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

Percentage Accurate: 99.8% → 99.8%

Time: 5.2s

Alternatives: 17

Speedup: N/A×

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

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% 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, 1.1× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = (fabs(x) * 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_0 * fabs(x)) * fabs(x)))) + ((1.0 / 21.0) * ((((x * x) * x) * x) * ((fabs(x) * x) * x))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * 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_0 * Math.abs(x)) * Math.abs(x)))) + ((1.0 / 21.0) * ((((x * x) * x) * x) * ((Math.abs(x) * x) * x))))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * 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_0 * math.fabs(x)) * math.fabs(x)))) + ((1.0 / 21.0) * ((((x * x) * x) * x) * ((math.fabs(x) * x) * x))))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * 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) * Float64(Float64(t_0 * abs(x)) * abs(x)))) + Float64(Float64(1.0 / 21.0) * Float64(Float64(Float64(Float64(x * x) * x) * x) * Float64(Float64(abs(x) * x) * x))))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * 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_0 * abs(x)) * abs(x)))) + ((1.0 / 21.0) * ((((x * x) * x) * x) * ((abs(x) * x) * 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]}, 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] * N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(N[(N[Abs[x], $MachinePrecision] * x), $MachinePrecision] * x), $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.047619047619047616), $MachinePrecision] * x + N[(x * 0.2), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(x + 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. Add Preprocessing

4. Applied rewrites 99.8%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f64 99.8

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

9. Applied rewrites 99.8%

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

Alternative 3: 99.8% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.047619047619047616 + 0.2), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(x + 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. Add Preprocessing

4. Applied rewrites 99.8%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f64 99.8

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

9. Applied rewrites 99.8%

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

10. Taylor expanded in x around 0

    \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(x \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right), \color{blue}{x} \cdot x, x + x\right)\right| \]
11. Step-by-step derivation

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. +-commutative N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 N/A

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

    6. +-commutative N/A

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

    7. *-commutative N/A

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

    8. lower-fma.f64 N/A

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

    9. pow2 N/A

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

    10. lift-*.f64 N/A

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

    11. pow2 N/A

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

    12. lift-*.f64 99.8

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

12. Applied rewrites 99.8%

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

Alternative 4: 93.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{\pi}}\\ t_1 := \left(x \cdot x\right) \cdot x\\ \mathbf{if}\;x \leq 2.7:\\ \;\;\;\;t\_0 \cdot \left|\mathsf{fma}\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \left(\left(t\_1 \cdot x\right) \cdot t\_1\right)\right) \cdot 0.047619047619047616\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt PI))) (t_1 (* (* x x) x)))
   (if (<= x 2.7)
     (*
      t_0
      (fabs (* (fma (fma (* 0.2 x) x 0.6666666666666666) (* x x) 2.0) x)))
     (* (* t_0 (* (* t_1 x) t_1)) 0.047619047619047616))))

C

double code(double x) {
	double t_0 = 1.0 / sqrt(((double) M_PI));
	double t_1 = (x * x) * x;
	double tmp;
	if (x <= 2.7) {
		tmp = t_0 * fabs((fma(fma((0.2 * x), x, 0.6666666666666666), (x * x), 2.0) * x));
	} else {
		tmp = (t_0 * ((t_1 * x) * t_1)) * 0.047619047619047616;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 / sqrt(pi))
	t_1 = Float64(Float64(x * x) * x)
	tmp = 0.0
	if (x <= 2.7)
		tmp = Float64(t_0 * abs(Float64(fma(fma(Float64(0.2 * x), x, 0.6666666666666666), Float64(x * x), 2.0) * x)));
	else
		tmp = Float64(Float64(t_0 * Float64(Float64(t_1 * x) * t_1)) * 0.047619047619047616);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, 2.7], N[(t$95$0 * N[Abs[N[(N[(N[(N[(0.2 * x), $MachinePrecision] * x + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(N[(t$95$1 * x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * 0.047619047619047616), $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. Add Preprocessing

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 93.6%

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

   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. Add Preprocessing

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{21} \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \color{blue}{\frac{1}{21}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \color{blue}{\frac{1}{21}} \]

   7. Applied rewrites 3.7%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \frac{1}{21} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \frac{1}{21} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \frac{1}{21} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \frac{1}{21} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \frac{1}{21} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \frac{1}{21} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \frac{1}{21} \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \frac{1}{21} \]

      9. lift-PI.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{21} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{21} \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{\sqrt{1}}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{21} \]

      12. sqrt-div N/A

          \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{21} \]

      13. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \frac{1}{21} \]

      14. pow3 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left({x}^{3} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \frac{1}{21} \]

      15. pow2 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left({x}^{3} \cdot {\left(x \cdot x\right)}^{2}\right)\right) \cdot \frac{1}{21} \]

      16. pow2 N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left({x}^{3} \cdot {\left({x}^{2}\right)}^{2}\right)\right) \cdot \frac{1}{21} \]

      17. pow-pow N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left({x}^{3} \cdot {x}^{\left(2 \cdot 2\right)}\right)\right) \cdot \frac{1}{21} \]

      18. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left({x}^{3} \cdot {x}^{4}\right)\right) \cdot \frac{1}{21} \]

      19. pow-prod-up N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot {x}^{\left(3 + 4\right)}\right) \cdot \frac{1}{21} \]

      20. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot {x}^{7}\right) \cdot \frac{1}{21} \]

   9. Applied rewrites 3.7%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right) \cdot 0.047619047619047616 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 93.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{\pi}}\\ \mathbf{if}\;x \leq 2.7:\\ \;\;\;\;t\_0 \cdot \left|\mathsf{fma}\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot t\_0\right) \cdot 0.047619047619047616\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt PI))))
   (if (<= x 2.7)
     (*
      t_0
      (fabs (* (fma (fma (* 0.2 x) x 0.6666666666666666) (* x x) 2.0) x)))
     (* (* (* (* (* x x) x) (* (* x x) (* x x))) t_0) 0.047619047619047616))))

C

double code(double x) {
	double t_0 = 1.0 / sqrt(((double) M_PI));
	double tmp;
	if (x <= 2.7) {
		tmp = t_0 * fabs((fma(fma((0.2 * x), x, 0.6666666666666666), (x * x), 2.0) * x));
	} else {
		tmp = ((((x * x) * x) * ((x * x) * (x * x))) * t_0) * 0.047619047619047616;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 / sqrt(pi))
	tmp = 0.0
	if (x <= 2.7)
		tmp = Float64(t_0 * abs(Float64(fma(fma(Float64(0.2 * x), x, 0.6666666666666666), Float64(x * x), 2.0) * x)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * x) * Float64(Float64(x * x) * Float64(x * x))) * t_0) * 0.047619047619047616);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.7], N[(t$95$0 * N[Abs[N[(N[(N[(N[(0.2 * x), $MachinePrecision] * x + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * 0.047619047619047616), $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. Add Preprocessing

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 93.6%

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

   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. Add Preprocessing

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{21} \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \color{blue}{\frac{1}{21}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \color{blue}{\frac{1}{21}} \]

   7. Applied rewrites 3.7%

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

Alternative 6: 99.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

10. Applied rewrites 99.8%

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

Alternative 7: 93.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.047619047619047616 \cdot x\right)}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.7)
   (*
    (/ 1.0 (sqrt PI))
    (fabs (* (fma (fma (* 0.2 x) x 0.6666666666666666) (* x x) 2.0) x)))
   (/
    (* (* (* x x) (* (* x x) (* x x))) (* 0.047619047619047616 x))
    (sqrt PI))))

C

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.7)
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(fma(fma(Float64(0.2 * x), x, 0.6666666666666666), Float64(x * x), 2.0) * x)));
	else
		tmp = Float64(Float64(Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(x * x))) * Float64(0.047619047619047616 * x)) / sqrt(pi));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.7], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(N[(N[(N[(0.2 * x), $MachinePrecision] * x + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.047619047619047616 * x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $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. Add Preprocessing

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 93.6%

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

   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. Add Preprocessing

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{21} \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \color{blue}{\frac{1}{21}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \color{blue}{\frac{1}{21}} \]

   7. Applied rewrites 3.7%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \frac{1}{21} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \frac{1}{21} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \frac{1}{21} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \frac{1}{21} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \frac{1}{21} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \frac{1}{21} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \frac{1}{21} \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \frac{1}{21} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 1}{\sqrt{\pi}} \cdot \frac{1}{21} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 1}{\sqrt{\pi}} \cdot \frac{1}{21} \]

   9. Applied rewrites 3.7%

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

   11. Applied rewrites 3.7%

       \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.047619047619047616 \cdot x\right)}{\color{blue}{\sqrt{\pi}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 93.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}{\sqrt{\pi}} \cdot 0.047619047619047616\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.7)
   (*
    (/ 1.0 (sqrt PI))
    (fabs (* (fma (fma (* 0.2 x) x 0.6666666666666666) (* x x) 2.0) x)))
   (*
    (/ (* (* (* x x) (* x x)) (* (* x x) x)) (sqrt PI))
    0.047619047619047616)))

C

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.7)
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(fma(fma(Float64(0.2 * x), x, 0.6666666666666666), Float64(x * x), 2.0) * x)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(Float64(x * x) * x)) / sqrt(pi)) * 0.047619047619047616);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.7], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(N[(N[(N[(0.2 * x), $MachinePrecision] * x + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * 0.047619047619047616), $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. Add Preprocessing

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 93.6%

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

   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. Add Preprocessing

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{21} \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \color{blue}{\frac{1}{21}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \color{blue}{\frac{1}{21}} \]

   7. Applied rewrites 3.7%

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

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \frac{1}{21} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \frac{1}{21} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \frac{1}{21} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \frac{1}{21} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \frac{1}{21} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \frac{1}{21} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \frac{1}{21} \]

      8. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \frac{1}{21} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 1}{\sqrt{\pi}} \cdot \frac{1}{21} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 1}{\sqrt{\pi}} \cdot \frac{1}{21} \]

   9. Applied rewrites 3.7%

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

       1. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 1}{\sqrt{\pi}} \cdot \frac{1}{21} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 1}{\sqrt{\pi}} \cdot \frac{1}{21} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 1}{\sqrt{\pi}} \cdot \frac{1}{21} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 1}{\sqrt{\pi}} \cdot \frac{1}{21} \]

       5. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 1}{\sqrt{\pi}} \cdot \frac{1}{21} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 1}{\sqrt{\pi}} \cdot \frac{1}{21} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 1}{\sqrt{\pi}} \cdot \frac{1}{21} \]

       8. *-rgt-identity N/A

          \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}{\sqrt{\pi}} \cdot \frac{1}{21} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}{\sqrt{\pi}} \cdot \frac{1}{21} \]

       10. lift-*.f64 N/A

           \[\leadsto \frac{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}{\sqrt{\pi}} \cdot \frac{1}{21} \]

       11. pow3 N/A

           \[\leadsto \frac{\left({x}^{3} \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}{\sqrt{\pi}} \cdot \frac{1}{21} \]

       12. pow-plus N/A

           \[\leadsto \frac{{x}^{\left(3 + 1\right)} \cdot \left(\left(x \cdot x\right) \cdot x\right)}{\sqrt{\pi}} \cdot \frac{1}{21} \]

       13. metadata-eval N/A

           \[\leadsto \frac{{x}^{4} \cdot \left(\left(x \cdot x\right) \cdot x\right)}{\sqrt{\pi}} \cdot \frac{1}{21} \]

       14. lower-*.f64 N/A

           \[\leadsto \frac{{x}^{4} \cdot \left(\left(x \cdot x\right) \cdot x\right)}{\sqrt{\pi}} \cdot \frac{1}{21} \]

       15. metadata-eval N/A

           \[\leadsto \frac{{x}^{\left(2 + 2\right)} \cdot \left(\left(x \cdot x\right) \cdot x\right)}{\sqrt{\pi}} \cdot \frac{1}{21} \]

       16. pow-prod-up N/A

           \[\leadsto \frac{\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}{\sqrt{\pi}} \cdot \frac{1}{21} \]

       17. lower-*.f64 N/A

           \[\leadsto \frac{\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}{\sqrt{\pi}} \cdot \frac{1}{21} \]

       18. pow2 N/A

           \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot {x}^{2}\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}{\sqrt{\pi}} \cdot \frac{1}{21} \]

       19. lift-*.f64 N/A

           \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot {x}^{2}\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}{\sqrt{\pi}} \cdot \frac{1}{21} \]

       20. pow2 N/A

           \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}{\sqrt{\pi}} \cdot \frac{1}{21} \]

       21. lift-*.f64 3.7

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

   11. Applied rewrites 3.7%

       \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}{\sqrt{\pi}} \cdot 0.047619047619047616 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 89.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{\pi}}\\ \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;t\_0 \cdot \left|\mathsf{fma}\left(0.6666666666666666 \cdot x, x \cdot x, x \cdot 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.2\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt PI))))
   (if (<= x 2.2)
     (* t_0 (fabs (fma (* 0.6666666666666666 x) (* x x) (* x 2.0))))
     (* t_0 (fabs (* (* (* (* x x) x) (* x x)) 0.2))))))

C

double code(double x) {
	double t_0 = 1.0 / sqrt(((double) M_PI));
	double tmp;
	if (x <= 2.2) {
		tmp = t_0 * fabs(fma((0.6666666666666666 * x), (x * x), (x * 2.0)));
	} else {
		tmp = t_0 * fabs(((((x * x) * x) * (x * x)) * 0.2));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 / sqrt(pi))
	tmp = 0.0
	if (x <= 2.2)
		tmp = Float64(t_0 * abs(fma(Float64(0.6666666666666666 * x), Float64(x * x), Float64(x * 2.0))));
	else
		tmp = Float64(t_0 * abs(Float64(Float64(Float64(Float64(x * x) * x) * Float64(x * x)) * 0.2)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.2], N[(t$95$0 * N[Abs[N[(N[(0.6666666666666666 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Abs[N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.2), $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. Add Preprocessing

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 89.8

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

   10. Applied rewrites 89.8%

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

   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. Add Preprocessing

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 93.6%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|{x}^{5} \cdot \frac{1}{5}\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|{x}^{5} \cdot \frac{1}{5}\right| \]

      3. metadata-eval N/A

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

      4. pow-prod-up N/A

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

      5. pow3 N/A

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

      6. pow2 N/A

         \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{5}\right| \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{5}\right| \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{5}\right| \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{5}\right| \]

      10. lift-*.f64 30.7

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

   10. Applied rewrites 30.7%

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

Alternative 10: 93.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(N[(N[(N[(0.2 * x), $MachinePrecision] * x + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * x), $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. Add Preprocessing

4. Applied rewrites 99.8%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

10. Applied rewrites 93.6%

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

Alternative 11: 34.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\pi}} \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) \cdot x\right) \cdot \left(2 \cdot x\right)}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e-17)
   (* (* (/ 1.0 (sqrt PI)) x) 2.0)
   (sqrt (/ (* (* (fma 0.6666666666666666 (* x x) 2.0) x) (* 2.0 x)) PI))))

C

double code(double x) {
	double tmp;
	if (x <= 2e-17) {
		tmp = ((1.0 / sqrt(((double) M_PI))) * x) * 2.0;
	} else {
		tmp = sqrt((((fma(0.6666666666666666, (x * x), 2.0) * x) * (2.0 * x)) / ((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e-17)
		tmp = Float64(Float64(Float64(1.0 / sqrt(pi)) * x) * 2.0);
	else
		tmp = sqrt(Float64(Float64(Float64(fma(0.6666666666666666, Float64(x * x), 2.0) * x) * Float64(2.0 * x)) / pi));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2e-17], N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * 2.0), $MachinePrecision], N[Sqrt[N[(N[(N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * x), $MachinePrecision] * N[(2.0 * x), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.00000000000000014e-17

   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. Add Preprocessing

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 33.4%

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

   if 2.00000000000000014e-17 < x

   1. Initial program 99.7%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 92.3%

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

      1. lift-/.f64 N/A

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

   9. Applied rewrites 92.7%

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

   10. Taylor expanded in x around 0

       \[\leadsto \sqrt{\frac{\left(\mathsf{fma}\left(\frac{2}{3}, x \cdot x, 2\right) \cdot x\right) \cdot \left(2 \cdot x\right)}{\pi}} \]
   11. Step-by-step derivation

   12. Applied rewrites 83.8%

       \[\leadsto \sqrt{\frac{\left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) \cdot x\right) \cdot \left(2 \cdot x\right)}{\pi}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 89.8% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in x around 0

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

   1. lower-*.f64 89.8

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

10. Applied rewrites 89.8%

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

Alternative 13: 34.3% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 34.3%

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

   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. Add Preprocessing

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 89.4%

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

   8. Taylor expanded in x around inf

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

      1. pow2 N/A

         \[\leadsto \frac{\left|\left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right|}{\sqrt{\pi}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left|\left(\frac{2}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right|}{\sqrt{\pi}} \]

      3. lift-*.f64 27.3

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

   10. Applied rewrites 27.3%

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

Alternative 14: 89.4% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 89.4%

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

Alternative 15: 34.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\left(\frac{1}{\sqrt{\pi}} \cdot x\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(x \cdot x\right) \cdot 4}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e-17)
   (* (* (/ 1.0 (sqrt PI)) x) 2.0)
   (sqrt (/ (* (* x x) 4.0) PI))))

C

double code(double x) {
	double tmp;
	if (x <= 2e-17) {
		tmp = ((1.0 / sqrt(((double) M_PI))) * x) * 2.0;
	} else {
		tmp = sqrt((((x * x) * 4.0) / ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2e-17) {
		tmp = ((1.0 / Math.sqrt(Math.PI)) * x) * 2.0;
	} else {
		tmp = Math.sqrt((((x * x) * 4.0) / Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2e-17:
		tmp = ((1.0 / math.sqrt(math.pi)) * x) * 2.0
	else:
		tmp = math.sqrt((((x * x) * 4.0) / math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e-17)
		tmp = Float64(Float64(Float64(1.0 / sqrt(pi)) * x) * 2.0);
	else
		tmp = sqrt(Float64(Float64(Float64(x * x) * 4.0) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2e-17)
		tmp = ((1.0 / sqrt(pi)) * x) * 2.0;
	else
		tmp = sqrt((((x * x) * 4.0) / pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2e-17], N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * 2.0), $MachinePrecision], N[Sqrt[N[(N[(N[(x * x), $MachinePrecision] * 4.0), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.00000000000000014e-17

   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. Add Preprocessing

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 33.4%

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

   if 2.00000000000000014e-17 < x

   1. Initial program 99.7%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 92.3%

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

      1. lift-/.f64 N/A

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

   9. Applied rewrites 92.7%

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

   10. Taylor expanded in x around 0

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

       1. unpow2 N/A

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

       2. sqr-abs-rev N/A

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

       3. pow2 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 83.2

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

   12. Applied rewrites 83.2%

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

Alternative 16: 34.3% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * 2.0), $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. Add Preprocessing

4. Applied rewrites 99.2%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

7. Applied rewrites 34.3%

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

Alternative 17: 67.8% accurate, 6.8× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (fabs (+ x x)) (sqrt PI)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Abs[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $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. Add Preprocessing

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 67.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f64 67.8

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

9. Applied rewrites 67.8%

   \[\leadsto \frac{\left|x + \color{blue}{x}\right|}{\sqrt{\pi}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025092 
(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)))))))