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

Percentage Accurate: 99.8% → 99.8%

Time: 3.7s

Alternatives: 12

Speedup: 1.6×

• 29.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)\right| \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

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

4. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)}\right| \]
5. 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: 98.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;\left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right) \cdot x, x + x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|{x}^{7} \cdot 0.047619047619047616\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.6)
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (fma (* x x) (* (fma (* x x) 0.2 0.6666666666666666) x) (+ x x))))
   (/ (fabs (* (pow x 7.0) 0.047619047619047616)) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (x <= 2.6) {
		tmp = fabs(((1.0 / sqrt(((double) M_PI))) * fma((x * x), (fma((x * x), 0.2, 0.6666666666666666) * x), (x + x))));
	} 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 = abs(Float64(Float64(1.0 / sqrt(pi)) * fma(Float64(x * x), Float64(fma(Float64(x * x), 0.2, 0.6666666666666666) * x), Float64(x + x))));
	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[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 0.2 + 0.6666666666666666), $MachinePrecision] * x), $MachinePrecision] + N[(x + x), $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 < 2.60000000000000009

   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 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 93.6%

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

   7. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval 93.6

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

   9. Applied rewrites 93.6%

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

   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 0

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-plus N/A

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

      6. pow3 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. metadata-eval 93.1

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

   6. Applied rewrites 93.1%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 37.6%

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

   10. Taylor expanded in x around 0

       \[\leadsto \frac{\left|{x}^{7} \cdot \frac{1}{21}\right|}{\sqrt{\pi}} \]
   11. Step-by-step derivation

       1. lift-pow.f64 37.6

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

   12. Applied rewrites 37.6%

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

Alternative 4: 98.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|{x}^{7} \cdot 0.047619047619047616\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.6)
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (* (fma (fma (* x x) 0.2 0.6666666666666666) (* x x) 2.0) x)))
   (/ (fabs (* (pow x 7.0) 0.047619047619047616)) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (x <= 2.6) {
		tmp = fabs(((1.0 / sqrt(((double) M_PI))) * (fma(fma((x * x), 0.2, 0.6666666666666666), (x * x), 2.0) * x)));
	} 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 = abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(fma(fma(Float64(x * x), 0.2, 0.6666666666666666), Float64(x * x), 2.0) * x)));
	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[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(x * x), $MachinePrecision] * 0.2 + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * x), $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 < 2.60000000000000009

   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 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 93.6%

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

   7. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   9. Applied rewrites 93.6%

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

   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 0

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-plus N/A

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

      6. pow3 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. metadata-eval 93.1

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

   6. Applied rewrites 93.1%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 37.6%

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

   10. Taylor expanded in x around 0

       \[\leadsto \frac{\left|{x}^{7} \cdot \frac{1}{21}\right|}{\sqrt{\pi}} \]
   11. Step-by-step derivation

       1. lift-pow.f64 37.6

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

   12. Applied rewrites 37.6%

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

Alternative 5: 93.6% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision] * x), $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 < 2.2000000000000002

   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 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 93.6%

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

   7. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. metadata-eval 89.0

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

   9. Applied rewrites 89.0%

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

   if 2.2000000000000002 < x

   1. Initial program 99.8%

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

   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

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-plus N/A

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

      6. pow3 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. metadata-eval 93.1

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

   6. Applied rewrites 93.1%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 37.6%

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

   10. Taylor expanded in x around 0

       \[\leadsto \frac{\left|{x}^{7} \cdot \frac{1}{21}\right|}{\sqrt{\pi}} \]
   11. Step-by-step derivation

       1. lift-pow.f64 37.6

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

   12. Applied rewrites 37.6%

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

Alternative 6: 93.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, 0.047619047619047616, x + x\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs (* (/ 1.0 (sqrt PI)) (fma (pow x 7.0) 0.047619047619047616 (+ x x)))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * fma(pow(x, 7.0), 0.047619047619047616, (x + x))));
}

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * fma((x ^ 7.0), 0.047619047619047616, Float64(x + x))))
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)}\right| \]

5. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lift-pow.f64 N/A

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

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

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

   6. pow2 N/A

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

   7. sqrt-pow1 N/A

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

   8. metadata-eval N/A

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

   9. unpow1 N/A

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

   10. metadata-eval N/A

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

   11. count-2-rev N/A

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

   12. lower-+.f64 N/A

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

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

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

   14. pow2 N/A

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

   15. sqrt-pow1 N/A

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

   16. metadata-eval N/A

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

   17. unpow1 N/A

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

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

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

   19. pow2 N/A

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

   20. sqrt-pow1 N/A

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

   21. metadata-eval N/A

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

   22. unpow1 98.9

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

7. Applied rewrites 98.9%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \color{blue}{0.047619047619047616}, x + x\right)\right| \]
8. Add Preprocessing

Alternative 7: 89.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1 \cdot \left(\mathsf{fma}\left({x}^{7}, 0.047619047619047616, x\right) + x\right)}{\sqrt{\pi}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs (/ (* 1.0 (+ (fma (pow x 7.0) 0.047619047619047616 x) x)) (sqrt PI))))

C

double code(double x) {
	return fabs(((1.0 * (fma(pow(x, 7.0), 0.047619047619047616, x) + x)) / sqrt(((double) M_PI))));
}

Julia

function code(x)
	return abs(Float64(Float64(1.0 * Float64(fma((x ^ 7.0), 0.047619047619047616, x) + x)) / sqrt(pi)))
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 * N[(N[(N[Power[x, 7.0], $MachinePrecision] * 0.047619047619047616 + x), $MachinePrecision] + x), $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

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

4. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)}\right| \]

5. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lift-pow.f64 N/A

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

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

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

   6. pow2 N/A

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

   7. sqrt-pow1 N/A

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

   8. metadata-eval N/A

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

   9. unpow1 N/A

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

   10. metadata-eval N/A

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

   11. count-2-rev N/A

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

   12. lower-+.f64 N/A

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

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

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

   14. pow2 N/A

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

   15. sqrt-pow1 N/A

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

   16. metadata-eval N/A

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

   17. unpow1 N/A

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

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

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

   19. pow2 N/A

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

   20. sqrt-pow1 N/A

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

   21. metadata-eval N/A

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

   22. unpow1 98.9

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

7. Applied rewrites 98.9%

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

9. Applied rewrites 98.4%

   \[\leadsto \left|\color{blue}{\frac{1 \cdot \left(\mathsf{fma}\left({x}^{7}, 0.047619047619047616, x\right) + x\right)}{\sqrt{\pi}}}\right| \]
10. Add Preprocessing

Alternative 8: 89.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{\pi}}\\ \mathbf{if}\;x \leq 2.3:\\ \;\;\;\;\left|t\_0 \cdot \left(\mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) \cdot x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_0 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(0.2 \cdot x\right)\right) \cdot x\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt PI))))
   (if (<= x 2.3)
     (fabs (* t_0 (* (fma (* x x) 0.6666666666666666 2.0) x)))
     (fabs (* t_0 (* (* (* (* x x) x) (* 0.2 x)) x))))))

C

double code(double x) {
	double t_0 = 1.0 / sqrt(((double) M_PI));
	double tmp;
	if (x <= 2.3) {
		tmp = fabs((t_0 * (fma((x * x), 0.6666666666666666, 2.0) * x)));
	} else {
		tmp = fabs((t_0 * ((((x * x) * x) * (0.2 * x)) * x)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 / sqrt(pi))
	tmp = 0.0
	if (x <= 2.3)
		tmp = abs(Float64(t_0 * Float64(fma(Float64(x * x), 0.6666666666666666, 2.0) * x)));
	else
		tmp = abs(Float64(t_0 * Float64(Float64(Float64(Float64(x * x) * x) * Float64(0.2 * x)) * x)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.3], N[Abs[N[(t$95$0 * N[(N[(N[(x * x), $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$0 * N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(0.2 * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2999999999999998

   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%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 93.6%

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

   7. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. metadata-eval 89.0

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

   9. Applied rewrites 89.0%

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

   if 2.2999999999999998 < 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

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

   4. Applied rewrites 99.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)}\right| \]

   5. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. associate-*r* N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\frac{1}{5} \cdot {x}^{4}\right) \cdot \left|x\right|\right)\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\frac{1}{5} \cdot {x}^{4}\right) \cdot \left|x\right|\right)\right| \]

      4. *-commutative N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left({x}^{4} \cdot \frac{1}{5}\right) \cdot \left|x\right|\right)\right| \]

      5. lower-*.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left({x}^{4} \cdot \frac{1}{5}\right) \cdot \left|x\right|\right)\right| \]

      6. metadata-eval N/A

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

      7. pow-plus N/A

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

      8. pow3 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. metadata-eval N/A

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

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

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

      14. pow2 N/A

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

      15. sqrt-pow1 N/A

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

      16. metadata-eval N/A

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

      17. unpow1 32.0

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

   7. Applied rewrites 32.0%

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

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. pow3 N/A

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

      8. unpow1 N/A

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

      9. metadata-eval N/A

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

      10. sqrt-pow1 N/A

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

      11. pow2 N/A

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

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

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. pow3 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

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

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

      21. pow2 N/A

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

      22. sqrt-pow1 N/A

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

      23. metadata-eval N/A

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

      24. unpow1 32.0

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

   9. Applied rewrites 32.0%

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

Alternative 9: 89.0% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 93.6%

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

7. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. pow2 N/A

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

   5. +-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. pow2 N/A

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

   8. lower-fma.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 N/A

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

   11. metadata-eval 89.0

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

9. Applied rewrites 89.0%

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

Alternative 10: 88.5% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

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

   1. metadata-eval N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval N/A

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

   5. pow-plus N/A

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

   6. pow3 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. metadata-eval 93.1

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

6. Applied rewrites 93.1%

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

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%

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

Alternative 11: 67.0% accurate, 9.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

6. Applied rewrites 89.0%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 67.0%

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

Alternative 12: 66.5% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

6. Applied rewrites 89.0%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 67.0%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. associate-*l/ N/A

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

    4. count-2-rev N/A

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

    5. lower-/.f64 N/A

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

    6. lift-+.f64 66.5

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

11. Applied rewrites 66.5%

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

Reproduce

herbie shell --seed 2025114 
(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)))))))