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

Percentage Accurate: 99.8% → 99.8%

Time: 19.1s

Alternatives: 15

Speedup: N/A×

• 28.7% 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, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right| \cdot \left(t\_0 \cdot t\_0\right), 0.047619047619047616, \mathsf{fma}\left(\left|x\right|, 0.2 \cdot \left(x \cdot t\_0\right), \left|x\right| \cdot \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (fma
      (* (fabs x) (* t_0 t_0))
      0.047619047619047616
      (fma
       (fabs x)
       (* 0.2 (* x t_0))
       (* (fabs x) (fma 0.6666666666666666 (* x x) 2.0))))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * fma((fabs(x) * (t_0 * t_0)), 0.047619047619047616, fma(fabs(x), (0.2 * (x * t_0)), (fabs(x) * fma(0.6666666666666666, (x * x), 2.0))))));
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * fma(Float64(abs(x) * Float64(t_0 * t_0)), 0.047619047619047616, fma(abs(x), Float64(0.2 * Float64(x * t_0)), Float64(abs(x) * fma(0.6666666666666666, Float64(x * x), 2.0))))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[x], $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.047619047619047616 + N[(N[Abs[x], $MachinePrecision] * N[(0.2 * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] * N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

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

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

4. Applied egg-rr 99.9%

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

Alternative 2: 99.2% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

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

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

7. Simplified 99.8%

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

   1. fabs-mul N/A

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

   2. fabs-div N/A

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

   3. metadata-eval N/A

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

   4. rem-sqrt-square N/A

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

   5. add-sqr-sqrt N/A

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

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

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

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

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

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

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

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

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

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

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

9. Applied egg-rr 99.9%

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

    1. associate-*r* N/A

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

    2. fabs-sqr N/A

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

    3. fabs-mul N/A

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

    4. pow3 N/A

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

    5. sqr-pow N/A

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

    6. fabs-sqr N/A

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

    7. sqr-pow N/A

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

    8. pow3 N/A

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

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

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

    10. *-lowering-*.f64 99.8

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

11. Applied egg-rr 99.8%

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

12. Final simplification 99.8%

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

Alternative 3: 99.4% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

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

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

7. Simplified 99.8%

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

   1. associate-*l/ N/A

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

   2. fabs-div N/A

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

   3. *-lft-identity N/A

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

   4. rem-sqrt-square N/A

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

   5. add-sqr-sqrt N/A

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

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

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

9. Applied egg-rr 99.4%

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

Alternative 4: 99.4% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

7. Simplified 99.4%

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

8. Final simplification 99.4%

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

Alternative 5: 99.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.04:\\ \;\;\;\;\left|x\right| \cdot \frac{\left|\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), 2\right)\right|}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.04)
   (*
    (fabs x)
    (/
     (fabs (fma x (* x (fma (* x x) 0.2 0.6666666666666666)) 2.0))
     (sqrt PI)))
   (*
    0.047619047619047616
    (/ (* (fabs x) (* (* x x) (* x (* x (* x x))))) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.04) {
		tmp = fabs(x) * (fabs(fma(x, (x * fma((x * x), 0.2, 0.6666666666666666)), 2.0)) / sqrt(((double) M_PI)));
	} else {
		tmp = 0.047619047619047616 * ((fabs(x) * ((x * x) * (x * (x * (x * x))))) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.04)
		tmp = Float64(abs(x) * Float64(abs(fma(x, Float64(x * fma(Float64(x * x), 0.2, 0.6666666666666666)), 2.0)) / sqrt(pi)));
	else
		tmp = Float64(0.047619047619047616 * Float64(Float64(abs(x) * Float64(Float64(x * x) * Float64(x * Float64(x * Float64(x * x))))) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.04], N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.2 + 0.6666666666666666), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0400000000000000008

   1. Initial program 99.9%

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

   7. Simplified 99.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. fabs-mul N/A

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

      4. fabs-fabs N/A

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

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

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

   9. Applied egg-rr 99.7%

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

   if 0.0400000000000000008 < (fabs.f64 x)

   1. Initial program 99.8%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. cube-prod N/A

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

      9. unpow2 N/A

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

      10. unpow3 N/A

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

      11. pow-sqr N/A

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

      12. metadata-eval N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

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

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

      16. metadata-eval N/A

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

      17. pow-sqr N/A

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

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

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

      19. unpow2 N/A

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

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

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

      21. unpow2 N/A

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

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

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

      23. *-commutative N/A

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

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

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

      25. unpow2 N/A

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

      26. *-lowering-*.f64 99.4

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

   7. Simplified 99.4%

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

      1. associate-*l/ N/A

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

      2. fabs-div N/A

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

   9. Applied egg-rr 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.04:\\ \;\;\;\;\left|x\right| \cdot \frac{\left|\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), 2\right)\right|}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.04:\\ \;\;\;\;\left|x\right| \cdot \frac{\left|\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), 2\right)\right|}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot \left|\frac{x}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.04)
   (*
    (fabs x)
    (/
     (fabs (fma x (* x (fma (* x x) 0.2 0.6666666666666666)) 2.0))
     (sqrt PI)))
   (*
    (* 0.047619047619047616 (* (* x x) (* x (* x (* x x)))))
    (fabs (/ x (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.04) {
		tmp = fabs(x) * (fabs(fma(x, (x * fma((x * x), 0.2, 0.6666666666666666)), 2.0)) / sqrt(((double) M_PI)));
	} else {
		tmp = (0.047619047619047616 * ((x * x) * (x * (x * (x * x))))) * fabs((x / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.04)
		tmp = Float64(abs(x) * Float64(abs(fma(x, Float64(x * fma(Float64(x * x), 0.2, 0.6666666666666666)), 2.0)) / sqrt(pi)));
	else
		tmp = Float64(Float64(0.047619047619047616 * Float64(Float64(x * x) * Float64(x * Float64(x * Float64(x * x))))) * abs(Float64(x / sqrt(pi))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.04], N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.2 + 0.6666666666666666), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.047619047619047616 * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0400000000000000008

   1. Initial program 99.9%

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

   7. Simplified 99.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. fabs-mul N/A

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

      4. fabs-fabs N/A

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

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

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

   9. Applied egg-rr 99.7%

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

   if 0.0400000000000000008 < (fabs.f64 x)

   1. Initial program 99.8%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. cube-prod N/A

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

      9. unpow2 N/A

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

      10. unpow3 N/A

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

      11. pow-sqr N/A

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

      12. metadata-eval N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

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

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

      16. metadata-eval N/A

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

      17. pow-sqr N/A

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

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

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

      19. unpow2 N/A

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

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

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

      21. unpow2 N/A

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

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

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

      23. *-commutative N/A

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

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

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

      25. unpow2 N/A

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

      26. *-lowering-*.f64 99.4

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

   7. Simplified 99.4%

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

   9. Applied egg-rr 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.04:\\ \;\;\;\;\left|x\right| \cdot \frac{\left|\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), 2\right)\right|}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left(0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot \left|\frac{x}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.04:\\ \;\;\;\;\left|x\right| \cdot \frac{\left|\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), 2\right)\right|}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{\left(x \cdot x\right) \cdot 0.047619047619047616}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.04)
   (*
    (fabs x)
    (/
     (fabs (fma x (* x (fma (* x x) 0.2 0.6666666666666666)) 2.0))
     (sqrt PI)))
   (*
    (* (* x x) (* x (* x x)))
    (/ (* (* x x) 0.047619047619047616) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.04) {
		tmp = fabs(x) * (fabs(fma(x, (x * fma((x * x), 0.2, 0.6666666666666666)), 2.0)) / sqrt(((double) M_PI)));
	} else {
		tmp = ((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) <= 0.04)
		tmp = Float64(abs(x) * Float64(abs(fma(x, Float64(x * fma(Float64(x * x), 0.2, 0.6666666666666666)), 2.0)) / sqrt(pi)));
	else
		tmp = Float64(Float64(Float64(x * x) * Float64(x * Float64(x * x))) * Float64(Float64(Float64(x * x) * 0.047619047619047616) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.04], N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.2 + 0.6666666666666666), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 0.047619047619047616), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0400000000000000008

   1. Initial program 99.9%

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

   7. Simplified 99.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. fabs-mul N/A

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

      4. fabs-fabs N/A

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

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

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

   9. Applied egg-rr 99.7%

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

   if 0.0400000000000000008 < (fabs.f64 x)

   1. Initial program 99.8%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. cube-prod N/A

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

      9. unpow2 N/A

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

      10. unpow3 N/A

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

      11. pow-sqr N/A

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

      12. metadata-eval N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

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

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

      16. metadata-eval N/A

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

      17. pow-sqr N/A

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

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

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

      19. unpow2 N/A

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

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

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

      21. unpow2 N/A

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

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

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

      23. *-commutative N/A

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

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

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

      25. unpow2 N/A

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

      26. *-lowering-*.f64 99.4

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

   7. Simplified 99.4%

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

      1. associate-*l/ N/A

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

      2. fabs-div N/A

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

      3. *-lft-identity N/A

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

      4. rem-sqrt-square N/A

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

      5. add-sqr-sqrt N/A

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

      6. div-inv N/A

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

   9. Applied egg-rr 0.1%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.04:\\ \;\;\;\;\left|x\right| \cdot \frac{\left|\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), 2\right)\right|}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{\left(x \cdot x\right) \cdot 0.047619047619047616}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

7. Simplified 99.4%

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

8. Taylor expanded in x around 0

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

10. Simplified 99.4%

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

Alternative 9: 66.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.04:\\ \;\;\;\;\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{0.2}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.04)
   (* (fabs x) (fabs (/ (fma x (* x 0.6666666666666666) 2.0) (sqrt PI))))
   (* (* (* x x) (* x (* x x))) (/ 0.2 (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.04) {
		tmp = fabs(x) * fabs((fma(x, (x * 0.6666666666666666), 2.0) / sqrt(((double) M_PI))));
	} else {
		tmp = ((x * x) * (x * (x * x))) * (0.2 / sqrt(((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.04)
		tmp = Float64(abs(x) * abs(Float64(fma(x, Float64(x * 0.6666666666666666), 2.0) / sqrt(pi))));
	else
		tmp = Float64(Float64(Float64(x * x) * Float64(x * Float64(x * x))) * Float64(0.2 / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.04], N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(N[(x * N[(x * 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.2 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0400000000000000008

   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. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 99.6

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

   10. Simplified 99.6%

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. fabs-mul N/A

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

       4. fabs-fabs N/A

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

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

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

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

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

       7. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

       13. PI-lowering-PI.f64 99.6

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

   12. Applied egg-rr 99.6%

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

   if 0.0400000000000000008 < (fabs.f64 x)

   1. Initial program 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

   7. Simplified 86.5%

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

   8. Taylor expanded in x around inf

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. unpow2 N/A

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

      12. cube-mult N/A

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

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

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

      14. cube-mult N/A

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

      15. unpow2 N/A

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

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

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

      17. unpow2 N/A

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

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

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

      19. *-commutative N/A

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

      20. associate-*r* N/A

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

   10. Simplified 86.5%

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

       1. associate-*r* N/A

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

       2. sqrt-div N/A

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

       3. metadata-eval N/A

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

       4. associate-*r* N/A

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

       5. un-div-inv N/A

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

       6. fabs-div N/A

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

   12. Applied egg-rr 0.1%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.04:\\ \;\;\;\;\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{0.2}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 94.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

7. Simplified 94.7%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. fabs-mul N/A

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

   4. fabs-fabs N/A

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

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

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

9. Applied egg-rr 95.1%

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

10. Final simplification 95.1%

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

Alternative 11: 93.5% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

7. Simplified 94.7%

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

Alternative 12: 93.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

7. Simplified 94.7%

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

8. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

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

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 94.3

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

10. Simplified 94.3%

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

Alternative 13: 89.7% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

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

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

7. Simplified 99.8%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-out N/A

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

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

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

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

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

   6. +-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

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

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

   11. *-commutative N/A

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

   12. *-lowering-*.f64 89.2

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

10. Simplified 89.2%

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

    1. *-commutative N/A

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

    2. associate-*l* N/A

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

    3. fabs-mul N/A

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

    4. fabs-fabs N/A

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

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

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

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

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

    7. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

    13. PI-lowering-PI.f64 89.2

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

12. Applied egg-rr 89.2%

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

Alternative 14: 89.3% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-in N/A

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

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

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

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

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

   6. +-commutative N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 88.8

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

7. Simplified 88.8%

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

Alternative 15: 67.8% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in x around 0

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

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

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

   2. fabs-lowering-fabs.f64 66.3

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

7. Simplified 66.3%

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

   1. fabs-div N/A

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

   2. neg-fabs N/A

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

   3. neg-fabs N/A

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

   4. fabs-mul N/A

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

   5. metadata-eval N/A

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

   6. fabs-fabs N/A

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

   7. *-commutative N/A

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

   8. rem-sqrt-square N/A

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

   9. add-sqr-sqrt N/A

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

   10. associate-/l* N/A

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

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

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

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

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

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

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

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

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

   15. PI-lowering-PI.f64 66.7

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

9. Applied egg-rr 66.7%

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

Reproduce

herbie shell --seed 2024194 
(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)))))))