fabs-sqr 49.2%

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

      7. sqr-pow 98.8%

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

      8. unpow1 98.8%

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

   6. Simplified 98.8%

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

      1. expm1-log1p-u 98.8%

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

      2. expm1-udef 6.7%

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

      3. associate-*r* 6.7%

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

      4. sqrt-div 6.7%

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

      5. metadata-eval 6.7%

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

      6. un-div-inv 6.7%

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

   8. Applied egg-rr 6.7%

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

      1. expm1-def 98.1%

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

      2. expm1-log1p 98.1%

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

   10. Simplified 98.1%

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

       1. expm1-log1p-u 98.1%

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

       2. expm1-udef 6.7%

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

       3. *-commutative 6.7%

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

       4. *-un-lft-identity 6.7%

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

       5. times-frac 6.7%

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

       6. metadata-eval 6.7%

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

   12. Applied egg-rr 6.7%

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

       1. expm1-def 98.1%

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

       2. expm1-log1p 98.1%

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

       3. associate-*r/ 98.1%

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

       4. *-commutative 98.1%

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

       5. associate-*r/ 98.8%

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

   14. Simplified 98.8%

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

4. Final simplification 80.0%

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

Alternative 17: 67.3% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. *-rgt-identity 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around 0 64.8%

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

   1. associate-*r* 64.8%

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

   2. *-commutative 64.8%

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

   3. associate-*l* 64.8%

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

   4. unpow1 64.8%

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

   5. sqr-pow 30.8%

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

   6. fabs-sqr 30.8%

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

   7. sqr-pow 64.8%

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

   8. unpow1 64.8%

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

6. Simplified 64.8%

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

   1. expm1-log1p-u 62.7%

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

   2. expm1-udef 5.2%

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

   3. associate-*r* 5.2%

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

   4. sqrt-div 5.2%

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

   5. metadata-eval 5.2%

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

   6. un-div-inv 5.2%

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

8. Applied egg-rr 5.2%

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

   1. expm1-def 62.3%

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

   2. expm1-log1p 64.4%

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

10. Simplified 64.4%

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

    1. expm1-log1p-u 62.3%

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

    2. expm1-udef 5.2%

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

    3. *-commutative 5.2%

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

    4. *-un-lft-identity 5.2%

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

    5. times-frac 5.2%

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

    6. metadata-eval 5.2%

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

12. Applied egg-rr 5.2%

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

    1. expm1-def 62.3%

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

    2. expm1-log1p 64.4%

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

    3. associate-*r/ 64.4%

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

    4. *-commutative 64.4%

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

    5. associate-*r/ 64.8%

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

14. Simplified 64.8%

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

15. Final simplification 64.8%

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

Reproduce

herbie shell --seed 2023171 
(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)))))))