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

Percentage Accurate: 99.8% → 99.8%

Time: 13.9s

Alternatives: 9

Speedup: N/A×

• 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, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.4%

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

   1. lift-fabs.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. inv-pow N/A

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

6. Applied rewrites 99.8%

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

Alternative 2: 99.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;\left|x\right| \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|x\right| \cdot \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right), 0.6666666666666666\right)}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.5)
   (*
    (fabs x)
    (/ (fma (* x x) (fma x (* x 0.2) 0.6666666666666666) 2.0) (sqrt PI)))
   (*
    (fabs x)
    (/
     (*
      (* x x)
      (fma x (* x (fma (* x x) 0.047619047619047616 0.2)) 0.6666666666666666))
     (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.5) {
		tmp = fabs(x) * (fma((x * x), fma(x, (x * 0.2), 0.6666666666666666), 2.0) / sqrt(((double) M_PI)));
	} else {
		tmp = fabs(x) * (((x * x) * fma(x, (x * fma((x * x), 0.047619047619047616, 0.2)), 0.6666666666666666)) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.5)
		tmp = Float64(abs(x) * Float64(fma(Float64(x * x), fma(x, Float64(x * 0.2), 0.6666666666666666), 2.0) / sqrt(pi)));
	else
		tmp = Float64(abs(x) * Float64(Float64(Float64(x * x) * fma(x, Float64(x * fma(Float64(x * x), 0.047619047619047616, 0.2)), 0.6666666666666666)) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.5], N[(N[Abs[x], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.047619047619047616 + 0.2), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.5

   1. Initial program 99.8%

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

   4. Applied rewrites 99.2%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. inv-pow N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 99.8

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-*.f64 99.5

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

   12. Applied rewrites 99.5%

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

   if 0.5 < (fabs.f64 x)

   1. Initial program 99.9%

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

   4. Applied rewrites 99.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. inv-pow N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 99.8

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in x around inf

       \[\leadsto \left|x\right| \cdot \frac{\color{blue}{{x}^{6} \cdot \left(\frac{1}{21} + \left(\frac{\frac{2}{3}}{{x}^{4}} + \frac{1}{5} \cdot \frac{1}{{x}^{2}}\right)\right)}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   12. Applied rewrites 99.8%

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

Alternative 3: 99.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;\left|x\right| \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|x\right| \cdot \frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right)\right)\right)\right)}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.5)
   (*
    (fabs x)
    (/ (fma (* x x) (fma x (* x 0.2) 0.6666666666666666) 2.0) (sqrt PI)))
   (*
    (fabs x)
    (/
     (* x (* x (* x (* x (fma (* x x) 0.047619047619047616 0.2)))))
     (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.5) {
		tmp = fabs(x) * (fma((x * x), fma(x, (x * 0.2), 0.6666666666666666), 2.0) / sqrt(((double) M_PI)));
	} else {
		tmp = fabs(x) * ((x * (x * (x * (x * fma((x * x), 0.047619047619047616, 0.2))))) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.5)
		tmp = Float64(abs(x) * Float64(fma(Float64(x * x), fma(x, Float64(x * 0.2), 0.6666666666666666), 2.0) / sqrt(pi)));
	else
		tmp = Float64(abs(x) * Float64(Float64(x * Float64(x * Float64(x * Float64(x * fma(Float64(x * x), 0.047619047619047616, 0.2))))) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.5], N[(N[Abs[x], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x], $MachinePrecision] * N[(N[(x * N[(x * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.047619047619047616 + 0.2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.5

   1. Initial program 99.8%

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

   4. Applied rewrites 99.2%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. inv-pow N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 99.8

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-*.f64 99.5

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

   12. Applied rewrites 99.5%

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

   if 0.5 < (fabs.f64 x)

   1. Initial program 99.9%

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

   4. Applied rewrites 99.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. inv-pow N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 99.8

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in x around inf

       \[\leadsto \left|x\right| \cdot \frac{\color{blue}{{x}^{6} \cdot \left(\frac{1}{21} + \frac{1}{5} \cdot \frac{1}{{x}^{2}}\right)}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   12. Applied rewrites 99.4%

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

Alternative 4: 99.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;\left|x\right| \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|x\right| \cdot \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.5)
   (*
    (fabs x)
    (/ (fma (* x x) (fma x (* x 0.2) 0.6666666666666666) 2.0) (sqrt PI)))
   (*
    (fabs x)
    (/ (* (* x (* x x)) (* x (* (* x x) 0.047619047619047616))) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.5) {
		tmp = fabs(x) * (fma((x * x), fma(x, (x * 0.2), 0.6666666666666666), 2.0) / sqrt(((double) M_PI)));
	} else {
		tmp = fabs(x) * (((x * (x * x)) * (x * ((x * x) * 0.047619047619047616))) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.5)
		tmp = Float64(abs(x) * Float64(fma(Float64(x * x), fma(x, Float64(x * 0.2), 0.6666666666666666), 2.0) / sqrt(pi)));
	else
		tmp = Float64(abs(x) * Float64(Float64(Float64(x * Float64(x * x)) * Float64(x * Float64(Float64(x * x) * 0.047619047619047616))) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.5], N[(N[Abs[x], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x], $MachinePrecision] * N[(N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * 0.047619047619047616), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.5

   1. Initial program 99.8%

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

   4. Applied rewrites 99.2%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. inv-pow N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 99.8

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*l* N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-*.f64 99.5

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

   12. Applied rewrites 99.5%

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

   if 0.5 < (fabs.f64 x)

   1. Initial program 99.9%

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

   4. Applied rewrites 99.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. inv-pow N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 99.8

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. cube-prod N/A

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

       5. unpow2 N/A

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

       6. cube-unmult N/A

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

       7. pow-sqr N/A

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

       8. metadata-eval N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. metadata-eval N/A

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

       12. pow-sqr N/A

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

       13. associate-*l* N/A

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

       14. *-commutative N/A

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

       15. unpow2 N/A

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

       16. associate-*l* N/A

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

       17. associate-*r* N/A

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

       18. unpow2 N/A

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

       19. unpow3 N/A

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

       20. lower-*.f64 N/A

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

   12. Applied rewrites 98.5%

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

Alternative 5: 99.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.4%

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

   1. lift-fabs.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. inv-pow N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 99.8

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

9. Applied rewrites 99.8%

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

Alternative 6: 93.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;\left|x\right| \cdot \frac{\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\left(x \cdot x\right) \cdot 0.2\right) \cdot \left(x \cdot \left(x \cdot \left|x\right|\right)\right)}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.5)
   (* (fabs x) (/ (fma 0.6666666666666666 (* x x) 2.0) (sqrt PI)))
   (fabs (/ (* (* (* x x) 0.2) (* x (* x (fabs x)))) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.5) {
		tmp = fabs(x) * (fma(0.6666666666666666, (x * x), 2.0) / sqrt(((double) M_PI)));
	} else {
		tmp = fabs(((((x * x) * 0.2) * (x * (x * fabs(x)))) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.5)
		tmp = Float64(abs(x) * Float64(fma(0.6666666666666666, Float64(x * x), 2.0) / sqrt(pi)));
	else
		tmp = abs(Float64(Float64(Float64(Float64(x * x) * 0.2) * Float64(x * Float64(x * abs(x)))) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.5], N[(N[Abs[x], $MachinePrecision] * N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[(N[(N[(x * x), $MachinePrecision] * 0.2), $MachinePrecision] * N[(x * N[(x * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.5

   1. Initial program 99.8%

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

   4. Applied rewrites 99.2%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. inv-pow N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 99.4

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

   9. Applied rewrites 99.4%

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

   if 0.5 < (fabs.f64 x)

   1. Initial program 99.9%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. pow-sqr N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

   7. Applied rewrites 82.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 82.5%

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

Alternative 7: 93.2% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.4%

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

   1. lift-fabs.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. inv-pow N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 99.8

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

9. Applied rewrites 99.8%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. unpow2 N/A

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

    4. lower-*.f64 N/A

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

    5. +-commutative N/A

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

    6. *-commutative N/A

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

    7. unpow2 N/A

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

    8. associate-*l* N/A

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

    9. lower-fma.f64 N/A

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

    10. lower-*.f64 93.6

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

12. Applied rewrites 93.6%

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

Alternative 8: 88.8% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.4%

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

   1. lift-fabs.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. inv-pow N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 88.7

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

9. Applied rewrites 88.7%

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

Alternative 9: 67.0% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Abs[x], $MachinePrecision] * N[(2.0 / 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. Add Preprocessing

4. Applied rewrites 99.4%

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

   1. lift-fabs.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. inv-pow N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 66.5%

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

Reproduce

herbie shell --seed 2024228 
(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)))))))