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

Percentage Accurate: 99.8% → 99.8%

Time: 14.3s

Alternatives: 9

Speedup: 2.7×

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

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

   3. unpow2 N/A

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

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

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

   5. +-commutative N/A

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

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

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

   7. unpow2 N/A

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

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

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

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

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

   14. *-lowering-*.f64 99.9

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

7. Simplified 99.9%

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

9. Applied egg-rr 99.9%

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

Alternative 2: 99.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left|x\right| \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.1)
   (*
    (/ 1.0 (sqrt PI))
    (fabs (* x (fma (* x x) (fma x (* x 0.2) 0.6666666666666666) 2.0))))
   (/
    (fabs
     (* (* (fabs x) (* x (* x x))) (* x (* (* x x) 0.047619047619047616))))
    (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.1) {
		tmp = (1.0 / sqrt(((double) M_PI))) * fabs((x * fma((x * x), fma(x, (x * 0.2), 0.6666666666666666), 2.0)));
	} else {
		tmp = fabs(((fabs(x) * (x * (x * x))) * (x * ((x * x) * 0.047619047619047616)))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.1)
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(x * fma(Float64(x * x), fma(x, Float64(x * 0.2), 0.6666666666666666), 2.0))));
	else
		tmp = Float64(abs(Float64(Float64(abs(x) * Float64(x * Float64(x * x))) * Float64(x * Float64(Float64(x * x) * 0.047619047619047616)))) / sqrt(pi));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.1], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(N[(N[Abs[x], $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * 0.047619047619047616), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.10000000000000001

   1. Initial program 99.9%

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

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 99.5

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

   7. Simplified 99.5%

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

      1. fabs-mul N/A

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

      2. fabs-div N/A

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

      3. metadata-eval N/A

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

      4. rem-sqrt-square N/A

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

      5. add-sqr-sqrt N/A

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

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

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

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

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

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

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

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

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

      10. fabs-mul N/A

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

      11. fabs-fabs N/A

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

      12. mul-fabs N/A

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

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

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

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

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

   9. Applied egg-rr 99.5%

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

   if 0.10000000000000001 < (fabs.f64 x)

   1. Initial program 99.9%

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

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. pow-sqr N/A

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

      12. associate-*r* N/A

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

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

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

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

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

      20. *-commutative N/A

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

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

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

      22. unpow2 N/A

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

      23. *-lowering-*.f64 99.3

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

   9. Simplified 99.3%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. pow3 N/A

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

       4. associate-*r* N/A

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

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

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

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

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

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

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

       8. cube-unmult N/A

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

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

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

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

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

       11. associate-*r* N/A

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

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

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

       13. associate-*r* N/A

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

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

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

       15. *-lowering-*.f64 99.3

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

   11. Applied egg-rr 99.3%

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

Alternative 3: 99.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(0.047619047619047616 \cdot \left|x\right|\right)}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.1)
   (*
    (/ 1.0 (sqrt PI))
    (fabs (* x (fma (* x x) (fma x (* x 0.2) 0.6666666666666666) 2.0))))
   (/
    (* (* (* x x) (* x (* x (* x x)))) (* 0.047619047619047616 (fabs x)))
    (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.1) {
		tmp = (1.0 / sqrt(((double) M_PI))) * fabs((x * fma((x * x), fma(x, (x * 0.2), 0.6666666666666666), 2.0)));
	} else {
		tmp = (((x * x) * (x * (x * (x * x)))) * (0.047619047619047616 * fabs(x))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.1)
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(x * fma(Float64(x * x), fma(x, Float64(x * 0.2), 0.6666666666666666), 2.0))));
	else
		tmp = Float64(Float64(Float64(Float64(x * x) * Float64(x * Float64(x * Float64(x * x)))) * Float64(0.047619047619047616 * abs(x))) / sqrt(pi));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.1], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.047619047619047616 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.10000000000000001

   1. Initial program 99.9%

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

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 99.5

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

   7. Simplified 99.5%

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

      1. fabs-mul N/A

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

      2. fabs-div N/A

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

      3. metadata-eval N/A

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

      4. rem-sqrt-square N/A

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

      5. add-sqr-sqrt N/A

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

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

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

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

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

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

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

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

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

      10. fabs-mul N/A

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

      11. fabs-fabs N/A

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

      12. mul-fabs N/A

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

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

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

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

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

   9. Applied egg-rr 99.5%

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

   if 0.10000000000000001 < (fabs.f64 x)

   1. Initial program 99.9%

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

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. pow-sqr N/A

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

      12. associate-*r* N/A

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

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

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

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

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

      20. *-commutative N/A

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

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

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

      22. unpow2 N/A

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

      23. *-lowering-*.f64 99.3

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

   9. Simplified 99.3%

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

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. fabs-mul N/A

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

       3. metadata-eval N/A

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

       4. fabs-mul N/A

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

       5. fabs-fabs N/A

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

       6. associate-*l* N/A

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

       7. metadata-eval N/A

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

       8. pow-sqr N/A

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

       9. fabs-sqr N/A

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

       10. pow-sqr N/A

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

       11. metadata-eval N/A

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

       12. *-commutative N/A

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

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

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

   12. Simplified 99.3%

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

Alternative 4: 99.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot \left|x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.1)
   (*
    (/ 1.0 (sqrt PI))
    (fabs (* x (fma (* x x) (fma x (* x 0.2) 0.6666666666666666) 2.0))))
   (/
    (* (* x x) (fabs (* x (* x (* x (* (* x x) 0.047619047619047616))))))
    (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.1) {
		tmp = (1.0 / sqrt(((double) M_PI))) * fabs((x * fma((x * x), fma(x, (x * 0.2), 0.6666666666666666), 2.0)));
	} else {
		tmp = ((x * x) * fabs((x * (x * (x * ((x * x) * 0.047619047619047616)))))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.1)
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(x * fma(Float64(x * x), fma(x, Float64(x * 0.2), 0.6666666666666666), 2.0))));
	else
		tmp = Float64(Float64(Float64(x * x) * abs(Float64(x * Float64(x * Float64(x * Float64(Float64(x * x) * 0.047619047619047616)))))) / sqrt(pi));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.1], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * N[Abs[N[(x * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.047619047619047616), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.10000000000000001

   1. Initial program 99.9%

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

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 99.5

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

   7. Simplified 99.5%

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

      1. fabs-mul N/A

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

      2. fabs-div N/A

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

      3. metadata-eval N/A

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

      4. rem-sqrt-square N/A

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

      5. add-sqr-sqrt N/A

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

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

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

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

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

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

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

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

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

      10. fabs-mul N/A

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

      11. fabs-fabs N/A

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

      12. mul-fabs N/A

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

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

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

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

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

   9. Applied egg-rr 99.5%

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

   if 0.10000000000000001 < (fabs.f64 x)

   1. Initial program 99.9%

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

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. pow-sqr N/A

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

      12. associate-*r* N/A

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

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

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

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

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

      20. *-commutative N/A

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

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

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

      22. unpow2 N/A

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

      23. *-lowering-*.f64 99.3

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

   9. Simplified 99.3%

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

       1. associate-*l* N/A

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

       2. fabs-mul N/A

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

       3. fabs-sqr N/A

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

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

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

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

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

       6. *-commutative N/A

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

       7. fabs-mul N/A

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

       8. fabs-fabs N/A

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

       9. fabs-mul N/A

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

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

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

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

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

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

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

       13. associate-*r* N/A

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

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

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

       15. associate-*r* N/A

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

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

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

       17. *-lowering-*.f64 99.2

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

   11. Applied egg-rr 99.2%

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

Alternative 5: 99.4% accurate, 3.0× speedup

Math

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

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(Float64(x * x), fma(x, Float64(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[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.047619047619047616), $MachinePrecision] + 0.2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

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

6. Applied egg-rr 99.4%

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

   1. fabs-mul N/A

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

   2. associate-*l* N/A

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

   3. associate-*r* N/A

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

   4. associate-+l+ N/A

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

   5. distribute-lft-in N/A

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

   6. fabs-fabs N/A

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

   7. mul-fabs N/A

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

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

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

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

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

8. Applied egg-rr 99.4%

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

9. Final simplification 99.4%

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

Alternative 6: 93.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

   3. unpow2 N/A

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

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

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 94.3

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

7. Simplified 94.3%

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

   1. fabs-mul N/A

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

   2. fabs-div N/A

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

   3. metadata-eval N/A

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

   4. rem-sqrt-square N/A

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

   5. add-sqr-sqrt N/A

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

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

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

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

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

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

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

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

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

   10. fabs-mul N/A

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

   11. fabs-fabs N/A

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

   12. mul-fabs N/A

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

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

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

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

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

9. Applied egg-rr 94.3%

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

Alternative 7: 93.2% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

   3. unpow2 N/A

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

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

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 94.3

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

7. Simplified 94.3%

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

   1. associate-*l/ N/A

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

   2. fabs-div N/A

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

   3. *-lft-identity N/A

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

   4. rem-sqrt-square N/A

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

   5. add-sqr-sqrt N/A

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

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

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

9. Applied egg-rr 93.9%

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

Alternative 8: 89.5% 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 99.9%

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

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

5. Taylor expanded in x around 0

   \[\leadsto \left|\color{blue}{\frac{2}{3} \cdot \left(\left({x}^{2} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + 2 \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)}\right| \]
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. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-in N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-in N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

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

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

7. Simplified 91.0%

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

   1. sqrt-div N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. fabs-mul N/A

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

   5. fabs-fabs N/A

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

   6. mul-fabs N/A

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

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

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

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

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

9. Applied egg-rr 91.0%

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

10. Final simplification 91.0%

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

Alternative 9: 68.2% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

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

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 63.4%

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

   1. associate-*l/ N/A

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

   2. fabs-div N/A

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

   3. *-lft-identity N/A

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

   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. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

   13. PI-lowering-PI.f64 63.0

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

9. Applied egg-rr 63.0%

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

    1. associate-/l* N/A

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

    2. *-commutative N/A

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

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

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

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

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

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

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

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

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

    7. fabs-lowering-fabs.f64 63.4

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

11. Applied egg-rr 63.4%

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

12. Final simplification 63.4%

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

Reproduce

herbie shell --seed 2024198 
(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)))))))