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

Percentage Accurate: 99.8% → 99.8%

Time: 4.7s

Alternatives: 15

Speedup: 1.5×

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

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{\pi}}\\ t_1 := \left(\left(x \cdot x\right) \cdot x\right) \cdot x\\ \left|\mathsf{fma}\left(t\_0 \cdot \left(\left(\left(t\_1 \cdot \left|x\right|\right) \cdot x\right) \cdot x\right), 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, t\_1, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right) \cdot t\_0\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt PI))) (t_1 (* (* (* x x) x) x)))
   (fabs
    (fma
     (* t_0 (* (* (* t_1 (fabs x)) x) x))
     0.047619047619047616
     (*
      (fma
       (* 0.2 (fabs x))
       t_1
       (* (fabs x) (fma (* x x) 0.6666666666666666 2.0)))
      t_0)))))

C

double code(double x) {
	double t_0 = 1.0 / sqrt(((double) M_PI));
	double t_1 = ((x * x) * x) * x;
	return fabs(fma((t_0 * (((t_1 * fabs(x)) * x) * x)), 0.047619047619047616, (fma((0.2 * fabs(x)), t_1, (fabs(x) * fma((x * x), 0.6666666666666666, 2.0))) * t_0)));
}

Julia

function code(x)
	t_0 = Float64(1.0 / sqrt(pi))
	t_1 = Float64(Float64(Float64(x * x) * x) * x)
	return abs(fma(Float64(t_0 * Float64(Float64(Float64(t_1 * abs(x)) * x) * x)), 0.047619047619047616, Float64(fma(Float64(0.2 * abs(x)), t_1, Float64(abs(x) * fma(Float64(x * x), 0.6666666666666666, 2.0))) * t_0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, N[Abs[N[(N[(t$95$0 * N[(N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * 0.047619047619047616 + N[(N[(N[(0.2 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $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%

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

Alternative 2: 99.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 99.8%

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{\pi}}\\ \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\mathsf{fma}\left(2 \cdot t\_0, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666666\right) \cdot t\_0\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left|{x}^{7} \cdot 0.047619047619047616\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt PI))))
   (if (<= x 2.2)
     (fabs (fma (* 2.0 t_0) x (* (* (* (* x x) x) 0.6666666666666666) t_0)))
     (* t_0 (fabs (* (pow x 7.0) 0.047619047619047616))))))

C

double code(double x) {
	double t_0 = 1.0 / sqrt(((double) M_PI));
	double tmp;
	if (x <= 2.2) {
		tmp = fabs(fma((2.0 * t_0), x, ((((x * x) * x) * 0.6666666666666666) * t_0)));
	} else {
		tmp = t_0 * fabs((pow(x, 7.0) * 0.047619047619047616));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 / sqrt(pi))
	tmp = 0.0
	if (x <= 2.2)
		tmp = abs(fma(Float64(2.0 * t_0), x, Float64(Float64(Float64(Float64(x * x) * x) * 0.6666666666666666) * t_0)));
	else
		tmp = Float64(t_0 * abs(Float64((x ^ 7.0) * 0.047619047619047616)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.2], N[Abs[N[(N[(2.0 * t$95$0), $MachinePrecision] * x + N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * 0.6666666666666666), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$0 * N[Abs[N[(N[Power[x, 7.0], $MachinePrecision] * 0.047619047619047616), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 89.6%

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

   if 2.2000000000000002 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 36.3%

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

   7. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. metadata-eval 36.3

         \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|{x}^{7} \cdot 0.047619047619047616\right| \]

   9. Applied rewrites 36.3%

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|{x}^{7} \cdot \color{blue}{0.047619047619047616}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{\pi}}\\ \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\left|\left(x + x\right) \cdot t\_0\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left|{x}^{7} \cdot 0.047619047619047616\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt PI))))
   (if (<= x 1.9)
     (fabs (* (+ x x) t_0))
     (* t_0 (fabs (* (pow x 7.0) 0.047619047619047616))))))

C

double code(double x) {
	double t_0 = 1.0 / sqrt(((double) M_PI));
	double tmp;
	if (x <= 1.9) {
		tmp = fabs(((x + x) * t_0));
	} else {
		tmp = t_0 * fabs((pow(x, 7.0) * 0.047619047619047616));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.sqrt(Math.PI);
	double tmp;
	if (x <= 1.9) {
		tmp = Math.abs(((x + x) * t_0));
	} else {
		tmp = t_0 * Math.abs((Math.pow(x, 7.0) * 0.047619047619047616));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 / math.sqrt(math.pi)
	tmp = 0
	if x <= 1.9:
		tmp = math.fabs(((x + x) * t_0))
	else:
		tmp = t_0 * math.fabs((math.pow(x, 7.0) * 0.047619047619047616))
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 / sqrt(pi))
	tmp = 0.0
	if (x <= 1.9)
		tmp = abs(Float64(Float64(x + x) * t_0));
	else
		tmp = Float64(t_0 * abs(Float64((x ^ 7.0) * 0.047619047619047616)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 / sqrt(pi);
	tmp = 0.0;
	if (x <= 1.9)
		tmp = abs(((x + x) * t_0));
	else
		tmp = t_0 * abs(((x ^ 7.0) * 0.047619047619047616));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.9], N[Abs[N[(N[(x + x), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision], N[(t$95$0 * N[Abs[N[(N[Power[x, 7.0], $MachinePrecision] * 0.047619047619047616), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   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%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 68.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. unpow1 N/A

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

      6. metadata-eval N/A

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

      7. sqrt-pow1 N/A

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

      8. pow2 N/A

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

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

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

      10. lower-*.f64 N/A

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

   8. Applied rewrites 68.3%

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

   if 1.8999999999999999 < 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

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

   6. Applied rewrites 36.3%

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

   7. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. metadata-eval 36.3

         \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|{x}^{7} \cdot 0.047619047619047616\right| \]

   9. Applied rewrites 36.3%

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|{x}^{7} \cdot \color{blue}{0.047619047619047616}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 92.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\left|\left(x + x\right) \cdot \frac{1}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.047619047619047616\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.9)
   (fabs (* (+ x x) (/ 1.0 (sqrt PI))))
   (/
    (fabs (* (* (* (* (* (* x x) x) x) x) (* x x)) 0.047619047619047616))
    (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = fabs(((x + x) * (1.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs(((((((x * x) * x) * x) * x) * (x * x)) * 0.047619047619047616)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = Math.abs(((x + x) * (1.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs(((((((x * x) * x) * x) * x) * (x * x)) * 0.047619047619047616)) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.9:
		tmp = math.fabs(((x + x) * (1.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs(((((((x * x) * x) * x) * x) * (x * x)) * 0.047619047619047616)) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.9)
		tmp = abs(Float64(Float64(x + x) * Float64(1.0 / sqrt(pi))));
	else
		tmp = Float64(abs(Float64(Float64(Float64(Float64(Float64(Float64(x * x) * x) * x) * x) * Float64(x * x)) * 0.047619047619047616)) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.9)
		tmp = abs(((x + x) * (1.0 / sqrt(pi))));
	else
		tmp = abs(((((((x * x) * x) * x) * x) * (x * x)) * 0.047619047619047616)) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.9], N[Abs[N[(N[(x + x), $MachinePrecision] * N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.047619047619047616), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   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%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 68.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. unpow1 N/A

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

      6. metadata-eval N/A

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

      7. sqrt-pow1 N/A

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

      8. pow2 N/A

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

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

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

      10. lower-*.f64 N/A

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

   8. Applied rewrites 68.3%

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

   if 1.8999999999999999 < 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%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 36.3%

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

   8. Applied rewrites 36.3%

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

Alternative 7: 89.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\left|\left(x + x\right) \cdot \frac{1}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|{x}^{7} \cdot 0.047619047619047616\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.9)
   (fabs (* (+ x x) (/ 1.0 (sqrt PI))))
   (/ (fabs (* (pow x 7.0) 0.047619047619047616)) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = fabs(((x + x) * (1.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((pow(x, 7.0) * 0.047619047619047616)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = Math.abs(((x + x) * (1.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs((Math.pow(x, 7.0) * 0.047619047619047616)) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.9:
		tmp = math.fabs(((x + x) * (1.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs((math.pow(x, 7.0) * 0.047619047619047616)) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.9)
		tmp = abs(Float64(Float64(x + x) * Float64(1.0 / sqrt(pi))));
	else
		tmp = Float64(abs(Float64((x ^ 7.0) * 0.047619047619047616)) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.9)
		tmp = abs(((x + x) * (1.0 / sqrt(pi))));
	else
		tmp = abs(((x ^ 7.0) * 0.047619047619047616)) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.9], N[Abs[N[(N[(x + x), $MachinePrecision] * N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(N[Power[x, 7.0], $MachinePrecision] * 0.047619047619047616), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   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%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 68.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. unpow1 N/A

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

      6. metadata-eval N/A

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

      7. sqrt-pow1 N/A

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

      8. pow2 N/A

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

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

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

      10. lower-*.f64 N/A

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

   8. Applied rewrites 68.3%

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

   if 1.8999999999999999 < 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%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 36.3%

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

   7. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. metadata-eval 36.3

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

   9. Applied rewrites 36.3%

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

Alternative 8: 89.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 99.4%

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

Alternative 9: 68.3% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 99.4%

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

7. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

9. Applied rewrites 99.4%

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

    1. metadata-eval N/A

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

    2. metadata-eval N/A

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

    3. metadata-eval N/A

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

11. Applied rewrites 99.4%

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

Alternative 10: 68.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8:\\ \;\;\;\;\left|\left(x + x\right) \cdot \frac{1}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.2\right) \cdot x\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.8)
   (fabs (* (+ x x) (/ 1.0 (sqrt PI))))
   (/ (fabs (* (* (* (* (* x x) x) x) 0.2) x)) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = fabs(((x + x) * (1.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((((((x * x) * x) * x) * 0.2) * x)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = Math.abs(((x + x) * (1.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs((((((x * x) * x) * x) * 0.2) * x)) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.8:
		tmp = math.fabs(((x + x) * (1.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs((((((x * x) * x) * x) * 0.2) * x)) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.8)
		tmp = abs(Float64(Float64(x + x) * Float64(1.0 / sqrt(pi))));
	else
		tmp = Float64(abs(Float64(Float64(Float64(Float64(Float64(x * x) * x) * x) * 0.2) * x)) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.8)
		tmp = abs(((x + x) * (1.0 / sqrt(pi))));
	else
		tmp = abs((((((x * x) * x) * x) * 0.2) * x)) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.8], N[Abs[N[(N[(x + x), $MachinePrecision] * N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 0.2), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.80000000000000004

   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%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 68.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. unpow1 N/A

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

      6. metadata-eval N/A

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

      7. sqrt-pow1 N/A

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

      8. pow2 N/A

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

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

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

      10. lower-*.f64 N/A

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

   8. Applied rewrites 68.3%

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

   if 1.80000000000000004 < 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%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   9. Applied rewrites 93.3%

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

   10. Taylor expanded in x around inf

       \[\leadsto \frac{\left|\left(\frac{1}{5} \cdot {x}^{4}\right) \cdot x\right|}{\sqrt{\pi}} \]
   11. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{\left|\left(\frac{1}{5} \cdot {x}^{4}\right) \cdot x\right|}{\sqrt{\pi}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\left|\left({x}^{4} \cdot \frac{1}{5}\right) \cdot x\right|}{\sqrt{\pi}} \]

       3. lower-*.f64 N/A

          \[\leadsto \frac{\left|\left({x}^{4} \cdot \frac{1}{5}\right) \cdot x\right|}{\sqrt{\pi}} \]

       4. sqr-pow N/A

          \[\leadsto \frac{\left|\left(\left({x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)}\right) \cdot \frac{1}{5}\right) \cdot x\right|}{\sqrt{\pi}} \]

       5. metadata-eval N/A

          \[\leadsto \frac{\left|\left(\left({x}^{2} \cdot {x}^{\left(\frac{4}{2}\right)}\right) \cdot \frac{1}{5}\right) \cdot x\right|}{\sqrt{\pi}} \]

       6. pow2 N/A

          \[\leadsto \frac{\left|\left(\left(\left(x \cdot x\right) \cdot {x}^{\left(\frac{4}{2}\right)}\right) \cdot \frac{1}{5}\right) \cdot x\right|}{\sqrt{\pi}} \]

       7. metadata-eval N/A

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

       8. pow2 N/A

          \[\leadsto \frac{\left|\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{5}\right) \cdot x\right|}{\sqrt{\pi}} \]

       9. associate-*l* N/A

          \[\leadsto \frac{\left|\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \frac{1}{5}\right) \cdot x\right|}{\sqrt{\pi}} \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{\left|\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \frac{1}{5}\right) \cdot x\right|}{\sqrt{\pi}} \]

       11. lower-*.f64 N/A

           \[\leadsto \frac{\left|\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \frac{1}{5}\right) \cdot x\right|}{\sqrt{\pi}} \]

       12. lift-*.f64 N/A

           \[\leadsto \frac{\left|\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \frac{1}{5}\right) \cdot x\right|}{\sqrt{\pi}} \]

       13. metadata-eval 30.9

           \[\leadsto \frac{\left|\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.2\right) \cdot x\right|}{\sqrt{\pi}} \]

   12. Applied rewrites 30.9%

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

Alternative 11: 68.3% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 99.4%

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

7. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

9. Applied rewrites 93.3%

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

10. Taylor expanded in x around inf

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

    1. metadata-eval N/A

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

    2. pow2 N/A

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

    3. associate-*l* N/A

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

    4. lower-*.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. metadata-eval 92.9

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

12. Applied rewrites 92.9%

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

Alternative 12: 68.3% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \frac{1}{5} \cdot {\color{blue}{x}}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, {x}^{4} \cdot \color{blue}{\frac{1}{5}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, {x}^{4} \cdot \color{blue}{\frac{1}{5}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   4. metadata-eval N/A

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

   5. pow-plus N/A

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

   6. pow3 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. metadata-eval 93.3

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

6. Applied rewrites 93.3%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 89.2%

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

Alternative 13: 68.3% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\left|\left(x + x\right) \cdot \frac{1}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(x + x\right) \cdot \left(x + x\right)}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e-10)
   (fabs (* (+ x x) (/ 1.0 (sqrt PI))))
   (/ (sqrt (* (+ x x) (+ x x))) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (x <= 2e-10) {
		tmp = fabs(((x + x) * (1.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = sqrt(((x + x) * (x + x))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2e-10) {
		tmp = Math.abs(((x + x) * (1.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.sqrt(((x + x) * (x + x))) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2e-10:
		tmp = math.fabs(((x + x) * (1.0 / math.sqrt(math.pi))))
	else:
		tmp = math.sqrt(((x + x) * (x + x))) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e-10)
		tmp = abs(Float64(Float64(x + x) * Float64(1.0 / sqrt(pi))));
	else
		tmp = Float64(sqrt(Float64(Float64(x + x) * Float64(x + x))) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2e-10)
		tmp = abs(((x + x) * (1.0 / sqrt(pi))));
	else
		tmp = sqrt(((x + x) * (x + x))) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2e-10], N[Abs[N[(N[(x + x), $MachinePrecision] * N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(x + x), $MachinePrecision] * N[(x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.00000000000000007e-10

   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%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 68.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. unpow1 N/A

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

      6. metadata-eval N/A

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

      7. sqrt-pow1 N/A

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

      8. pow2 N/A

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

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

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

      10. lower-*.f64 N/A

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

   8. Applied rewrites 68.3%

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

   if 2.00000000000000007e-10 < 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%

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

   4. Taylor expanded in x around 0

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

      1. metadata-eval N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \frac{1}{5} \cdot {\color{blue}{x}}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, {x}^{4} \cdot \color{blue}{\frac{1}{5}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, {x}^{4} \cdot \color{blue}{\frac{1}{5}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

      4. metadata-eval N/A

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

      5. pow-plus N/A

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

      6. pow3 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. metadata-eval 93.3

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

   6. Applied rewrites 93.3%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 67.9%

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

   11. Applied rewrites 53.2%

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

Alternative 14: 68.3% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 68.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. unpow1 N/A

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

   6. metadata-eval N/A

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

   7. sqrt-pow1 N/A

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

   8. pow2 N/A

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

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

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

   10. lower-*.f64 N/A

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

8. Applied rewrites 68.3%

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

Alternative 15: 67.9% accurate, 9.4× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (fabs (+ x x)) (sqrt PI)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Abs[N[(x + 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%

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

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \frac{1}{5} \cdot {\color{blue}{x}}^{4}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, {x}^{4} \cdot \color{blue}{\frac{1}{5}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, {x}^{4} \cdot \color{blue}{\frac{1}{5}}, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right)\right)\right|}{\sqrt{\pi}} \]

   4. metadata-eval N/A

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

   5. pow-plus N/A

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

   6. pow3 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. metadata-eval 93.3

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

6. Applied rewrites 93.3%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 67.9%

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

Reproduce

herbie shell --seed 2025130 
(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)))))))