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

Percentage Accurate: 99.8% → 99.8%

Time: 5.1s

Alternatives: 14

Speedup: 1.6×

• 29.0% 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, 1.0× 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]]]

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

Alternative 2: 99.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

Alternative 3: 99.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left|\frac{\mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)}{\sqrt{\pi}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (/
   (fma
    (pow (fabs x) 7.0)
    0.047619047619047616
    (fma
     (* 0.2 (fabs x))
     (* (* (* x x) x) x)
     (* (fabs x) (fma (* x x) 0.6666666666666666 2.0))))
   (sqrt PI))))

C

double code(double x) {
	return fabs((fma(pow(fabs(x), 7.0), 0.047619047619047616, fma((0.2 * fabs(x)), (((x * x) * x) * x), (fabs(x) * fma((x * x), 0.6666666666666666, 2.0)))) / sqrt(((double) M_PI))));
}

Julia

function code(x)
	return abs(Float64(fma((abs(x) ^ 7.0), 0.047619047619047616, fma(Float64(0.2 * abs(x)), Float64(Float64(Float64(x * x) * x) * x), Float64(abs(x) * fma(Float64(x * x), 0.6666666666666666, 2.0)))) / sqrt(pi)))
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0

   \[\leadsto \left|\color{blue}{\frac{\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

4. Applied rewrites 99.4%

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

Alternative 4: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

Alternative 5: 99.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

   \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \color{blue}{{x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]
5. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, {x}^{4} \cdot \left(\frac{1}{5} + \color{blue}{\frac{1}{21}} \cdot {x}^{2}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, {x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {\color{blue}{x}}^{2}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   5. +-commutative N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{21} \cdot {x}^{2} + \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   6. pow2 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{21} \cdot \left(x \cdot x\right) + \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\left(x \cdot x\right) \cdot \frac{1}{21} + \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   12. metadata-eval N/A

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

   13. pow-prod-up N/A

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

   14. pow-prod-down N/A

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

   15. pow2 N/A

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

   16. lower-*.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. lift-*.f64 99.4

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

6. Applied rewrites 99.4%

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

8. Applied rewrites 99.4%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

11. Applied rewrites 99.4%

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

Alternative 6: 98.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\left|x\right|, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.2, \left|x\right| \cdot 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|{x}^{7} \cdot 0.047619047619047616\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.4)
   (*
    (/ 1.0 (sqrt PI))
    (fabs (fma (fabs x) (* (* (* x x) (* x x)) 0.2) (* (fabs x) 2.0))))
   (/ (fabs (* (pow x 7.0) 0.047619047619047616)) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = (1.0 / sqrt(((double) M_PI))) * fabs(fma(fabs(x), (((x * x) * (x * x)) * 0.2), (fabs(x) * 2.0)));
	} else {
		tmp = fabs((pow(x, 7.0) * 0.047619047619047616)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.4)
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(fma(abs(x), Float64(Float64(Float64(x * x) * Float64(x * x)) * 0.2), Float64(abs(x) * 2.0))));
	else
		tmp = Float64(abs(Float64((x ^ 7.0) * 0.047619047619047616)) / sqrt(pi));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.4], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.2), $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(N[Power[x, 7.0], $MachinePrecision] * 0.047619047619047616), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\left|x\right|, \color{blue}{\frac{1}{5} \cdot {x}^{4}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right| \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\left|x\right|, \frac{1}{5} \cdot {\color{blue}{x}}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right| \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\left|x\right|, {x}^{4} \cdot \color{blue}{\frac{1}{5}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\left|x\right|, {x}^{4} \cdot \color{blue}{\frac{1}{5}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right| \]

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow-prod-down N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. metadata-eval 93.4

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

   6. Applied rewrites 93.4%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 92.9%

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

   if 2.39999999999999991 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\left|\left({x}^{\left(4 + 2\right)} \cdot \left|x\right|\right) \cdot \frac{1}{21}\right|}{\sqrt{\pi}} \]

      4. pow-prod-up N/A

         \[\leadsto \frac{\left|\left(\left({x}^{4} \cdot {x}^{2}\right) \cdot \left|x\right|\right) \cdot \frac{1}{21}\right|}{\sqrt{\pi}} \]

      5. pow2 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow-prod-down N/A

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

      9. pow2 N/A

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

      10. sqr-abs-rev N/A

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

      11. associate-*l* N/A

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

      12. sqr-abs-rev N/A

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

      13. sqr-abs-rev N/A

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

      14. associate-*l* N/A

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

   6. Applied rewrites 36.9%

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

   8. Applied rewrites 37.0%

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

Alternative 7: 93.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;\frac{\left|\left|x\right| \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), x \cdot x, 2\right)\right|}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|{x}^{7} \cdot 0.047619047619047616\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.6)
   (/
    (fabs (* (fabs x) (fma (fma (* x x) 0.2 0.6666666666666666) (* x x) 2.0)))
    (sqrt PI))
   (/ (fabs (* (pow x 7.0) 0.047619047619047616)) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (x <= 2.6) {
		tmp = fabs((fabs(x) * fma(fma((x * x), 0.2, 0.6666666666666666), (x * x), 2.0))) / sqrt(((double) M_PI));
	} else {
		tmp = fabs((pow(x, 7.0) * 0.047619047619047616)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.6)
		tmp = Float64(abs(Float64(abs(x) * fma(fma(Float64(x * x), 0.2, 0.6666666666666666), Float64(x * x), 2.0))) / sqrt(pi));
	else
		tmp = Float64(abs(Float64((x ^ 7.0) * 0.047619047619047616)) / sqrt(pi));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.6], N[(N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 0.2 + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(N[Power[x, 7.0], $MachinePrecision] * 0.047619047619047616), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.60000000000000009

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \color{blue}{{x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, {x}^{4} \cdot \left(\frac{1}{5} + \color{blue}{\frac{1}{21}} \cdot {x}^{2}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, {x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {\color{blue}{x}}^{2}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{21} \cdot {x}^{2} + \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      6. pow2 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{21} \cdot \left(x \cdot x\right) + \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\left(x \cdot x\right) \cdot \frac{1}{21} + \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      12. metadata-eval N/A

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

      13. pow-prod-up N/A

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

      14. pow-prod-down N/A

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

      15. pow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 99.4

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

   6. Applied rewrites 99.4%

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 93.0

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

   11. Applied rewrites 93.0%

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

   if 2.60000000000000009 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\left|\left({x}^{\left(4 + 2\right)} \cdot \left|x\right|\right) \cdot \frac{1}{21}\right|}{\sqrt{\pi}} \]

      4. pow-prod-up N/A

         \[\leadsto \frac{\left|\left(\left({x}^{4} \cdot {x}^{2}\right) \cdot \left|x\right|\right) \cdot \frac{1}{21}\right|}{\sqrt{\pi}} \]

      5. pow2 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow-prod-down N/A

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

      9. pow2 N/A

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

      10. sqr-abs-rev N/A

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

      11. associate-*l* N/A

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

      12. sqr-abs-rev N/A

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

      13. sqr-abs-rev N/A

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

      14. associate-*l* N/A

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

   6. Applied rewrites 36.9%

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

   8. Applied rewrites 37.0%

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

Alternative 8: 92.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.1:\\ \;\;\;\;\frac{\left|\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) \cdot \left|x\right|\right|}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|{x}^{7} \cdot 0.047619047619047616\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.1)
   (/ (fabs (* (fma 0.6666666666666666 (* x x) 2.0) (fabs x))) (sqrt PI))
   (/ (fabs (* (pow x 7.0) 0.047619047619047616)) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (x <= 2.1) {
		tmp = fabs((fma(0.6666666666666666, (x * x), 2.0) * fabs(x))) / sqrt(((double) M_PI));
	} else {
		tmp = fabs((pow(x, 7.0) * 0.047619047619047616)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.1)
		tmp = Float64(abs(Float64(fma(0.6666666666666666, Float64(x * x), 2.0) * abs(x))) / sqrt(pi));
	else
		tmp = Float64(abs(Float64((x ^ 7.0) * 0.047619047619047616)) / sqrt(pi));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.1], N[(N[Abs[N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(N[Power[x, 7.0], $MachinePrecision] * 0.047619047619047616), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.10000000000000009

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \color{blue}{{x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, {x}^{4} \cdot \left(\frac{1}{5} + \color{blue}{\frac{1}{21}} \cdot {x}^{2}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, {x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {\color{blue}{x}}^{2}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{21} \cdot {x}^{2} + \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      6. pow2 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{21} \cdot \left(x \cdot x\right) + \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\left(x \cdot x\right) \cdot \frac{1}{21} + \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      12. metadata-eval N/A

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

      13. pow-prod-up N/A

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

      14. pow-prod-down N/A

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

      15. pow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 99.4

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 88.5%

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

   if 2.10000000000000009 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{\left|\left({x}^{\left(4 + 2\right)} \cdot \left|x\right|\right) \cdot \frac{1}{21}\right|}{\sqrt{\pi}} \]

      4. pow-prod-up N/A

         \[\leadsto \frac{\left|\left(\left({x}^{4} \cdot {x}^{2}\right) \cdot \left|x\right|\right) \cdot \frac{1}{21}\right|}{\sqrt{\pi}} \]

      5. pow2 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow-prod-down N/A

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

      9. pow2 N/A

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

      10. sqr-abs-rev N/A

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

      11. associate-*l* N/A

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

      12. sqr-abs-rev N/A

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

      13. sqr-abs-rev N/A

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

      14. associate-*l* N/A

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

   6. Applied rewrites 36.9%

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

   8. Applied rewrites 37.0%

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

Alternative 9: 88.5% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0

   \[\leadsto \left|\color{blue}{\frac{\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

   \[\leadsto \left|\frac{\mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left|\frac{\mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, \frac{1}{21}, \left|x\right| \cdot 2\right)}{\sqrt{\pi}}\right| \]

   2. lower-*.f64 N/A

      \[\leadsto \left|\frac{\mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, \frac{1}{21}, \left|x\right| \cdot 2\right)}{\sqrt{\pi}}\right| \]

   3. lift-fabs.f64 98.4

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

7. Applied rewrites 98.4%

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

Alternative 10: 88.5% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[(N[Abs[N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * 0.2), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \color{blue}{{x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, {x}^{4} \cdot \left(\frac{1}{5} + \color{blue}{\frac{1}{21}} \cdot {x}^{2}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, {x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {\color{blue}{x}}^{2}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{21} \cdot {x}^{2} + \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      6. pow2 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{21} \cdot \left(x \cdot x\right) + \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\left(x \cdot x\right) \cdot \frac{1}{21} + \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      12. metadata-eval N/A

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

      13. pow-prod-up N/A

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

      14. pow-prod-down N/A

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

      15. pow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 99.4

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 88.5%

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

   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. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{\frac{\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in x around inf

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

      1. metadata-eval N/A

         \[\leadsto \left|\frac{1}{5} \cdot \frac{{x}^{4} \cdot \left|x\right|}{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}}\right| \]

      2. *-commutative N/A

         \[\leadsto \left|\frac{{x}^{4} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{1}{\color{blue}{5}}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{{x}^{4} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{1}{\color{blue}{5}}\right| \]

   7. Applied rewrites 31.2%

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

Alternative 11: 88.5% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

   \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \color{blue}{{x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]
5. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, {x}^{4} \cdot \left(\frac{1}{5} + \color{blue}{\frac{1}{21}} \cdot {x}^{2}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, {x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {\color{blue}{x}}^{2}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   5. +-commutative N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{21} \cdot {x}^{2} + \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   6. pow2 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{21} \cdot \left(x \cdot x\right) + \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\left(x \cdot x\right) \cdot \frac{1}{21} + \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   12. metadata-eval N/A

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

   13. pow-prod-up N/A

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

   14. pow-prod-down N/A

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

   15. pow2 N/A

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

   16. lower-*.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. lift-*.f64 99.4

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

6. Applied rewrites 99.4%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 88.5%

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

Alternative 12: 67.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot 2\\ \mathbf{if}\;x \leq 5 \cdot 10^{-155}:\\ \;\;\;\;\frac{\left|t\_0\right|}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{t\_0 \cdot t\_0}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) 2.0)))
   (if (<= x 5e-155) (/ (fabs t_0) (sqrt PI)) (sqrt (/ (* t_0 t_0) PI)))))

C

double code(double x) {
	double t_0 = fabs(x) * 2.0;
	double tmp;
	if (x <= 5e-155) {
		tmp = fabs(t_0) / sqrt(((double) M_PI));
	} else {
		tmp = sqrt(((t_0 * t_0) / ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * 2.0;
	double tmp;
	if (x <= 5e-155) {
		tmp = Math.abs(t_0) / Math.sqrt(Math.PI);
	} else {
		tmp = Math.sqrt(((t_0 * t_0) / Math.PI));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.fabs(x) * 2.0
	tmp = 0
	if x <= 5e-155:
		tmp = math.fabs(t_0) / math.sqrt(math.pi)
	else:
		tmp = math.sqrt(((t_0 * t_0) / math.pi))
	return tmp

Julia

function code(x)
	t_0 = Float64(abs(x) * 2.0)
	tmp = 0.0
	if (x <= 5e-155)
		tmp = Float64(abs(t_0) / sqrt(pi));
	else
		tmp = sqrt(Float64(Float64(t_0 * t_0) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = abs(x) * 2.0;
	tmp = 0.0;
	if (x <= 5e-155)
		tmp = abs(t_0) / sqrt(pi);
	else
		tmp = sqrt(((t_0 * t_0) / pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[x, 5e-155], N[(N[Abs[t$95$0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.9999999999999999e-155

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \color{blue}{{x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, {x}^{4} \cdot \left(\frac{1}{5} + \color{blue}{\frac{1}{21}} \cdot {x}^{2}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, {x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {\color{blue}{x}}^{2}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{21} \cdot {x}^{2} + \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      6. pow2 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{21} \cdot \left(x \cdot x\right) + \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\left(x \cdot x\right) \cdot \frac{1}{21} + \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      12. metadata-eval N/A

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

      13. pow-prod-up N/A

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

      14. pow-prod-down N/A

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

      15. pow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 99.4

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 67.2%

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

   if 4.9999999999999999e-155 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \color{blue}{{x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, {x}^{4} \cdot \left(\frac{1}{5} + \color{blue}{\frac{1}{21}} \cdot {x}^{2}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, {x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {\color{blue}{x}}^{2}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{21} \cdot {x}^{2} + \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      6. pow2 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{21} \cdot \left(x \cdot x\right) + \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\left(x \cdot x\right) \cdot \frac{1}{21} + \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      12. metadata-eval N/A

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

      13. pow-prod-up N/A

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

      14. pow-prod-down N/A

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

      15. pow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 99.4

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 67.2%

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

   11. Applied rewrites 54.2%

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

Alternative 13: 67.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left|\left|x\right| \cdot 2\right|}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\sqrt{x \cdot x} \cdot 2\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e-7)
   (/ (fabs (* (fabs x) 2.0)) (sqrt PI))
   (/ (fabs (* (sqrt (* x x)) 2.0)) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (x <= 2e-7) {
		tmp = fabs((fabs(x) * 2.0)) / sqrt(((double) M_PI));
	} else {
		tmp = fabs((sqrt((x * x)) * 2.0)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2e-7) {
		tmp = Math.abs((Math.abs(x) * 2.0)) / Math.sqrt(Math.PI);
	} else {
		tmp = Math.abs((Math.sqrt((x * x)) * 2.0)) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2e-7:
		tmp = math.fabs((math.fabs(x) * 2.0)) / math.sqrt(math.pi)
	else:
		tmp = math.fabs((math.sqrt((x * x)) * 2.0)) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e-7)
		tmp = Float64(abs(Float64(abs(x) * 2.0)) / sqrt(pi));
	else
		tmp = Float64(abs(Float64(sqrt(Float64(x * x)) * 2.0)) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2e-7)
		tmp = abs((abs(x) * 2.0)) / sqrt(pi);
	else
		tmp = abs((sqrt((x * x)) * 2.0)) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2e-7], N[(N[Abs[N[(N[Abs[x], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(N[Sqrt[N[(x * x), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9999999999999999e-7

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \color{blue}{{x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, {x}^{4} \cdot \left(\frac{1}{5} + \color{blue}{\frac{1}{21}} \cdot {x}^{2}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, {x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {\color{blue}{x}}^{2}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{21} \cdot {x}^{2} + \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      6. pow2 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{21} \cdot \left(x \cdot x\right) + \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\left(x \cdot x\right) \cdot \frac{1}{21} + \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      12. metadata-eval N/A

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

      13. pow-prod-up N/A

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

      14. pow-prod-down N/A

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

      15. pow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 99.4

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 67.2%

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

   if 1.9999999999999999e-7 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \color{blue}{{x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, {x}^{4} \cdot \left(\frac{1}{5} + \color{blue}{\frac{1}{21}} \cdot {x}^{2}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, {x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {\color{blue}{x}}^{2}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{21} \cdot {x}^{2} + \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      6. pow2 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{21} \cdot \left(x \cdot x\right) + \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\left(x \cdot x\right) \cdot \frac{1}{21} + \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      12. metadata-eval N/A

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

      13. pow-prod-up N/A

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

      14. pow-prod-down N/A

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

      15. pow2 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 99.4

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 67.2%

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

       1. lift-fabs.f64 N/A

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

       2. rem-sqrt-square-rev N/A

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

       3. pow2 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 54.0

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

   11. Applied rewrites 54.0%

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

Alternative 14: 67.2% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

   \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \color{blue}{{x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]
5. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, {x}^{4} \cdot \left(\frac{1}{5} + \color{blue}{\frac{1}{21}} \cdot {x}^{2}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, {x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {\color{blue}{x}}^{2}\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   5. +-commutative N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{21} \cdot {x}^{2} + \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   6. pow2 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\frac{1}{21} \cdot \left(x \cdot x\right) + \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \left(\left(x \cdot x\right) \cdot \frac{1}{21} + \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   12. metadata-eval N/A

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

   13. pow-prod-up N/A

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

   14. pow-prod-down N/A

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

   15. pow2 N/A

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

   16. lower-*.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. lift-*.f64 99.4

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

6. Applied rewrites 99.4%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 67.2%

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

Reproduce

herbie shell --seed 2025142 
(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)))))))