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

Percentage Accurate: 99.9% → 99.9%

Time: 12.7s

Alternatives: 18

Speedup: 0.7×

• 29.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-fabs.f64 N/A

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

   4. +-commutative N/A

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

   5. +-commutative N/A

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

   6. unpow3 N/A

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

   7. associate-*r* N/A

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

   8. distribute-rgt-out N/A

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

   9. lower-fma.f64 N/A

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

5. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.4%

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

   1. lift-fabs.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-lft-out N/A

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

   6. associate-/l* N/A

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

6. Applied rewrites 99.8%

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

8. Applied rewrites 99.8%

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

9. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.4%

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

   1. lift-fabs.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-lft-out N/A

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

   6. associate-/l* N/A

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

6. Applied rewrites 99.8%

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

Alternative 4: 99.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.4%

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

   1. lift-fabs.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-lft-out N/A

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

   6. associate-/l* N/A

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

6. Applied rewrites 99.8%

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

8. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.8

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

   4. lift-fma.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. associate-+r+ N/A

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

   7. distribute-lft-out N/A

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

   8. lower-fma.f64 N/A

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

10. Applied rewrites 99.8%

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

Alternative 5: 99.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.05:\\ \;\;\;\;\left|x \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x\right| \cdot \left|\frac{x \cdot \left(x \cdot \left(x \cdot \left(0.047619047619047616 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.05)
   (fabs
    (* x (/ (fma (* x x) (fma x (* x 0.2) 0.6666666666666666) 2.0) (sqrt PI))))
   (*
    (fabs x)
    (fabs
     (/ (* x (* x (* x (* 0.047619047619047616 (* x (* x x)))))) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.05) {
		tmp = fabs((x * (fma((x * x), fma(x, (x * 0.2), 0.6666666666666666), 2.0) / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs(x) * fabs(((x * (x * (x * (0.047619047619047616 * (x * (x * x)))))) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.05)
		tmp = abs(Float64(x * Float64(fma(Float64(x * x), fma(x, Float64(x * 0.2), 0.6666666666666666), 2.0) / sqrt(pi))));
	else
		tmp = Float64(abs(x) * abs(Float64(Float64(x * Float64(x * Float64(x * Float64(0.047619047619047616 * Float64(x * Float64(x * x)))))) / sqrt(pi))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.05], N[Abs[N[(x * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(N[(x * N[(x * N[(x * N[(0.047619047619047616 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.050000000000000003

   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 rewrites 99.2%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. associate-/l* N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 99.6

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

   9. Applied rewrites 99.6%

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

       1. lift-*.f64 N/A

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

       2. lift-fabs.f64 N/A

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

       3. lift-fabs.f64 N/A

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

       4. mul-fabs N/A

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

       5. lower-fabs.f64 N/A

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

       6. lower-*.f64 99.6

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

   11. Applied rewrites 99.6%

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

   if 0.050000000000000003 < (fabs.f64 x)

   1. Initial program 99.9%

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

   4. Applied rewrites 99.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. associate-/l* N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. pow-plus N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 N/A

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

      25. unpow2 N/A

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

      26. lower-*.f64 99.5

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

   9. Applied rewrites 99.5%

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

   11. Applied rewrites 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.05:\\ \;\;\;\;\left|x \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x\right| \cdot \left|\frac{x \cdot \left(x \cdot \left(x \cdot \left(0.047619047619047616 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.05:\\ \;\;\;\;\left|x \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x\right| \cdot \left|\frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)\right)\right)}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.05)
   (fabs
    (* x (/ (fma (* x x) (fma x (* x 0.2) 0.6666666666666666) 2.0) (sqrt PI))))
   (*
    (fabs x)
    (fabs
     (/ (* x (* x (* x (* x (* 0.047619047619047616 (* x x)))))) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.05) {
		tmp = fabs((x * (fma((x * x), fma(x, (x * 0.2), 0.6666666666666666), 2.0) / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs(x) * fabs(((x * (x * (x * (x * (0.047619047619047616 * (x * x)))))) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.05)
		tmp = abs(Float64(x * Float64(fma(Float64(x * x), fma(x, Float64(x * 0.2), 0.6666666666666666), 2.0) / sqrt(pi))));
	else
		tmp = Float64(abs(x) * abs(Float64(Float64(x * Float64(x * Float64(x * Float64(x * Float64(0.047619047619047616 * Float64(x * x)))))) / sqrt(pi))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.05], N[Abs[N[(x * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(N[(x * N[(x * N[(x * N[(x * N[(0.047619047619047616 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.050000000000000003

   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 rewrites 99.2%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. associate-/l* N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 99.6

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

   9. Applied rewrites 99.6%

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

       1. lift-*.f64 N/A

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

       2. lift-fabs.f64 N/A

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

       3. lift-fabs.f64 N/A

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

       4. mul-fabs N/A

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

       5. lower-fabs.f64 N/A

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

       6. lower-*.f64 99.6

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

   11. Applied rewrites 99.6%

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

   if 0.050000000000000003 < (fabs.f64 x)

   1. Initial program 99.9%

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

   4. Applied rewrites 99.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. associate-/l* N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. pow-plus N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 N/A

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

      25. unpow2 N/A

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

      26. lower-*.f64 99.5

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

   9. Applied rewrites 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.05:\\ \;\;\;\;\left|x \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x\right| \cdot \left|\frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)\right)\right)}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;\left|x\right| \leq 0.05:\\ \;\;\;\;\left|x \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{t\_0 \cdot \left(t\_0 \cdot \left(0.047619047619047616 \cdot \left|x\right|\right)\right)}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= (fabs x) 0.05)
     (fabs
      (*
       x
       (/ (fma (* x x) (fma x (* x 0.2) 0.6666666666666666) 2.0) (sqrt PI))))
     (fabs (/ (* t_0 (* t_0 (* 0.047619047619047616 (fabs x)))) (sqrt PI))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double tmp;
	if (fabs(x) <= 0.05) {
		tmp = fabs((x * (fma((x * x), fma(x, (x * 0.2), 0.6666666666666666), 2.0) / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs(((t_0 * (t_0 * (0.047619047619047616 * fabs(x)))) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (abs(x) <= 0.05)
		tmp = abs(Float64(x * Float64(fma(Float64(x * x), fma(x, Float64(x * 0.2), 0.6666666666666666), 2.0) / sqrt(pi))));
	else
		tmp = abs(Float64(Float64(t_0 * Float64(t_0 * Float64(0.047619047619047616 * abs(x)))) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.05], N[Abs[N[(x * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(t$95$0 * N[(t$95$0 * N[(0.047619047619047616 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.050000000000000003

   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 rewrites 99.2%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. associate-/l* N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 99.6

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

   9. Applied rewrites 99.6%

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

       1. lift-*.f64 N/A

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

       2. lift-fabs.f64 N/A

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

       3. lift-fabs.f64 N/A

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

       4. mul-fabs N/A

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

       5. lower-fabs.f64 N/A

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

       6. lower-*.f64 99.6

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

   11. Applied rewrites 99.6%

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

   if 0.050000000000000003 < (fabs.f64 x)

   1. Initial program 99.9%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fabs.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. cube-prod N/A

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

      9. unpow2 N/A

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

      10. cube-unmult N/A

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

      11. pow-sqr N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval N/A

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

      17. pow-sqr N/A

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

      18. lower-*.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 N/A

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

      21. unpow2 N/A

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

      22. lower-*.f64 99.4

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

   7. Applied rewrites 99.4%

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

   9. Applied rewrites 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.05:\\ \;\;\;\;\left|x \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.047619047619047616 \cdot \left|x\right|\right)\right)}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.05:\\ \;\;\;\;\left|x \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(0.047619047619047616 \cdot \left|x\right|\right)\right)\right)\right)}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.05)
   (fabs
    (* x (/ (fma (* x x) (fma x (* x 0.2) 0.6666666666666666) 2.0) (sqrt PI))))
   (fabs
    (/
     (* (* x x) (* (* x x) (* x (* x (* 0.047619047619047616 (fabs x))))))
     (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.05) {
		tmp = fabs((x * (fma((x * x), fma(x, (x * 0.2), 0.6666666666666666), 2.0) / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((((x * x) * ((x * x) * (x * (x * (0.047619047619047616 * fabs(x)))))) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.05)
		tmp = abs(Float64(x * Float64(fma(Float64(x * x), fma(x, Float64(x * 0.2), 0.6666666666666666), 2.0) / sqrt(pi))));
	else
		tmp = abs(Float64(Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(x * Float64(x * Float64(0.047619047619047616 * abs(x)))))) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.05], N[Abs[N[(x * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(0.047619047619047616 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.050000000000000003

   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 rewrites 99.2%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. associate-/l* N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 99.6

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

   9. Applied rewrites 99.6%

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

       1. lift-*.f64 N/A

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

       2. lift-fabs.f64 N/A

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

       3. lift-fabs.f64 N/A

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

       4. mul-fabs N/A

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

       5. lower-fabs.f64 N/A

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

       6. lower-*.f64 99.6

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

   11. Applied rewrites 99.6%

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

   if 0.050000000000000003 < (fabs.f64 x)

   1. Initial program 99.9%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fabs.f64 5.9

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

   7. Applied rewrites 5.9%

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

   8. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. pow-plus N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. pow-sqr N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. associate-*r* N/A

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

      15. associate-*r* N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

   10. Applied rewrites 99.4%

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

Alternative 9: 99.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.05:\\ \;\;\;\;\left|x \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)\right)\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.05)
   (fabs
    (* x (/ (fma (* x x) (fma x (* x 0.2) 0.6666666666666666) 2.0) (sqrt PI))))
   (/
    (fabs (* x (* (* x (* x x)) (* x (* 0.047619047619047616 (* x x))))))
    (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.05) {
		tmp = fabs((x * (fma((x * x), fma(x, (x * 0.2), 0.6666666666666666), 2.0) / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((x * ((x * (x * x)) * (x * (0.047619047619047616 * (x * x)))))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.05)
		tmp = abs(Float64(x * Float64(fma(Float64(x * x), fma(x, Float64(x * 0.2), 0.6666666666666666), 2.0) / sqrt(pi))));
	else
		tmp = Float64(abs(Float64(x * Float64(Float64(x * Float64(x * x)) * Float64(x * Float64(0.047619047619047616 * Float64(x * x)))))) / sqrt(pi));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.05], N[Abs[N[(x * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(x * N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * N[(0.047619047619047616 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.050000000000000003

   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 rewrites 99.2%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. associate-/l* N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 99.6

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

   9. Applied rewrites 99.6%

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

       1. lift-*.f64 N/A

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

       2. lift-fabs.f64 N/A

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

       3. lift-fabs.f64 N/A

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

       4. mul-fabs N/A

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

       5. lower-fabs.f64 N/A

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

       6. lower-*.f64 99.6

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

   11. Applied rewrites 99.6%

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

   if 0.050000000000000003 < (fabs.f64 x)

   1. Initial program 99.9%

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

   4. Applied rewrites 99.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. associate-/l* N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. pow-plus N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. metadata-eval N/A

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

      10. pow-sqr N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. associate-*l* N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 N/A

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

      25. unpow2 N/A

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

      26. lower-*.f64 99.5

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

   9. Applied rewrites 99.5%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

   11. Applied rewrites 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.05:\\ \;\;\;\;\left|x \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \left(0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)\right)\right|}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.4% accurate, 2.9× speedup

Math

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

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(Float64(x * x), fma(x, Float64(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[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.047619047619047616), $MachinePrecision] + 0.2), $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 rewrites 99.4%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-fabs.f64 66.4

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

7. Applied rewrites 66.4%

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

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. Applied rewrites 99.4%

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

Alternative 11: 99.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.4%

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

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. lower-*.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. lower-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. lower-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. lower-*.f64 91.1

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

7. Applied rewrites 91.1%

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

   1. lift-fabs.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. fabs-div N/A

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

   4. rem-sqrt-square N/A

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

   5. sqrt-prod N/A

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

   6. rem-square-sqrt N/A

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

9. Applied rewrites 91.1%

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

10. Taylor expanded in x around 0

    \[\leadsto \frac{\color{blue}{\left|\left|x\right| \cdot \left(2 + \frac{2}{3} \cdot {x}^{2}\right) + \left|x\right| \cdot \left(\frac{1}{21} \cdot {x}^{6} + \frac{1}{5} \cdot {x}^{4}\right)\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

12. Applied rewrites 99.4%

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

Alternative 12: 94.0% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.4%

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

   1. lift-fabs.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-lft-out N/A

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

   6. associate-/l* N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 95.1

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

9. Applied rewrites 95.1%

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

    1. lift-*.f64 N/A

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

    2. lift-fabs.f64 N/A

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

    3. lift-fabs.f64 N/A

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

    4. mul-fabs N/A

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

    5. lower-fabs.f64 N/A

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

    6. lower-*.f64 95.1

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

11. Applied rewrites 95.1%

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

Alternative 13: 89.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.05:\\ \;\;\;\;\left|x\right| \cdot \left|\frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left|x\right| \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.05)
   (* (fabs x) (fabs (/ 2.0 (sqrt PI))))
   (fabs (/ (* (fabs x) (* 0.6666666666666666 (* x x))) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.05) {
		tmp = fabs(x) * fabs((2.0 / sqrt(((double) M_PI))));
	} else {
		tmp = fabs(((fabs(x) * (0.6666666666666666 * (x * x))) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.05) {
		tmp = Math.abs(x) * Math.abs((2.0 / Math.sqrt(Math.PI)));
	} else {
		tmp = Math.abs(((Math.abs(x) * (0.6666666666666666 * (x * x))) / Math.sqrt(Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.05:
		tmp = math.fabs(x) * math.fabs((2.0 / math.sqrt(math.pi)))
	else:
		tmp = math.fabs(((math.fabs(x) * (0.6666666666666666 * (x * x))) / math.sqrt(math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.05)
		tmp = Float64(abs(x) * abs(Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(Float64(Float64(abs(x) * Float64(0.6666666666666666 * Float64(x * x))) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.05)
		tmp = abs(x) * abs((2.0 / sqrt(pi)));
	else
		tmp = abs(((abs(x) * (0.6666666666666666 * (x * x))) / sqrt(pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.05], N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[(N[Abs[x], $MachinePrecision] * N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.050000000000000003

   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 rewrites 99.2%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-out N/A

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

      6. associate-/l* N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 98.8%

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

   if 0.050000000000000003 < (fabs.f64 x)

   1. Initial program 99.9%

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

   4. Applied rewrites 99.9%

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

   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. lower-*.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. lower-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. lower-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. lower-*.f64 76.7

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

   7. Applied rewrites 76.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 76.7%

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

Alternative 14: 89.5% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(N[(0.6666666666666666 * N[(x * x), $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

4. Applied rewrites 99.4%

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

   1. lift-fabs.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-lft-out N/A

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

   6. associate-/l* N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 91.5

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

9. Applied rewrites 91.5%

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

Alternative 15: 89.1% 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 rewrites 99.4%

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

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. lower-*.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. lower-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. lower-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. lower-*.f64 91.1

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

7. Applied rewrites 91.1%

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

Alternative 16: 67.3% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-fabs.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 66.9

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

7. Applied rewrites 66.9%

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

8. Final simplification 66.9%

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

Alternative 17: 67.5% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

4. Applied rewrites 99.4%

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

   1. lift-fabs.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-lft-out N/A

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

   6. associate-/l* N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 66.9%

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

Alternative 18: 67.1% accurate, 6.3× speedup

Math

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

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(Float64(x * 2.0)) / sqrt(pi))
end

MATLAB

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

Wolfram

code[x_] := N[(N[Abs[N[(x * 2.0), $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

4. Applied rewrites 99.4%

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

   1. lift-fabs.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-lft-out N/A

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

   6. associate-/l* N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 66.9%

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

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lift-fabs.f64 N/A

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

    4. lift-/.f64 N/A

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

    5. fabs-div N/A

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

    6. rem-sqrt-square N/A

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

    7. sqrt-prod N/A

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

    8. rem-square-sqrt N/A

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

11. Applied rewrites 66.4%

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

12. Final simplification 66.4%

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

Reproduce

herbie shell --seed 2024226 
(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)))))))