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

Percentage Accurate: 99.8% → 99.9%

Time: 15.4s

Alternatives: 11

Speedup: 2.3×

• 29.2% 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, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l+ N/A

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

   2. +-commutative N/A

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

7. Simplified 99.8%

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

   1. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. associate-+l+ N/A

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

   5. associate-*r* N/A

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

   6. associate-*r* N/A

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

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

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

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

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

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

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

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

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

   11. associate-*r* N/A

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

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

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

   13. *-lowering-*.f64 99.9

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

9. Applied egg-rr 99.9%

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

Alternative 2: 98.7% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 1e-5)
   (fabs
    (* (fabs x) (* (sqrt (/ 1.0 PI)) (fma 0.6666666666666666 (* x x) 2.0))))
   (*
    (fabs (/ x (sqrt PI)))
    (fabs (* (* x x) (* x (* (* x (* x x)) 0.047619047619047616)))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 1e-5) {
		tmp = fabs((fabs(x) * (sqrt((1.0 / ((double) M_PI))) * fma(0.6666666666666666, (x * x), 2.0))));
	} else {
		tmp = fabs((x / sqrt(((double) M_PI)))) * fabs(((x * x) * (x * ((x * (x * x)) * 0.047619047619047616))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 1e-5)
		tmp = abs(Float64(abs(x) * Float64(sqrt(Float64(1.0 / pi)) * fma(0.6666666666666666, Float64(x * x), 2.0))));
	else
		tmp = Float64(abs(Float64(x / sqrt(pi))) * abs(Float64(Float64(x * x) * Float64(x * Float64(Float64(x * Float64(x * x)) * 0.047619047619047616)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 1e-5], N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.047619047619047616), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000008e-5

   1. Initial program 99.8%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

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

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

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

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

   7. Simplified 99.8%

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

   if 1.00000000000000008e-5 < (fabs.f64 x)

   1. Initial program 99.8%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. cube-prod N/A

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

      8. unpow2 N/A

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

      9. unpow3 N/A

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

      10. pow-sqr N/A

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

      11. metadata-eval N/A

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

      12. associate-*l* N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. metadata-eval N/A

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

      20. pow-sqr N/A

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

      21. associate-*l* N/A

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

      22. *-commutative N/A

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

      23. unpow2 N/A

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

      24. associate-*l* N/A

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

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

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

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

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

      27. *-commutative N/A

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

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

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

      29. unpow2 N/A

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

      30. *-lowering-*.f64 98.2

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

   7. Simplified 98.2%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. sqrt-div N/A

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

      5. fabs-mul N/A

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

   9. Applied egg-rr 98.2%

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

4. Final simplification 99.3%

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

Alternative 3: 99.8% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (* (fabs x) (sqrt (/ 1.0 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(((fabs(x) * sqrt((1.0 / ((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 abs(Float64(Float64(abs(x) * sqrt(Float64(1.0 / pi))) * fma(x, Float64(x * fma(Float64(x * x), fma(Float64(x * x), 0.047619047619047616, 0.2), 0.6666666666666666)), 2.0)))
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l+ N/A

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

   2. +-commutative N/A

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

7. Simplified 99.8%

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

Alternative 4: 98.7% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 1e-5)
   (fabs
    (* (fabs x) (* (sqrt (/ 1.0 PI)) (fma 0.6666666666666666 (* x x) 2.0))))
   (*
    (fabs (/ x (sqrt PI)))
    (* 0.047619047619047616 (* (* x x) (* x (* x (* x x))))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 1e-5) {
		tmp = fabs((fabs(x) * (sqrt((1.0 / ((double) M_PI))) * fma(0.6666666666666666, (x * x), 2.0))));
	} else {
		tmp = fabs((x / sqrt(((double) M_PI)))) * (0.047619047619047616 * ((x * x) * (x * (x * (x * x)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 1e-5)
		tmp = abs(Float64(abs(x) * Float64(sqrt(Float64(1.0 / pi)) * fma(0.6666666666666666, Float64(x * x), 2.0))));
	else
		tmp = Float64(abs(Float64(x / sqrt(pi))) * Float64(0.047619047619047616 * Float64(Float64(x * x) * Float64(x * Float64(x * Float64(x * x))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 1e-5], N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.047619047619047616 * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000008e-5

   1. Initial program 99.8%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

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

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

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

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

   7. Simplified 99.8%

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

   if 1.00000000000000008e-5 < (fabs.f64 x)

   1. Initial program 99.8%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. cube-prod N/A

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

      8. unpow2 N/A

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

      9. unpow3 N/A

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

      10. pow-sqr N/A

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

      11. metadata-eval N/A

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

      12. associate-*l* N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. metadata-eval N/A

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

      20. pow-sqr N/A

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

      21. associate-*l* N/A

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

      22. *-commutative N/A

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

      23. unpow2 N/A

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

      24. associate-*l* N/A

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

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

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

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

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

      27. *-commutative N/A

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

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

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

      29. unpow2 N/A

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

      30. *-lowering-*.f64 98.2

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

   7. Simplified 98.2%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. sqrt-div N/A

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

      5. fabs-mul N/A

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

   9. Applied egg-rr 98.2%

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

       3. fabs-mul N/A

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

       4. cube-unmult N/A

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

       5. sqr-pow N/A

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

       6. fabs-sqr N/A

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

       7. sqr-pow N/A

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

       8. cube-unmult N/A

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

       9. *-commutative N/A

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

       10. fabs-mul N/A

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

       11. cube-unmult N/A

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

       12. sqr-pow N/A

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

       13. fabs-sqr N/A

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

       14. sqr-pow N/A

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

       15. cube-unmult N/A

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

       16. metadata-eval N/A

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

       17. associate-*r* N/A

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

   11. Applied egg-rr 98.1%

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

4. Final simplification 99.3%

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

Alternative 5: 65.2% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 1e-5)
   (fabs
    (* (fabs x) (* (sqrt (/ 1.0 PI)) (fma 0.6666666666666666 (* x x) 2.0))))
   (*
    (fabs (* (* x (* x x)) (/ x (sqrt PI))))
    (* x (* (* x x) 0.047619047619047616)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 1e-5) {
		tmp = fabs((fabs(x) * (sqrt((1.0 / ((double) M_PI))) * fma(0.6666666666666666, (x * x), 2.0))));
	} else {
		tmp = fabs(((x * (x * x)) * (x / sqrt(((double) M_PI))))) * (x * ((x * x) * 0.047619047619047616));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 1e-5)
		tmp = abs(Float64(abs(x) * Float64(sqrt(Float64(1.0 / pi)) * fma(0.6666666666666666, Float64(x * x), 2.0))));
	else
		tmp = Float64(abs(Float64(Float64(x * Float64(x * x)) * Float64(x / sqrt(pi)))) * Float64(x * Float64(Float64(x * x) * 0.047619047619047616)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 1e-5], N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * 0.047619047619047616), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000008e-5

   1. Initial program 99.8%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

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

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

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

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

   7. Simplified 99.8%

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

   if 1.00000000000000008e-5 < (fabs.f64 x)

   1. Initial program 99.8%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. cube-prod N/A

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

      8. unpow2 N/A

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

      9. unpow3 N/A

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

      10. pow-sqr N/A

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

      11. metadata-eval N/A

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

      12. associate-*l* N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

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

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

      19. metadata-eval N/A

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

      20. pow-sqr N/A

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

      21. associate-*l* N/A

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

      22. *-commutative N/A

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

      23. unpow2 N/A

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

      24. associate-*l* N/A

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

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

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

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

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

      27. *-commutative N/A

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

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

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

      29. unpow2 N/A

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

      30. *-lowering-*.f64 98.2

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

   7. Simplified 98.2%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. sqrt-div N/A

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

      5. fabs-mul N/A

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

   9. Applied egg-rr 98.2%

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

       1. mul-fabs N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

       4. associate-*r* N/A

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

       5. fabs-mul N/A

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

       6. fabs-mul N/A

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

       7. metadata-eval N/A

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

       8. cube-unmult N/A

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

       9. sqr-pow N/A

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

       10. fabs-sqr N/A

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

       11. sqr-pow N/A

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

       12. cube-unmult N/A

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

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

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

   11. Applied egg-rr 0.1%

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

4. Final simplification 65.6%

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

Alternative 6: 99.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(abs(Float64(x * fma(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[Abs[N[(x * 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] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. associate-+l+ N/A

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

   2. +-commutative N/A

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

4. Applied egg-rr 99.8%

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

   1. distribute-rgt-in N/A

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

   2. associate-*l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. distribute-rgt-in N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

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

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

6. Applied egg-rr 99.8%

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

   1. associate-*l/ N/A

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

   2. fabs-div N/A

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

   3. *-lft-identity N/A

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

   4. rem-sqrt-square N/A

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

   5. add-sqr-sqrt N/A

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

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

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

8. Applied egg-rr 99.4%

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

Alternative 7: 93.9% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l+ N/A

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

   2. +-commutative N/A

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

7. Simplified 93.4%

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

Alternative 8: 93.9% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l+ N/A

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

   2. +-commutative N/A

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

7. Simplified 93.4%

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

   1. *-commutative N/A

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

   2. fabs-mul N/A

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

   3. fabs-fabs N/A

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

   4. mul-fabs N/A

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

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

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

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

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

9. Applied egg-rr 93.4%

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

10. Final simplification 93.4%

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

Alternative 9: 89.6% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l+ N/A

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

   2. +-commutative N/A

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-in N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

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

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

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

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

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

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

7. Simplified 89.3%

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

Alternative 10: 89.6% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l+ N/A

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

   2. +-commutative N/A

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

7. Simplified 93.4%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-out N/A

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

   4. *-commutative N/A

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

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

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

   6. +-commutative N/A

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

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

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

   8. unpow2 N/A

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

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

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

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

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

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

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

   12. PI-lowering-PI.f64 89.3

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

10. Simplified 89.3%

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

    1. *-commutative N/A

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

    2. fabs-mul N/A

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

    3. fabs-fabs N/A

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

    4. mul-fabs N/A

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

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

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

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

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

12. Applied egg-rr 89.3%

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

13. Final simplification 89.3%

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

Alternative 11: 67.4% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l+ N/A

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

   2. +-commutative N/A

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in x around 0

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

7. Simplified 67.3%

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

   1. *-commutative N/A

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

   2. un-div-inv N/A

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

   3. div-fabs N/A

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

   4. fabs-mul N/A

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

   5. fabs-fabs N/A

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

   6. metadata-eval N/A

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

   7. rem-sqrt-square N/A

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

   8. add-sqr-sqrt N/A

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

   9. un-div-inv N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

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

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

   14. associate-*l/ N/A

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

   15. metadata-eval N/A

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

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

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

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

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

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

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

   19. fabs-lowering-fabs.f64 67.3

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

9. Applied egg-rr 67.3%

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

10. Final simplification 67.3%

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

Reproduce

herbie shell --seed 2024196 
(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)))))))