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

Percentage Accurate: 99.8% → 99.9%

Time: 17.0s

Alternatives: 13

Speedup: 2.7×

• 28.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Abs[x], $MachinePrecision] * N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.047619047619047616 + 0.2), $MachinePrecision]), $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 99.5%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0

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

9. Simplified 99.9%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.047619047619047616, 0.2\right), 0.6666666666666666\right), 2\right)} \]
10. Step-by-step derivation

    1. *-commutative N/A

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

    2. associate-*l* N/A

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

    3. *-lowering-*.f64 N/A

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

    4. fabs-lowering-fabs.f64 N/A

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

    5. *-lowering-*.f64 N/A

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

    6. sqrt-div N/A

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

    7. metadata-eval N/A

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

    8. /-lowering-/.f64 N/A

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

    9. sqrt-lowering-sqrt.f64 N/A

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

    10. PI-lowering-PI.f64 N/A

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

    11. accelerator-lowering-fma.f64 N/A

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

11. Applied egg-rr 99.9%

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

Alternative 2: 93.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := \left|x\right| \cdot \left|x \cdot t\_0\right|\\ \mathbf{if}\;\left(\left(\left|x\right| \cdot 2 + \frac{2}{3} \cdot \left|t\_0\right|\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot t\_1\right)\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left|x\right| \cdot \mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot \left(\mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right) \cdot \left(x \cdot \left|x\right|\right)\right)}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))) (t_1 (* (fabs x) (fabs (* x t_0)))))
   (if (<=
        (+
         (+
          (+ (* (fabs x) 2.0) (* (/ 2.0 3.0) (fabs t_0)))
          (* (/ 1.0 5.0) t_1))
         (* (/ 1.0 21.0) (* (fabs x) (* (fabs x) t_1))))
        4e-5)
     (fabs
      (* (/ 1.0 (sqrt PI)) (* (fabs x) (fma x (* x 0.6666666666666666) 2.0))))
     (fabs
      (/
       (* x (* (fma x (* x 0.2) 0.6666666666666666) (* x (fabs x))))
       (sqrt PI))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = fabs(x) * fabs((x * t_0));
	double tmp;
	if (((((fabs(x) * 2.0) + ((2.0 / 3.0) * fabs(t_0))) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * (fabs(x) * (fabs(x) * t_1)))) <= 4e-5) {
		tmp = fabs(((1.0 / sqrt(((double) M_PI))) * (fabs(x) * fma(x, (x * 0.6666666666666666), 2.0))));
	} else {
		tmp = fabs(((x * (fma(x, (x * 0.2), 0.6666666666666666) * (x * fabs(x)))) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(abs(x) * abs(Float64(x * t_0)))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(abs(x) * 2.0) + Float64(Float64(2.0 / 3.0) * abs(t_0))) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(abs(x) * Float64(abs(x) * t_1)))) <= 4e-5)
		tmp = abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(abs(x) * fma(x, Float64(x * 0.6666666666666666), 2.0))));
	else
		tmp = abs(Float64(Float64(x * Float64(fma(x, Float64(x * 0.2), 0.6666666666666666) * Float64(x * abs(x)))) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(x * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[Abs[x], $MachinePrecision] * 2.0), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e-5], N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[(x * N[(x * 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x * N[(N[(x * N[(x * 0.2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[(x * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) < 4.00000000000000033e-5

   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%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lowering-*.f64 N/A

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

      5. fabs-lowering-fabs.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 99.9

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

   7. Simplified 99.9%

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

   if 4.00000000000000033e-5 < (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (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%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

   7. Simplified 84.6%

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

   8. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r/ N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. associate-/l* N/A

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

      15. *-inverses N/A

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

      16. *-rgt-identity N/A

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

   10. Simplified 84.6%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left|x\right| \cdot 2 + \frac{2}{3} \cdot \left|x \cdot \left(x \cdot x\right)\right|\right) + \frac{1}{5} \cdot \left(\left|x\right| \cdot \left|x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right|\right)\right) + \frac{1}{21} \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left|x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right|\right)\right)\right) \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left|x\right| \cdot \mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot \left(\mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right) \cdot \left(x \cdot \left|x\right|\right)\right)}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left|x\right| \cdot \mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\left|\frac{x}{\sqrt{\pi}}\right| \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \mathsf{fma}\left(0.047619047619047616, x \cdot x, 0.2\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 2e-5)
   (fabs
    (* (/ 1.0 (sqrt PI)) (* (fabs x) (fma x (* x 0.6666666666666666) 2.0))))
   (fabs
    (*
     (* (fabs (/ x (sqrt PI))) (* x (* x (* x x))))
     (fma 0.047619047619047616 (* x x) 0.2)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 2e-5) {
		tmp = fabs(((1.0 / sqrt(((double) M_PI))) * (fabs(x) * fma(x, (x * 0.6666666666666666), 2.0))));
	} else {
		tmp = fabs(((fabs((x / sqrt(((double) M_PI)))) * (x * (x * (x * x)))) * fma(0.047619047619047616, (x * x), 0.2)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 2e-5)
		tmp = abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(abs(x) * fma(x, Float64(x * 0.6666666666666666), 2.0))));
	else
		tmp = abs(Float64(Float64(abs(Float64(x / sqrt(pi))) * Float64(x * Float64(x * Float64(x * x)))) * fma(0.047619047619047616, Float64(x * x), 0.2)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 2e-5], N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[(x * N[(x * 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Abs[N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.047619047619047616 * N[(x * x), $MachinePrecision] + 0.2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2.00000000000000016e-5

   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%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lowering-*.f64 N/A

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

      5. fabs-lowering-fabs.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 99.9

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

   7. Simplified 99.9%

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

   if 2.00000000000000016e-5 < (fabs.f64 x)

   1. Initial program 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 99.3%

      \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot \mathsf{fma}\left(0.047619047619047616, x \cdot x, 0.2\right)}\right| \]
   8. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. un-div-inv N/A

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

      10. add-sqr-sqrt N/A

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

      11. rem-sqrt-square N/A

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

      12. div-fabs N/A

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

      13. fabs-lowering-fabs.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. sqrt-lowering-sqrt.f64 N/A

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

      16. PI-lowering-PI.f64 99.3

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

   9. Applied egg-rr 99.3%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left|x\right| \cdot \mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\left|\frac{x}{\sqrt{\pi}}\right| \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \mathsf{fma}\left(0.047619047619047616, x \cdot x, 0.2\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(Float64(abs(x) * fma(x, Float64(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[(N[Abs[x], $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.047619047619047616 + 0.2), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision]), $MachinePrecision] + 2.0), $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 99.5%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0

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

9. Simplified 99.9%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.047619047619047616, 0.2\right), 0.6666666666666666\right), 2\right)} \]
10. Step-by-step derivation

    1. associate-*l* N/A

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

    2. *-commutative N/A

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

    3. sqrt-div N/A

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

    4. metadata-eval N/A

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

    5. un-div-inv N/A

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

    6. /-lowering-/.f64 N/A

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

11. Applied egg-rr 99.5%

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

Alternative 5: 99.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(x * N[(x * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.047619047619047616 + 0.2), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[Abs[N[(x / N[Sqrt[Pi], $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 99.5%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0

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

9. Simplified 99.9%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.047619047619047616, 0.2\right), 0.6666666666666666\right), 2\right)} \]
10. Step-by-step derivation

    1. *-commutative N/A

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

    2. *-lowering-*.f64 N/A

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

    3. accelerator-lowering-fma.f64 N/A

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

    4. *-lowering-*.f64 N/A

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

    5. associate-*l* N/A

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

    6. accelerator-lowering-fma.f64 N/A

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

    7. *-lowering-*.f64 N/A

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

    8. associate-*r* N/A

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

    9. accelerator-lowering-fma.f64 N/A

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

    10. *-lowering-*.f64 N/A

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

    11. *-commutative N/A

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

    12. sqrt-div N/A

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

11. Applied egg-rr 99.5%

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

Alternative 6: 66.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left|x\right| \cdot \mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.2 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{\sqrt{\pi}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 2e-5)
   (fabs
    (* (/ 1.0 (sqrt PI)) (* (fabs x) (fma x (* x 0.6666666666666666) 2.0))))
   (* x (* x (/ (* 0.2 (* x (* x x))) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 2e-5) {
		tmp = fabs(((1.0 / sqrt(((double) M_PI))) * (fabs(x) * fma(x, (x * 0.6666666666666666), 2.0))));
	} else {
		tmp = x * (x * ((0.2 * (x * (x * x))) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 2e-5)
		tmp = abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(abs(x) * fma(x, Float64(x * 0.6666666666666666), 2.0))));
	else
		tmp = Float64(x * Float64(x * Float64(Float64(0.2 * Float64(x * Float64(x * x))) / sqrt(pi))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 2e-5], N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[(x * N[(x * 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(x * N[(x * N[(N[(0.2 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2.00000000000000016e-5

   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%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lowering-*.f64 N/A

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

      5. fabs-lowering-fabs.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 99.9

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

   7. Simplified 99.9%

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

   if 2.00000000000000016e-5 < (fabs.f64 x)

   1. Initial program 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

   7. Simplified 84.6%

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

   8. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. *-lowering-*.f64 N/A

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

      18. fabs-lowering-fabs.f64 84.6

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

   10. Simplified 84.6%

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

       1. fabs-div N/A

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

       2. rem-sqrt-square N/A

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

       3. add-sqr-sqrt N/A

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

   12. Applied egg-rr 0.1%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left|x\right| \cdot \mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.2 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{\sqrt{\pi}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.4% accurate, 3.5× speedup

Math

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

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

Wolfram

code[x_] := N[Abs[N[(N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / 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 99.5%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. distribute-lft-in N/A

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

   9. +-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

7. Simplified 93.0%

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

Alternative 8: 92.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[Abs[N[(N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.2), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / 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 99.5%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. distribute-lft-in N/A

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

   9. +-commutative N/A

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

   10. associate-*r* N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

7. Simplified 93.0%

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

8. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 92.8

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

10. Simplified 92.8%

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

Alternative 9: 89.8% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(abs(x) * fma(x, Float64(x * 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[(x * N[(x * 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%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lowering-*.f64 N/A

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

   5. fabs-lowering-fabs.f64 N/A

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

   6. +-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. *-lowering-*.f64 88.9

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

7. Simplified 88.9%

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

8. Final simplification 88.9%

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

Alternative 10: 66.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left|x\right| \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.6666666666666666 \cdot \frac{x \cdot \left(x \cdot x\right)}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 2e-5)
   (* (fabs x) (/ 2.0 (sqrt PI)))
   (* 0.6666666666666666 (/ (* x (* x x)) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 2e-5) {
		tmp = fabs(x) * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = 0.6666666666666666 * ((x * (x * x)) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 2e-5) {
		tmp = Math.abs(x) * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = 0.6666666666666666 * ((x * (x * x)) / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 2e-5:
		tmp = math.fabs(x) * (2.0 / math.sqrt(math.pi))
	else:
		tmp = 0.6666666666666666 * ((x * (x * x)) / math.sqrt(math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 2e-5)
		tmp = Float64(abs(x) * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(0.6666666666666666 * Float64(Float64(x * Float64(x * x)) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 2e-5)
		tmp = abs(x) * (2.0 / sqrt(pi));
	else
		tmp = 0.6666666666666666 * ((x * (x * x)) / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 2e-5], N[(N[Abs[x], $MachinePrecision] * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.6666666666666666 * N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2.00000000000000016e-5

   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 99.2%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. fabs-lowering-fabs.f64 98.8

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

   7. Simplified 98.8%

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

      1. fabs-div N/A

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

      2. rem-sqrt-square N/A

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

      3. add-sqr-sqrt N/A

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

      4. fabs-mul N/A

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

      5. fabs-fabs N/A

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

      6. metadata-eval N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. PI-lowering-PI.f64 N/A

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

      13. fabs-lowering-fabs.f64 99.5

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

   9. Applied egg-rr 99.5%

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

   if 2.00000000000000016e-5 < (fabs.f64 x)

   1. Initial program 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lowering-*.f64 N/A

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

      5. fabs-lowering-fabs.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 73.9

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

   7. Simplified 73.9%

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

   8. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. fabs-lowering-fabs.f64 73.9

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

   10. Simplified 73.9%

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

       1. fabs-div N/A

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

       2. *-commutative N/A

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

       3. fabs-mul N/A

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

       4. fabs-mul N/A

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

       5. metadata-eval N/A

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

       6. fabs-fabs N/A

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

       7. fabs-sqr N/A

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

       8. associate-*l* N/A

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

       9. fabs-sqr N/A

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

       10. fabs-mul N/A

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

       11. cube-unmult N/A

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

       12. sqr-pow N/A

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

       13. fabs-sqr N/A

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

       14. sqr-pow N/A

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

       15. cube-unmult N/A

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

       16. rem-sqrt-square N/A

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

       17. add-sqr-sqrt N/A

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

       18. *-lft-identity N/A

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

       19. times-frac N/A

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

   12. Applied egg-rr 0.1%

       \[\leadsto \color{blue}{0.6666666666666666 \cdot \frac{x \cdot \left(x \cdot x\right)}{\sqrt{\pi}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left|x\right| \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.6666666666666666 \cdot \frac{x \cdot \left(x \cdot x\right)}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.8% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(N[(x * N[(x * 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $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 99.5%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lowering-*.f64 N/A

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

   5. fabs-lowering-fabs.f64 N/A

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

   6. +-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. *-lowering-*.f64 88.5

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

7. Simplified 88.5%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. fabs-mul N/A

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

   4. fabs-fabs N/A

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

   5. *-lowering-*.f64 N/A

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

   6. fabs-lowering-fabs.f64 N/A

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

   7. associate-*r* N/A

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

   8. /-lowering-/.f64 N/A

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

   9. associate-*r* N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. sqrt-lowering-sqrt.f64 N/A

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

   13. PI-lowering-PI.f64 N/A

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

   14. fabs-lowering-fabs.f64 88.9

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

9. Applied egg-rr 88.9%

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

10. Final simplification 88.9%

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

Alternative 12: 89.3% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[Abs[N[(N[(N[Abs[x], $MachinePrecision] * N[(x * N[(x * 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / 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 99.5%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lowering-*.f64 N/A

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

   5. fabs-lowering-fabs.f64 N/A

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

   6. +-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. *-lowering-*.f64 88.5

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

7. Simplified 88.5%

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

Alternative 13: 68.3% 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 99.5%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

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

   3. fabs-lowering-fabs.f64 59.7

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

7. Simplified 59.7%

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

   1. fabs-div N/A

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

   2. rem-sqrt-square N/A

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

   3. add-sqr-sqrt N/A

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

   4. fabs-mul N/A

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

   5. fabs-fabs N/A

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

   6. metadata-eval N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. *-lowering-*.f64 N/A

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

   10. /-lowering-/.f64 N/A

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

   11. sqrt-lowering-sqrt.f64 N/A

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

   12. PI-lowering-PI.f64 N/A

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

   13. fabs-lowering-fabs.f64 60.1

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

9. Applied egg-rr 60.1%

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

10. Final simplification 60.1%

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

Reproduce

herbie shell --seed 2024204 
(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)))))))