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

Percentage Accurate: 99.8% → 99.8%

Time: 17.9s

Alternatives: 8

Speedup: 3.0×

• 28.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fabs x)
  (/
   (fma
    (* x (* x (* x x)))
    (fma 0.047619047619047616 (* x x) 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(0.047619047619047616, (x * x), 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(0.047619047619047616, Float64(x * x), 0.2), fma(Float64(x * 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[(0.047619047619047616 * N[(x * x), $MachinePrecision] + 0.2), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * 0.6666666666666666 + 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 99.9%

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

6. Applied rewrites 99.9%

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

Alternative 2: 99.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.20000000000000001

   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%

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 99.2%

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

   if 0.20000000000000001 < (fabs.f64 x)

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fabs.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-plus N/A

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

      9. pow-plus N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. pow-sqr N/A

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

      19. unpow2 N/A

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

      20. associate-*l* N/A

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

      21. unpow2 N/A

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

      22. cube-mult N/A

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

      23. lower-*.f64 N/A

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

      24. cube-mult N/A

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

      25. unpow2 N/A

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

      26. lower-*.f64 N/A

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

      27. unpow2 N/A

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

      28. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

Alternative 3: 99.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.20000000000000001

   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%

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      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. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.2

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

   9. Applied rewrites 99.2%

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

   if 0.20000000000000001 < (fabs.f64 x)

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fabs.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-plus N/A

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

      9. pow-plus N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. pow-sqr N/A

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

      19. unpow2 N/A

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

      20. associate-*l* N/A

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

      21. unpow2 N/A

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

      22. cube-mult N/A

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

      23. lower-*.f64 N/A

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

      24. cube-mult N/A

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

      25. unpow2 N/A

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

      26. lower-*.f64 N/A

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

      27. unpow2 N/A

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

      28. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

Alternative 4: 99.8% accurate, 3.0× speedup

Math

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

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(x, Float64(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[(x * N[(x * 0.047619047619047616), $MachinePrecision] + 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 99.9%

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

6. Applied rewrites 99.9%

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

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. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.9

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

9. Applied rewrites 99.9%

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

Alternative 5: 93.4% accurate, 3.6× speedup

Math

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

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(Float64(x * 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[(N[(x * x), $MachinePrecision] * 0.2 + 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 99.9%

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

6. Applied rewrites 99.9%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   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. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 93.3

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

9. Applied rewrites 93.3%

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

Alternative 6: 89.0% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Abs[x], $MachinePrecision] * N[(N[(x * N[(x * 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 99.9%

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

6. Applied rewrites 99.9%

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

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. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 87.9

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

9. Applied rewrites 87.9%

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

Alternative 7: 88.6% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.9%

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

6. Applied rewrites 99.9%

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

8. Applied rewrites 87.5%

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

Alternative 8: 67.9% 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 99.9%

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

6. Applied rewrites 99.9%

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

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 68.1%

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

Reproduce

herbie shell --seed 2024212 
(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)))))))