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

Percentage Accurate: 99.8% → 99.8%

Time: 5.5s

Alternatives: 18

Speedup: 1.7×

• 29.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.8%

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

Alternative 2: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Abs[N[(N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision] * 0.047619047619047616 + N[(N[Abs[x], $MachinePrecision] * N[(N[(N[(0.2 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x + N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $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. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)\right)\right| \]

4. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{5}, \mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right|\right)\right)\right)}\right| \]

6. Applied rewrites 99.4%

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

Alternative 3: 99.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.4%

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

Alternative 4: 98.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left|x\right| + \left(\left|x\right| + \left(0.6666666666666666 \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+ (fabs x) (+ (fabs x) (* (* 0.6666666666666666 (fabs x)) (* x x))))))
   (fabs (* 0.047619047619047616 (/ (* (pow x 6.0) (fabs x)) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = fabs(((1.0 / sqrt(((double) M_PI))) * (fabs(x) + (fabs(x) + ((0.6666666666666666 * fabs(x)) * (x * x))))));
	} else {
		tmp = fabs((0.047619047619047616 * ((pow(x, 6.0) * fabs(x)) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = Math.abs(((1.0 / Math.sqrt(Math.PI)) * (Math.abs(x) + (Math.abs(x) + ((0.6666666666666666 * Math.abs(x)) * (x * x))))));
	} else {
		tmp = Math.abs((0.047619047619047616 * ((Math.pow(x, 6.0) * Math.abs(x)) / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.2:
		tmp = math.fabs(((1.0 / math.sqrt(math.pi)) * (math.fabs(x) + (math.fabs(x) + ((0.6666666666666666 * math.fabs(x)) * (x * x))))))
	else:
		tmp = math.fabs((0.047619047619047616 * ((math.pow(x, 6.0) * math.fabs(x)) / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(abs(x) + Float64(abs(x) + Float64(Float64(0.6666666666666666 * abs(x)) * Float64(x * x))))));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64(Float64((x ^ 6.0) * abs(x)) / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.2)
		tmp = abs(((1.0 / sqrt(pi)) * (abs(x) + (abs(x) + ((0.6666666666666666 * abs(x)) * (x * x))))));
	else
		tmp = abs((0.047619047619047616 * (((x ^ 6.0) * abs(x)) / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] + N[(N[(0.6666666666666666 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[(N[Power[x, 6.0], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. lower-fma.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-fabs.f64 88.9

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

   6. Applied rewrites 88.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. associate-+l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 88.9

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

   8. Applied rewrites 88.9%

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

   if 2.2000000000000002 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-PI.f64 37.6

         \[\leadsto \left|0.047619047619047616 \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

   6. Applied rewrites 37.6%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left|x\right| + \left(\left|x\right| + \left(0.6666666666666666 \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{\left|x\right|}{\sqrt{\pi}}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+ (fabs x) (+ (fabs x) (* (* 0.6666666666666666 (fabs x)) (* x x))))))
   (fabs
    (*
     0.047619047619047616
     (* (* (* (* x x) x) x) (* (* x x) (/ (fabs x) (sqrt PI))))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = fabs(((1.0 / sqrt(((double) M_PI))) * (fabs(x) + (fabs(x) + ((0.6666666666666666 * fabs(x)) * (x * x))))));
	} else {
		tmp = fabs((0.047619047619047616 * ((((x * x) * x) * x) * ((x * x) * (fabs(x) / sqrt(((double) M_PI)))))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = Math.abs(((1.0 / Math.sqrt(Math.PI)) * (Math.abs(x) + (Math.abs(x) + ((0.6666666666666666 * Math.abs(x)) * (x * x))))));
	} else {
		tmp = Math.abs((0.047619047619047616 * ((((x * x) * x) * x) * ((x * x) * (Math.abs(x) / Math.sqrt(Math.PI))))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.2:
		tmp = math.fabs(((1.0 / math.sqrt(math.pi)) * (math.fabs(x) + (math.fabs(x) + ((0.6666666666666666 * math.fabs(x)) * (x * x))))))
	else:
		tmp = math.fabs((0.047619047619047616 * ((((x * x) * x) * x) * ((x * x) * (math.fabs(x) / math.sqrt(math.pi))))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(abs(x) + Float64(abs(x) + Float64(Float64(0.6666666666666666 * abs(x)) * Float64(x * x))))));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64(Float64(Float64(Float64(x * x) * x) * x) * Float64(Float64(x * x) * Float64(abs(x) / sqrt(pi))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.2)
		tmp = abs(((1.0 / sqrt(pi)) * (abs(x) + (abs(x) + ((0.6666666666666666 * abs(x)) * (x * x))))));
	else
		tmp = abs((0.047619047619047616 * ((((x * x) * x) * x) * ((x * x) * (abs(x) / sqrt(pi))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] + N[(N[(0.6666666666666666 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. lower-fma.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-fabs.f64 88.9

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

   6. Applied rewrites 88.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. associate-+l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 88.9

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

   8. Applied rewrites 88.9%

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

   if 2.2000000000000002 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-PI.f64 37.6

         \[\leadsto \left|0.047619047619047616 \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

   6. Applied rewrites 37.6%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}}\right| \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\color{blue}{\sqrt{\pi}}}\right| \]

      2. lift-*.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\color{blue}{\pi}}}\right| \]

      3. lift-pow.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

      4. metadata-eval N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{\left(3 + 3\right)} \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

      5. pow-prod-up N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{\left({x}^{3} \cdot {x}^{3}\right) \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

      6. pow3 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot {x}^{3}\right) \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

      7. pow3 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

      8. lift-*.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

      9. lift-*.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

      10. lift-*.f64 N/A

          \[\leadsto \left|\frac{1}{21} \cdot \frac{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

      11. lift-*.f64 N/A

          \[\leadsto \left|\frac{1}{21} \cdot \frac{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

      12. lift-*.f64 N/A

          \[\leadsto \left|\frac{1}{21} \cdot \frac{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

      13. associate-/l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

          \[\leadsto \left|\frac{1}{21} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot \frac{\left|x\right|}{\sqrt{\pi}}\right)\right| \]

      18. lift-*.f64 N/A

          \[\leadsto \left|\frac{1}{21} \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot \frac{\left|x\right|}{\sqrt{\pi}}\right)\right| \]

      19. *-commutative N/A

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

      20. mult-flip N/A

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

   8. Applied rewrites 37.6%

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

Alternative 6: 98.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left|x\right| + \left(\left|x\right| + \left(0.6666666666666666 \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.047619047619047616\right)\right) \cdot x\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+ (fabs x) (+ (fabs x) (* (* 0.6666666666666666 (fabs x)) (* x x))))))
   (fabs
    (*
     (*
      (/ (fabs x) (sqrt PI))
      (* (* (* (* (* x x) x) x) x) 0.047619047619047616))
     x))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = fabs(((1.0 / sqrt(((double) M_PI))) * (fabs(x) + (fabs(x) + ((0.6666666666666666 * fabs(x)) * (x * x))))));
	} else {
		tmp = fabs((((fabs(x) / sqrt(((double) M_PI))) * (((((x * x) * x) * x) * x) * 0.047619047619047616)) * x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = Math.abs(((1.0 / Math.sqrt(Math.PI)) * (Math.abs(x) + (Math.abs(x) + ((0.6666666666666666 * Math.abs(x)) * (x * x))))));
	} else {
		tmp = Math.abs((((Math.abs(x) / Math.sqrt(Math.PI)) * (((((x * x) * x) * x) * x) * 0.047619047619047616)) * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.2:
		tmp = math.fabs(((1.0 / math.sqrt(math.pi)) * (math.fabs(x) + (math.fabs(x) + ((0.6666666666666666 * math.fabs(x)) * (x * x))))))
	else:
		tmp = math.fabs((((math.fabs(x) / math.sqrt(math.pi)) * (((((x * x) * x) * x) * x) * 0.047619047619047616)) * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(abs(x) + Float64(abs(x) + Float64(Float64(0.6666666666666666 * abs(x)) * Float64(x * x))))));
	else
		tmp = abs(Float64(Float64(Float64(abs(x) / sqrt(pi)) * Float64(Float64(Float64(Float64(Float64(x * x) * x) * x) * x) * 0.047619047619047616)) * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.2)
		tmp = abs(((1.0 / sqrt(pi)) * (abs(x) + (abs(x) + ((0.6666666666666666 * abs(x)) * (x * x))))));
	else
		tmp = abs((((abs(x) / sqrt(pi)) * (((((x * x) * x) * x) * x) * 0.047619047619047616)) * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] + N[(N[(0.6666666666666666 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[Abs[x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 0.047619047619047616), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. lower-fma.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-fabs.f64 88.9

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

   6. Applied rewrites 88.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. associate-+l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 88.9

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

   8. Applied rewrites 88.9%

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

   if 2.2000000000000002 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-PI.f64 37.6

         \[\leadsto \left|0.047619047619047616 \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

   6. Applied rewrites 37.6%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}}\right| \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \color{blue}{\frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}}\right| \]

      2. lift-/.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\color{blue}{\sqrt{\pi}}}\right| \]

      3. lift-*.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\color{blue}{\pi}}}\right| \]

      4. lift-pow.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

      5. metadata-eval N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{\left(3 + 3\right)} \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

      6. pow-prod-up N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{\left({x}^{3} \cdot {x}^{3}\right) \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

      7. pow3 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot {x}^{3}\right) \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

      8. pow3 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

      9. lift-*.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

      10. lift-*.f64 N/A

          \[\leadsto \left|\frac{1}{21} \cdot \frac{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

      11. lift-*.f64 N/A

          \[\leadsto \left|\frac{1}{21} \cdot \frac{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

      12. lift-*.f64 N/A

          \[\leadsto \left|\frac{1}{21} \cdot \frac{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

      13. lift-*.f64 N/A

          \[\leadsto \left|\frac{1}{21} \cdot \frac{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

      14. associate-/l* N/A

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

      15. mult-flip N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

   8. Applied rewrites 37.6%

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

Alternative 7: 91.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left|x\right| + \left(\left|x\right| + \left(0.6666666666666666 \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \frac{\left|x\right|}{\sqrt{\pi}}\right) \cdot 0.047619047619047616\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+ (fabs x) (+ (fabs x) (* (* 0.6666666666666666 (fabs x)) (* x x))))))
   (fabs
    (*
     (* (* (* (* (* (* x x) x) x) x) x) (/ (fabs x) (sqrt PI)))
     0.047619047619047616))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = fabs(((1.0 / sqrt(((double) M_PI))) * (fabs(x) + (fabs(x) + ((0.6666666666666666 * fabs(x)) * (x * x))))));
	} else {
		tmp = fabs((((((((x * x) * x) * x) * x) * x) * (fabs(x) / sqrt(((double) M_PI)))) * 0.047619047619047616));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = Math.abs(((1.0 / Math.sqrt(Math.PI)) * (Math.abs(x) + (Math.abs(x) + ((0.6666666666666666 * Math.abs(x)) * (x * x))))));
	} else {
		tmp = Math.abs((((((((x * x) * x) * x) * x) * x) * (Math.abs(x) / Math.sqrt(Math.PI))) * 0.047619047619047616));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.2:
		tmp = math.fabs(((1.0 / math.sqrt(math.pi)) * (math.fabs(x) + (math.fabs(x) + ((0.6666666666666666 * math.fabs(x)) * (x * x))))))
	else:
		tmp = math.fabs((((((((x * x) * x) * x) * x) * x) * (math.fabs(x) / math.sqrt(math.pi))) * 0.047619047619047616))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(abs(x) + Float64(abs(x) + Float64(Float64(0.6666666666666666 * abs(x)) * Float64(x * x))))));
	else
		tmp = abs(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * x) * x) * x) * x) * x) * Float64(abs(x) / sqrt(pi))) * 0.047619047619047616));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.2)
		tmp = abs(((1.0 / sqrt(pi)) * (abs(x) + (abs(x) + ((0.6666666666666666 * abs(x)) * (x * x))))));
	else
		tmp = abs((((((((x * x) * x) * x) * x) * x) * (abs(x) / sqrt(pi))) * 0.047619047619047616));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] + N[(N[(0.6666666666666666 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.047619047619047616), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. lower-fma.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-fabs.f64 88.9

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

   6. Applied rewrites 88.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. associate-+l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 88.9

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

   8. Applied rewrites 88.9%

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

   if 2.2000000000000002 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-PI.f64 37.6

         \[\leadsto \left|0.047619047619047616 \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

   6. Applied rewrites 37.6%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}}\right| \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \color{blue}{\frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}}\right| \]

      2. *-commutative N/A

         \[\leadsto \left|\frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}} \cdot \color{blue}{\frac{1}{21}}\right| \]

      3. lower-*.f64 37.6

         \[\leadsto \left|\frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}} \cdot \color{blue}{0.047619047619047616}\right| \]

   8. Applied rewrites 37.6%

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

Alternative 8: 88.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left|x\right| + \left(\left|x\right| + \left(0.6666666666666666 \cdot \left|x\right|\right) \cdot \left(x \cdot x\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.047619047619047616\right) \cdot \frac{\left|x\right|}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+ (fabs x) (+ (fabs x) (* (* 0.6666666666666666 (fabs x)) (* x x))))))
   (fabs
    (*
     (* (* (* (* (* (* x x) x) x) x) x) 0.047619047619047616)
     (/ (fabs x) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = fabs(((1.0 / sqrt(((double) M_PI))) * (fabs(x) + (fabs(x) + ((0.6666666666666666 * fabs(x)) * (x * x))))));
	} else {
		tmp = fabs((((((((x * x) * x) * x) * x) * x) * 0.047619047619047616) * (fabs(x) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = Math.abs(((1.0 / Math.sqrt(Math.PI)) * (Math.abs(x) + (Math.abs(x) + ((0.6666666666666666 * Math.abs(x)) * (x * x))))));
	} else {
		tmp = Math.abs((((((((x * x) * x) * x) * x) * x) * 0.047619047619047616) * (Math.abs(x) / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.2:
		tmp = math.fabs(((1.0 / math.sqrt(math.pi)) * (math.fabs(x) + (math.fabs(x) + ((0.6666666666666666 * math.fabs(x)) * (x * x))))))
	else:
		tmp = math.fabs((((((((x * x) * x) * x) * x) * x) * 0.047619047619047616) * (math.fabs(x) / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(abs(x) + Float64(abs(x) + Float64(Float64(0.6666666666666666 * abs(x)) * Float64(x * x))))));
	else
		tmp = abs(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * x) * x) * x) * x) * x) * 0.047619047619047616) * Float64(abs(x) / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.2)
		tmp = abs(((1.0 / sqrt(pi)) * (abs(x) + (abs(x) + ((0.6666666666666666 * abs(x)) * (x * x))))));
	else
		tmp = abs((((((((x * x) * x) * x) * x) * x) * 0.047619047619047616) * (abs(x) / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] + N[(N[(0.6666666666666666 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 0.047619047619047616), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. lower-fma.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-fabs.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-fabs.f64 88.9

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

   6. Applied rewrites 88.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. associate-+l+ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 88.9

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

   8. Applied rewrites 88.9%

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

   if 2.2000000000000002 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-fabs.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-PI.f64 37.6

         \[\leadsto \left|0.047619047619047616 \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

   6. Applied rewrites 37.6%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}}\right| \]

   8. Applied rewrites 37.6%

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

Alternative 9: 88.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

   \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \frac{1}{21}\right), \left|x\right| \cdot \color{blue}{2}\right)\right|}{\sqrt{\pi}} \]
5. Step-by-step derivation

6. Applied rewrites 98.5%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 98.5

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

8. Applied rewrites 98.6%

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

Alternative 10: 88.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

   \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(\frac{1}{5} \cdot \left(x \cdot x\right), x \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \frac{1}{21}\right), \left|x\right| \cdot \color{blue}{2}\right)\right|}{\sqrt{\pi}} \]
5. Step-by-step derivation

6. Applied rewrites 98.5%

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

8. Applied rewrites 98.6%

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

Alternative 11: 88.9% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around 0

   \[\leadsto \left|\frac{\mathsf{fma}\left(\frac{1}{21} \cdot \left(x \cdot x\right), \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{2}{3}}, 2\right)\right) \cdot \left|x\right|}{\sqrt{\pi}}\right| \]
9. Step-by-step derivation

10. Applied rewrites 98.6%

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

Alternative 12: 88.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. lower-fma.f64 N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-fabs.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-fabs.f64 88.9

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

6. Applied rewrites 88.9%

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

   1. lift-fabs.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

8. Applied rewrites 88.4%

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

   1. rem-square-sqrt N/A

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

   2. sqrt-unprod N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lower-sqrt.f64 91.1

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

10. Applied rewrites 91.1%

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

Alternative 13: 88.9% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. lower-fma.f64 N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-fabs.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-fabs.f64 88.9

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

6. Applied rewrites 88.9%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow2 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{2}{3} \cdot \left(\left(x \cdot x\right) \cdot \left|x\right|\right) + 2 \cdot \left|x\right|\right)\right| \]

   5. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{2}{3} \cdot \left(\left(x \cdot x\right) \cdot \left|x\right|\right) + 2 \cdot \left|x\right|\right)\right| \]

   6. associate-*r* N/A

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

   7. *-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(x \cdot x\right) \cdot \frac{2}{3}\right) \cdot \left|x\right| + 2 \cdot \left|x\right|\right)\right| \]

   8. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(x \cdot x\right) \cdot \frac{2}{3}\right) \cdot \left|x\right| + 2 \cdot \left|x\right|\right)\right| \]

   9. *-commutative N/A

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

   10. associate-*l* N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left|x\right| \cdot \left(x \cdot \left(x \cdot \frac{2}{3}\right)\right) + 2 \cdot \left|x\right|\right)\right| \]

   11. associate-*r* N/A

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

   12. *-commutative N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left|x\right| \cdot x\right) \cdot \left(\frac{2}{3} \cdot x\right) + 2 \cdot \left|x\right|\right)\right| \]

   13. lower-fma.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 88.9

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

8. Applied rewrites 88.9%

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

Alternative 14: 88.4% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. lower-fma.f64 N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-fabs.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-fabs.f64 88.9

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

6. Applied rewrites 88.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. div-flip N/A

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

   5. lower-/.f64 N/A

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

8. Applied rewrites 88.4%

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

Alternative 15: 88.4% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. lower-fma.f64 N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-fabs.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-fabs.f64 88.9

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

6. Applied rewrites 88.9%

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

   1. lift-fabs.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

8. Applied rewrites 88.4%

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

Alternative 16: 82.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{if}\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \leq 4 \cdot 10^{-37}:\\ \;\;\;\;\left|\left|x\right| \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|2 \cdot \frac{\sqrt{x \cdot x}}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (if (<=
        (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))))))
        4e-37)
     (fabs (* (fabs x) (/ 2.0 (sqrt PI))))
     (fabs (* 2.0 (/ (sqrt (* x x)) (sqrt PI)))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	double tmp;
	if (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)))))) <= 4e-37) {
		tmp = fabs((fabs(x) * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((2.0 * (sqrt((x * x)) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

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);
	double tmp;
	if (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)))))) <= 4e-37) {
		tmp = Math.abs((Math.abs(x) * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs((2.0 * (Math.sqrt((x * x)) / Math.sqrt(Math.PI))));
	}
	return tmp;
}

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)
	tmp = 0
	if 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)))))) <= 4e-37:
		tmp = math.fabs((math.fabs(x) * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs((2.0 * (math.sqrt((x * x)) / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	tmp = 0.0
	if (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)))))) <= 4e-37)
		tmp = abs(Float64(abs(x) * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(Float64(2.0 * Float64(sqrt(Float64(x * x)) / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = 0.0;
	if (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)))))) <= 4e-37)
		tmp = abs((abs(x) * (2.0 / sqrt(pi))));
	else
		tmp = abs((2.0 * (sqrt((x * x)) / sqrt(pi))));
	end
	tmp_2 = tmp;
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]}, If[LessEqual[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], 4e-37], N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(2.0 * N[(N[Sqrt[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))))) < 4.00000000000000027e-37

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 66.5

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

   6. Applied rewrites 66.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 67.0

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

   8. Applied rewrites 67.0%

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

   if 4.00000000000000027e-37 < (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))))))

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 66.5

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

   6. Applied rewrites 66.5%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. lift-*.f64 N/A

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

      4. lower-sqrt.f64 52.3

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

   8. Applied rewrites 52.3%

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

Alternative 17: 67.0% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-fabs.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-PI.f64 66.5

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

6. Applied rewrites 66.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 67.0

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

8. Applied rewrites 67.0%

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

Alternative 18: 66.5% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-fabs.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-PI.f64 66.5

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

6. Applied rewrites 66.5%

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

Reproduce

herbie shell --seed 2025148 
(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)))))))