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

Percentage Accurate: 99.9% → 99.9%

Time: 13.9s

Alternatives: 16

Speedup: 0.7×

• 29.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% 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: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right), 0.2 \cdot {\left(\left|x\right|\right)}^{5}\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt PI))
   (fma
    0.047619047619047616
    (pow (fabs x) 7.0)
    (fma
     (fabs x)
     (fma x (* x 0.6666666666666666) 2.0)
     (* 0.2 (pow (fabs x) 5.0)))))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * fma(0.047619047619047616, pow(fabs(x), 7.0), fma(fabs(x), fma(x, (x * 0.6666666666666666), 2.0), (0.2 * pow(fabs(x), 5.0))))));
}

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * fma(0.047619047619047616, (abs(x) ^ 7.0), fma(abs(x), fma(x, Float64(x * 0.6666666666666666), 2.0), Float64(0.2 * (abs(x) ^ 5.0))))))
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(0.047619047619047616 * N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] * N[(x * N[(x * 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision] + N[(0.2 * N[Power[N[Abs[x], $MachinePrecision], 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

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

3. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)\right)}\right| \]
4. Step-by-step derivation

   1. lower-fma.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-fabs.f64 N/A

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

   4. +-commutative N/A

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

   5. +-commutative N/A

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

   6. unpow3 N/A

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

   7. associate-*r* N/A

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

   8. distribute-rgt-out N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

5. Simplified 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right), 0.2 \cdot {\left(\left|x\right|\right)}^{5}\right)\right)}\right| \]
6. Add Preprocessing

Alternative 2: 99.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 egg-rr 99.9%

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

Alternative 3: 99.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|\left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \left(\mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left|\frac{x}{\sqrt{\pi}}\right|\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.4)
   (fabs
    (*
     (* (fabs x) (sqrt (/ 1.0 PI)))
     (fma x (* x (fma x (* x 0.2) 0.6666666666666666)) 2.0)))
   (fabs
    (*
     x
     (*
      (fma (* x x) 0.047619047619047616 0.2)
      (* (* x (* x x)) (fabs (/ x (sqrt PI)))))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.4) {
		tmp = fabs(((fabs(x) * sqrt((1.0 / ((double) M_PI)))) * fma(x, (x * fma(x, (x * 0.2), 0.6666666666666666)), 2.0)));
	} else {
		tmp = fabs((x * (fma((x * x), 0.047619047619047616, 0.2) * ((x * (x * x)) * fabs((x / sqrt(((double) M_PI))))))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.4)
		tmp = abs(Float64(Float64(abs(x) * sqrt(Float64(1.0 / pi))) * fma(x, Float64(x * fma(x, Float64(x * 0.2), 0.6666666666666666)), 2.0)));
	else
		tmp = abs(Float64(x * Float64(fma(Float64(x * x), 0.047619047619047616, 0.2) * Float64(Float64(x * Float64(x * x)) * abs(Float64(x / sqrt(pi)))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.4], N[Abs[N[(N[(N[Abs[x], $MachinePrecision] * N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 0.2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(N[(N[(x * x), $MachinePrecision] * 0.047619047619047616 + 0.2), $MachinePrecision] * N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[Abs[N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.40000000000000002

   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 egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 99.6%

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

   if 0.40000000000000002 < (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 egg-rr 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 99.0%

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

   9. Applied egg-rr 99.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|\left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \left(\mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left|\frac{x}{\sqrt{\pi}}\right|\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 5: 99.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|\left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left(\left|x\right| \cdot \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\sqrt{\pi}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.4)
   (fabs
    (*
     (* (fabs x) (sqrt (/ 1.0 PI)))
     (fma x (* x (fma x (* x 0.2) 0.6666666666666666)) 2.0)))
   (*
    0.047619047619047616
    (* (fabs x) (/ (* (* x x) (* x (* x (* x x)))) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.4) {
		tmp = fabs(((fabs(x) * sqrt((1.0 / ((double) M_PI)))) * fma(x, (x * fma(x, (x * 0.2), 0.6666666666666666)), 2.0)));
	} else {
		tmp = 0.047619047619047616 * (fabs(x) * (((x * x) * (x * (x * (x * x)))) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.4)
		tmp = abs(Float64(Float64(abs(x) * sqrt(Float64(1.0 / pi))) * fma(x, Float64(x * fma(x, Float64(x * 0.2), 0.6666666666666666)), 2.0)));
	else
		tmp = Float64(0.047619047619047616 * Float64(abs(x) * Float64(Float64(Float64(x * x) * Float64(x * Float64(x * Float64(x * x)))) / sqrt(pi))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.4], N[Abs[N[(N[(N[Abs[x], $MachinePrecision] * N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 0.2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(0.047619047619047616 * N[(N[Abs[x], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.40000000000000002

   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 egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 99.6%

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

   if 0.40000000000000002 < (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 egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. cube-prod N/A

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

      5. unpow2 N/A

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

      6. cube-unmult N/A

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

      7. pow-sqr N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 98.9

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

   7. Simplified 98.9%

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

   9. Applied egg-rr 98.9%

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

       1. lift-fabs.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-PI.f64 N/A

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

       8. lift-sqrt.f64 N/A

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

       9. associate-/l* N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

   11. Applied egg-rr 98.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|\left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left(\left|x\right| \cdot \frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{\sqrt{\pi}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|\left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left|\frac{x}{\sqrt{\pi}}\right| \cdot \left(0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.4)
   (fabs
    (*
     (* (fabs x) (sqrt (/ 1.0 PI)))
     (fma x (* x (fma x (* x 0.2) 0.6666666666666666)) 2.0)))
   (*
    (* x (* x (* x x)))
    (* (fabs (/ x (sqrt PI))) (* 0.047619047619047616 (* x x))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.4) {
		tmp = fabs(((fabs(x) * sqrt((1.0 / ((double) M_PI)))) * fma(x, (x * fma(x, (x * 0.2), 0.6666666666666666)), 2.0)));
	} else {
		tmp = (x * (x * (x * x))) * (fabs((x / sqrt(((double) M_PI)))) * (0.047619047619047616 * (x * x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.4)
		tmp = abs(Float64(Float64(abs(x) * sqrt(Float64(1.0 / pi))) * fma(x, Float64(x * fma(x, Float64(x * 0.2), 0.6666666666666666)), 2.0)));
	else
		tmp = Float64(Float64(x * Float64(x * Float64(x * x))) * Float64(abs(Float64(x / sqrt(pi))) * Float64(0.047619047619047616 * Float64(x * x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.4], N[Abs[N[(N[(N[Abs[x], $MachinePrecision] * N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 0.2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.047619047619047616 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.40000000000000002

   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 egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 99.6%

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

   if 0.40000000000000002 < (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 egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. cube-prod N/A

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

      5. unpow2 N/A

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

      6. cube-unmult N/A

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

      7. pow-sqr N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 98.9

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

   7. Simplified 98.9%

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

   9. Applied egg-rr 98.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|\left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left|\frac{x}{\sqrt{\pi}}\right| \cdot \left(0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

7. Simplified 99.9%

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

9. Applied egg-rr 99.3%

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

11. Applied egg-rr 99.9%

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

Alternative 8: 99.4% accurate, 2.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fabs
   (*
    (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((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(Float64(abs(x) * fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), 0.047619047619047616, 0.2)), 0.6666666666666666), 2.0))) / sqrt(pi))
end

Wolfram

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

Derivation

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 egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

7. Simplified 99.9%

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

9. Applied egg-rr 99.3%

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

Alternative 9: 99.4% accurate, 3.0× speedup

Math

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

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(Float64(x * 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[N[(x * 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]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

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 egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

7. Simplified 99.9%

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

9. Applied egg-rr 99.3%

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

    1. lift-fabs.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-fma.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-fma.f64 N/A

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

    7. lift-fma.f64 N/A

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

    8. *-commutative N/A

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

    9. fabs-mul N/A

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

    10. lift-fabs.f64 N/A

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

    11. fabs-fabs N/A

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

11. Applied egg-rr 99.3%

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

12. Final simplification 99.3%

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

Alternative 10: 93.5% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

7. Simplified 99.9%

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

8. Taylor expanded in x around 0

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

10. Simplified 94.0%

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

11. Final simplification 94.0%

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

Alternative 11: 93.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left|x\right| \cdot \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.2\right) \cdot \frac{x \cdot \left(x \cdot \left(x \cdot \left|x\right|\right)\right)}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.4)
   (fabs
    (* (/ 1.0 (sqrt PI)) (* (fabs x) (fma 0.6666666666666666 (* x x) 2.0))))
   (* (* x 0.2) (/ (* x (* x (* x (fabs x)))) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.4) {
		tmp = fabs(((1.0 / sqrt(((double) M_PI))) * (fabs(x) * fma(0.6666666666666666, (x * x), 2.0))));
	} else {
		tmp = (x * 0.2) * ((x * (x * (x * fabs(x)))) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.4)
		tmp = abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(abs(x) * fma(0.6666666666666666, Float64(x * x), 2.0))));
	else
		tmp = Float64(Float64(x * 0.2) * Float64(Float64(x * Float64(x * Float64(x * abs(x)))) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.4], N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(x * 0.2), $MachinePrecision] * N[(N[(x * 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) < 0.40000000000000002

   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 egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.5

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

   7. Simplified 99.5%

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

   if 0.40000000000000002 < (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 egg-rr 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 99.0%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 81.4%

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

   12. Applied egg-rr 81.4%

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

Alternative 12: 93.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Wolfram

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

Derivation

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 egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

7. Simplified 99.9%

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

9. Applied egg-rr 99.3%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. +-commutative N/A

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

    4. associate-*r* N/A

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

    5. distribute-rgt-out N/A

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

    6. associate-*l* N/A

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

    7. *-commutative N/A

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

    8. distribute-lft-out N/A

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

12. Simplified 93.5%

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

Alternative 13: 89.2% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-fabs.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 90.5

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

7. Simplified 90.5%

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

Alternative 14: 88.7% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-fabs.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 90.5

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

7. Simplified 90.5%

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

   1. lift-PI.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. associate-*l/ N/A

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

   10. fabs-div N/A

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

   11. *-lft-identity N/A

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

   12. rem-sqrt-square N/A

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

   13. sqrt-prod N/A

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

   14. rem-square-sqrt N/A

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

9. Applied egg-rr 90.0%

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

    1. lift-fabs.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-fma.f64 N/A

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

    4. lift-fabs.f64 N/A

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

    5. *-commutative N/A

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

    6. lift-PI.f64 N/A

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

    7. lift-sqrt.f64 N/A

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

    8. associate-/l* N/A

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

    9. un-div-inv N/A

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

    10. metadata-eval N/A

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

    11. lift-sqrt.f64 N/A

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

    12. sqrt-div N/A

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

    13. lift-/.f64 N/A

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

    14. lift-sqrt.f64 N/A

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

    15. lift-*.f64 N/A

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

    16. lower-*.f64 90.5

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

11. Applied egg-rr 90.0%

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

12. Final simplification 90.0%

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

Alternative 15: 88.7% accurate, 4.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/ (fabs (* x (fma x (* x 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(Float64(x * fma(x, Float64(x * 0.6666666666666666), 2.0))) / sqrt(pi))
end

Wolfram

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

Derivation

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 egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-fabs.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 90.5

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

7. Simplified 90.5%

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

   1. lift-PI.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. associate-*l/ N/A

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

   10. fabs-div N/A

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

   11. *-lft-identity N/A

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

   12. rem-sqrt-square N/A

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

   13. sqrt-prod N/A

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

   14. rem-square-sqrt N/A

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

9. Applied egg-rr 90.0%

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

    1. lift-fabs.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-fma.f64 N/A

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

    4. lift-fabs.f64 N/A

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

    5. lift-PI.f64 N/A

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

    6. lift-sqrt.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. lift-/.f64 90.0

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

11. Applied egg-rr 90.0%

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

Alternative 16: 67.3% 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 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-fabs.f64 70.3

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

7. Simplified 70.3%

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

   1. lift-PI.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. un-div-inv N/A

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

   13. div-fabs N/A

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

   14. rem-sqrt-square N/A

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

   15. sqrt-prod N/A

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

   16. rem-square-sqrt N/A

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

9. Applied egg-rr 70.3%

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

Reproduce

herbie shell --seed 2024211 
(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)))))))