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

Percentage Accurate: 99.8% → 99.9%

Time: 4.8s

Alternatives: 15

Speedup: 1.2×

• 28.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{5}, \mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right|\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt PI))
   (fma
    0.047619047619047616
    (pow (fabs x) 7.0)
    (fma
     0.2
     (pow (fabs x) 5.0)
     (fma 0.6666666666666666 (pow (fabs x) 3.0) (* 2.0 (fabs x))))))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * fma(0.047619047619047616, pow(fabs(x), 7.0), fma(0.2, pow(fabs(x), 5.0), fma(0.6666666666666666, pow(fabs(x), 3.0), (2.0 * fabs(x)))))));
}

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * fma(0.047619047619047616, (abs(x) ^ 7.0), fma(0.2, (abs(x) ^ 5.0), fma(0.6666666666666666, (abs(x) ^ 3.0), Float64(2.0 * abs(x)))))))
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(0.047619047619047616 * N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision] + N[(0.2 * N[Power[N[Abs[x], $MachinePrecision], 5.0], $MachinePrecision] + N[(0.6666666666666666 * N[Power[N[Abs[x], $MachinePrecision], 3.0], $MachinePrecision] + N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

2. 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%

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

Alternative 2: 99.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

Alternative 3: 99.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

Alternative 4: 99.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|\frac{2}{\sqrt{\pi}} \cdot \left(-x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (fabs (* (/ 2.0 (sqrt PI)) (- x)))
   (*
    (/ 1.0 (sqrt PI))
    (fabs (* 0.047619047619047616 (* (pow x 6.0) (fabs x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = fabs(((2.0 / sqrt(((double) M_PI))) * -x));
	} else {
		tmp = (1.0 / sqrt(((double) M_PI))) * fabs((0.047619047619047616 * (pow(x, 6.0) * fabs(x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = Math.abs(((2.0 / Math.sqrt(Math.PI)) * -x));
	} else {
		tmp = (1.0 / Math.sqrt(Math.PI)) * Math.abs((0.047619047619047616 * (Math.pow(x, 6.0) * Math.abs(x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = math.fabs(((2.0 / math.sqrt(math.pi)) * -x))
	else:
		tmp = (1.0 / math.sqrt(math.pi)) * math.fabs((0.047619047619047616 * (math.pow(x, 6.0) * math.fabs(x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = abs(Float64(Float64(2.0 / sqrt(pi)) * Float64(-x)));
	else
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(0.047619047619047616 * Float64((x ^ 6.0) * abs(x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = abs(((2.0 / sqrt(pi)) * -x));
	else
		tmp = (1.0 / sqrt(pi)) * abs((0.047619047619047616 * ((x ^ 6.0) * abs(x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[Abs[N[(N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * (-x)), $MachinePrecision]], $MachinePrecision], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(0.047619047619047616 * N[(N[Power[x, 6.0], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   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%

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

   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 67.3

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

   6. Applied rewrites 67.3%

      \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\pi}}}\right| \]
   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 67.8

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

   8. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.8

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

      4. lift-fabs.f64 N/A

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

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

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

      6. sqr-neg-rev N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sqrt-unprod N/A

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

      10. rem-square-sqrt 67.8

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

   10. Applied rewrites 67.8%

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

   if 1.8500000000000001 < 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%

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

   4. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-fabs.f64 36.7

         \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right| \]

   6. Applied rewrites 36.7%

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\color{blue}{0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|\frac{2}{\sqrt{\pi}} \cdot \left(-x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|0.047619047619047616 \cdot {x}^{7}\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (fabs (* (/ 2.0 (sqrt PI)) (- x)))
   (/ (fabs (* 0.047619047619047616 (pow x 7.0))) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = fabs(((2.0 / sqrt(((double) M_PI))) * -x));
	} else {
		tmp = fabs((0.047619047619047616 * pow(x, 7.0))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = Math.abs(((2.0 / Math.sqrt(Math.PI)) * -x));
	} else {
		tmp = Math.abs((0.047619047619047616 * Math.pow(x, 7.0))) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = math.fabs(((2.0 / math.sqrt(math.pi)) * -x))
	else:
		tmp = math.fabs((0.047619047619047616 * math.pow(x, 7.0))) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = abs(Float64(Float64(2.0 / sqrt(pi)) * Float64(-x)));
	else
		tmp = Float64(abs(Float64(0.047619047619047616 * (x ^ 7.0))) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = abs(((2.0 / sqrt(pi)) * -x));
	else
		tmp = abs((0.047619047619047616 * (x ^ 7.0))) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[Abs[N[(N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * (-x)), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   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%

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

   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 67.3

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

   6. Applied rewrites 67.3%

      \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\pi}}}\right| \]
   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 67.8

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

   8. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.8

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

      4. lift-fabs.f64 N/A

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

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

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

      6. sqr-neg-rev N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sqrt-unprod N/A

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

      10. rem-square-sqrt 67.8

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

   10. Applied rewrites 67.8%

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

   if 1.8500000000000001 < 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 99.4%

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\left|\color{blue}{\frac{1}{21} \cdot {x}^{7}}\right|}{\sqrt{\pi}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\left|\frac{1}{21} \cdot \color{blue}{{x}^{7}}\right|}{\sqrt{\pi}} \]

      2. lower-pow.f64 36.7

         \[\leadsto \frac{\left|0.047619047619047616 \cdot {x}^{\color{blue}{7}}\right|}{\sqrt{\pi}} \]

   10. Applied rewrites 36.7%

       \[\leadsto \frac{\left|\color{blue}{0.047619047619047616 \cdot {x}^{7}}\right|}{\sqrt{\pi}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

5. Applied rewrites 99.4%

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

Alternative 7: 98.7% accurate, 2.1× speedup

Math

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

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))
    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)) * x)) / sqrt(((double) M_PI));
}

Julia

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

Wolfram

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

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

5. Applied rewrites 99.4%

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

7. Applied rewrites 99.4%

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

Alternative 8: 89.6% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Applied rewrites 99.4%

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.4%

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

Alternative 9: 89.1% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fabs
   (*
    (fma
     (* 0.047619047619047616 (* x x))
     (* (* (* x x) x) x)
     (fma (* x x) 0.6666666666666666 2.0))
    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)) * x)) / sqrt(((double) M_PI));
}

Julia

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

Wolfram

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

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

5. Applied rewrites 99.4%

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 98.7%

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

Alternative 10: 89.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{\pi}} \cdot \left|\left(\left(-x\right) \cdot 0.6666666666666666\right) \cdot \left(x \cdot x\right) - \left(x + x\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ 1.0 (sqrt PI))
  (fabs (- (* (* (- x) 0.6666666666666666) (* x x)) (+ x x)))))

C

double code(double x) {
	return (1.0 / sqrt(((double) M_PI))) * fabs((((-x * 0.6666666666666666) * (x * x)) - (x + x)));
}

Java

public static double code(double x) {
	return (1.0 / Math.sqrt(Math.PI)) * Math.abs((((-x * 0.6666666666666666) * (x * x)) - (x + x)));
}

Python

def code(x):
	return (1.0 / math.sqrt(math.pi)) * math.fabs((((-x * 0.6666666666666666) * (x * x)) - (x + x)))

Julia

function code(x)
	return Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(Float64(Float64(Float64(-x) * 0.6666666666666666) * Float64(x * x)) - Float64(x + x))))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / sqrt(pi)) * abs((((-x * 0.6666666666666666) * (x * x)) - (x + x)));
end

Wolfram

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

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

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. lower-fma.f64 N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-fabs.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-fabs.f64 89.6

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

6. Applied rewrites 89.6%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. associate-*l* N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 89.6

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

8. Applied rewrites 89.6%

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

   1. lift-fma.f64 N/A

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

   2. add-flip N/A

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

   3. lower--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lift-fabs.f64 N/A

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

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

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

   17. sqr-neg-rev N/A

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

   18. lift-neg.f64 N/A

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

   19. lift-neg.f64 N/A

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

   20. sqrt-unprod N/A

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

   21. rem-square-sqrt N/A

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

   22. lift-*.f64 N/A

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

   23. count-2-rev N/A

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

10. Applied rewrites 89.6%

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

Alternative 11: 83.4% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(1.0 / N[(N[Sqrt[Pi], $MachinePrecision] / N[Abs[N[((-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%

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

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. lower-fma.f64 N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-fabs.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-fabs.f64 89.6

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

6. Applied rewrites 89.6%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. associate-*l* N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 89.6

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

8. Applied rewrites 89.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. div-flip N/A

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

   5. lower-/.f64 N/A

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

10. Applied rewrites 89.1%

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

Alternative 12: 67.8% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. lower-fma.f64 N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-fabs.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-fabs.f64 89.6

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

6. Applied rewrites 89.6%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. associate-*l* N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 89.6

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

8. Applied rewrites 89.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip-rev N/A

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

   5. lower-/.f64 89.1

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

10. Applied rewrites 89.1%

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

Alternative 13: 67.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{if}\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \leq 10^{+89}:\\ \;\;\;\;\left|\frac{2}{\sqrt{\pi}} \cdot \left(-x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|2 \cdot \sqrt{\frac{x \cdot x}{\pi}}\right|\\ \end{array} \end{array} \]

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))))))
        1e+89)
     (fabs (* (/ 2.0 (sqrt PI)) (- x)))
     (fabs (* 2.0 (sqrt (/ (* x x) 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)))))) <= 1e+89) {
		tmp = fabs(((2.0 / sqrt(((double) M_PI))) * -x));
	} else {
		tmp = fabs((2.0 * sqrt(((x * x) / ((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)))))) <= 1e+89) {
		tmp = Math.abs(((2.0 / Math.sqrt(Math.PI)) * -x));
	} else {
		tmp = Math.abs((2.0 * Math.sqrt(((x * x) / 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)))))) <= 1e+89:
		tmp = math.fabs(((2.0 / math.sqrt(math.pi)) * -x))
	else:
		tmp = math.fabs((2.0 * math.sqrt(((x * x) / 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)))))) <= 1e+89)
		tmp = abs(Float64(Float64(2.0 / sqrt(pi)) * Float64(-x)));
	else
		tmp = abs(Float64(2.0 * sqrt(Float64(Float64(x * x) / 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)))))) <= 1e+89)
		tmp = abs(((2.0 / sqrt(pi)) * -x));
	else
		tmp = abs((2.0 * sqrt(((x * x) / 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], 1e+89], N[Abs[N[(N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * (-x)), $MachinePrecision]], $MachinePrecision], N[Abs[N[(2.0 * N[Sqrt[N[(N[(x * x), $MachinePrecision] / 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)))))) < 9.99999999999999995e88

   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%

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

   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 67.3

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

   6. Applied rewrites 67.3%

      \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\pi}}}\right| \]
   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 67.8

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

   8. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.8

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

      4. lift-fabs.f64 N/A

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

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

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

      6. sqr-neg-rev N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sqrt-unprod N/A

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

      10. rem-square-sqrt 67.8

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

   10. Applied rewrites 67.8%

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

   if 9.99999999999999995e88 < (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%

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

   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 67.3

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

   6. Applied rewrites 67.3%

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

      1. lift-/.f64 N/A

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

      2. lift-fabs.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-undiv N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 53.5

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

   8. Applied rewrites 53.5%

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

Alternative 14: 67.8% accurate, 8.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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 67.3

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

6. Applied rewrites 67.3%

   \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\pi}}}\right| \]
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 67.8

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

8. Applied rewrites 67.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 67.8

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

   4. lift-fabs.f64 N/A

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

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

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

   6. sqr-neg-rev N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-neg.f64 N/A

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

   9. sqrt-unprod N/A

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

   10. rem-square-sqrt 67.8

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

10. Applied rewrites 67.8%

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

Alternative 15: 67.3% accurate, 9.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Applied rewrites 99.4%

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 67.3%

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

Reproduce

herbie shell --seed 2025145 
(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)))))))