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

Percentage Accurate: 99.8% → 99.8%

Time: 12.1s

Alternatives: 11

Speedup: 3.0×

• 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, 2.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left|x\_m\right| \cdot \left(x\_m \cdot x\_m\right)\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\_m\right|, 0.6666666666666666 \cdot t\_0\right) + 0.2 \cdot {x\_m}^{5}\right) + 0.047619047619047616 \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left(\left(x\_m \cdot x\_m\right) \cdot t\_0\right)\right)\right)\right| \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (fabs x_m) (* x_m x_m))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (fma 2.0 (fabs x_m) (* 0.6666666666666666 t_0)) (* 0.2 (pow x_m 5.0)))
      (* 0.047619047619047616 (* (* x_m x_m) (* (* x_m x_m) t_0))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fabs(x_m) * (x_m * x_m);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((fma(2.0, fabs(x_m), (0.6666666666666666 * t_0)) + (0.2 * pow(x_m, 5.0))) + (0.047619047619047616 * ((x_m * x_m) * ((x_m * x_m) * t_0))))));
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(abs(x_m) * Float64(x_m * x_m))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(fma(2.0, abs(x_m), Float64(0.6666666666666666 * t_0)) + Float64(0.2 * (x_m ^ 5.0))) + Float64(0.047619047619047616 * Float64(Float64(x_m * x_m) * Float64(Float64(x_m * x_m) * t_0))))))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Abs[x$95$m], $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(2.0 * N[Abs[x$95$m], $MachinePrecision] + N[(0.6666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + \color{blue}{0.2 \cdot \left({x}^{4} \cdot \left|x\right|\right)}\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]
6. Step-by-step derivation

   1. rem-square-sqrt 34.0%

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

   2. fabs-sqr 34.0%

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

   3. rem-square-sqrt 77.6%

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

   4. pow-plus 77.6%

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

   5. metadata-eval 77.6%

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

7. Simplified 77.6%

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

8. Final simplification 77.6%

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

Alternative 2: 99.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \left|x\_m\right| \cdot \left|\frac{{x\_m}^{4} \cdot \left(0.2 + 0.047619047619047616 \cdot {x\_m}^{2}\right) + \mathsf{fma}\left(0.6666666666666666, x\_m \cdot x\_m, 2\right)}{\sqrt{\pi}}\right| \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (fabs x_m)
  (fabs
   (/
    (+
     (* (pow x_m 4.0) (+ 0.2 (* 0.047619047619047616 (pow x_m 2.0))))
     (fma 0.6666666666666666 (* x_m x_m) 2.0))
    (sqrt PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	return fabs(x_m) * fabs((((pow(x_m, 4.0) * (0.2 + (0.047619047619047616 * pow(x_m, 2.0)))) + fma(0.6666666666666666, (x_m * x_m), 2.0)) / sqrt(((double) M_PI))));
}

Julia

x_m = abs(x)
function code(x_m)
	return Float64(abs(x_m) * abs(Float64(Float64(Float64((x_m ^ 4.0) * Float64(0.2 + Float64(0.047619047619047616 * (x_m ^ 2.0)))) + fma(0.6666666666666666, Float64(x_m * x_m), 2.0)) / sqrt(pi))))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Abs[x$95$m], $MachinePrecision] * N[Abs[N[(N[(N[(N[Power[x$95$m, 4.0], $MachinePrecision] * N[(0.2 + N[(0.047619047619047616 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 99.9%

   \[\leadsto \left|x\right| \cdot \left|\frac{\color{blue}{{x}^{4} \cdot \left(0.2 + 0.047619047619047616 \cdot {x}^{2}\right)} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]
6. Step-by-step derivation

   1. *-commutative 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 3: 99.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \left|x\_m\right| \cdot \left|\frac{\mathsf{fma}\left(0.6666666666666666, x\_m \cdot x\_m, 2\right) + 0.047619047619047616 \cdot {x\_m}^{6}}{\sqrt{\pi}}\right| \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (fabs x_m)
  (fabs
   (/
    (+
     (fma 0.6666666666666666 (* x_m x_m) 2.0)
     (* 0.047619047619047616 (pow x_m 6.0)))
    (sqrt PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	return fabs(x_m) * fabs(((fma(0.6666666666666666, (x_m * x_m), 2.0) + (0.047619047619047616 * pow(x_m, 6.0))) / sqrt(((double) M_PI))));
}

Julia

x_m = abs(x)
function code(x_m)
	return Float64(abs(x_m) * abs(Float64(Float64(fma(0.6666666666666666, Float64(x_m * x_m), 2.0) + Float64(0.047619047619047616 * (x_m ^ 6.0))) / sqrt(pi))))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Abs[x$95$m], $MachinePrecision] * N[Abs[N[(N[(N[(0.6666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 2.0), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 99.7%

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

6. Final simplification 99.7%

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

Alternative 4: 98.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \left|x\_m \cdot \frac{\mathsf{fma}\left(0.047619047619047616, {x\_m}^{6}, 2\right)}{\sqrt{\pi}}\right| \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (fabs (* x_m (/ (fma 0.047619047619047616 (pow x_m 6.0) 2.0) (sqrt PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	return fabs((x_m * (fma(0.047619047619047616, pow(x_m, 6.0), 2.0) / sqrt(((double) M_PI)))));
}

Julia

x_m = abs(x)
function code(x_m)
	return abs(Float64(x_m * Float64(fma(0.047619047619047616, (x_m ^ 6.0), 2.0) / sqrt(pi))))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Abs[N[(x$95$m * N[(N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 99.7%

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

6. Taylor expanded in x around 0 99.3%

   \[\leadsto \left|x\right| \cdot \left|\frac{0.047619047619047616 \cdot {x}^{6} + \color{blue}{2}}{\sqrt{\pi}}\right| \]
7. Step-by-step derivation

   1. *-commutative 99.3%

      \[\leadsto \color{blue}{\left|\frac{0.047619047619047616 \cdot {x}^{6} + 2}{\sqrt{\pi}}\right| \cdot \left|x\right|} \]

   2. neg-fabs 99.3%

      \[\leadsto \left|\frac{0.047619047619047616 \cdot {x}^{6} + 2}{\sqrt{\pi}}\right| \cdot \color{blue}{\left|-x\right|} \]

   3. mul-fabs 99.3%

      \[\leadsto \color{blue}{\left|\frac{0.047619047619047616 \cdot {x}^{6} + 2}{\sqrt{\pi}} \cdot \left(-x\right)\right|} \]

   4. fma-define 99.3%

      \[\leadsto \left|\frac{\color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}}{\sqrt{\pi}} \cdot \left(-x\right)\right| \]

8. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}{\sqrt{\pi}} \cdot \left(-x\right)\right|} \]

9. Final simplification 99.3%

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

Alternative 5: 98.4% accurate, 6.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \mathsf{fma}\left(0.047619047619047616, {x\_m}^{6}, 2\right) \cdot \frac{x\_m}{\sqrt{\pi}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (* (fma 0.047619047619047616 (pow x_m 6.0) 2.0) (/ x_m (sqrt PI))))

C

x_m = fabs(x);
double code(double x_m) {
	return fma(0.047619047619047616, pow(x_m, 6.0), 2.0) * (x_m / sqrt(((double) M_PI)));
}

Julia

x_m = abs(x)
function code(x_m)
	return Float64(fma(0.047619047619047616, (x_m ^ 6.0), 2.0) * Float64(x_m / sqrt(pi)))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[(x$95$m / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 99.7%

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

6. Taylor expanded in x around 0 99.3%

   \[\leadsto \left|x\right| \cdot \left|\frac{0.047619047619047616 \cdot {x}^{6} + \color{blue}{2}}{\sqrt{\pi}}\right| \]
7. Step-by-step derivation

   1. add-sqr-sqrt 33.5%

      \[\leadsto \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left|\frac{0.047619047619047616 \cdot {x}^{6} + 2}{\sqrt{\pi}}\right| \]

   2. fabs-sqr 33.5%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left|\frac{0.047619047619047616 \cdot {x}^{6} + 2}{\sqrt{\pi}}\right| \]

   3. add-sqr-sqrt 35.2%

      \[\leadsto \color{blue}{x} \cdot \left|\frac{0.047619047619047616 \cdot {x}^{6} + 2}{\sqrt{\pi}}\right| \]

   4. add-sqr-sqrt 34.6%

      \[\leadsto x \cdot \left|\color{blue}{\sqrt{\frac{0.047619047619047616 \cdot {x}^{6} + 2}{\sqrt{\pi}}} \cdot \sqrt{\frac{0.047619047619047616 \cdot {x}^{6} + 2}{\sqrt{\pi}}}}\right| \]

   5. fabs-sqr 34.6%

      \[\leadsto x \cdot \color{blue}{\left(\sqrt{\frac{0.047619047619047616 \cdot {x}^{6} + 2}{\sqrt{\pi}}} \cdot \sqrt{\frac{0.047619047619047616 \cdot {x}^{6} + 2}{\sqrt{\pi}}}\right)} \]

   6. add-sqr-sqrt 35.2%

      \[\leadsto x \cdot \color{blue}{\frac{0.047619047619047616 \cdot {x}^{6} + 2}{\sqrt{\pi}}} \]

   7. clear-num 35.2%

      \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\sqrt{\pi}}{0.047619047619047616 \cdot {x}^{6} + 2}}} \]

   8. un-div-inv 34.9%

      \[\leadsto \color{blue}{\frac{x}{\frac{\sqrt{\pi}}{0.047619047619047616 \cdot {x}^{6} + 2}}} \]

   9. fma-define 34.9%

      \[\leadsto \frac{x}{\frac{\sqrt{\pi}}{\color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}}} \]

8. Applied egg-rr 34.9%

   \[\leadsto \color{blue}{\frac{x}{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}}} \]
9. Step-by-step derivation

   1. associate-/r/ 34.9%

      \[\leadsto \color{blue}{\frac{x}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)} \]

   2. *-commutative 34.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right) \cdot \frac{x}{\sqrt{\pi}}} \]

10. Simplified 34.9%

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

Alternative 6: 99.2% accurate, 8.4× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (if (<= x_m 2.2)
     (* t_0 (+ (* x_m (* 0.6666666666666666 (pow x_m 2.0))) (* 2.0 x_m)))
     (* t_0 (* 0.047619047619047616 (pow x_m 7.0))))))

C

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	double tmp;
	if (x_m <= 2.2) {
		tmp = t_0 * ((x_m * (0.6666666666666666 * Math.pow(x_m, 2.0))) + (2.0 * x_m));
	} else {
		tmp = t_0 * (0.047619047619047616 * Math.pow(x_m, 7.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = math.sqrt((1.0 / math.pi))
	tmp = 0
	if x_m <= 2.2:
		tmp = t_0 * ((x_m * (0.6666666666666666 * math.pow(x_m, 2.0))) + (2.0 * x_m))
	else:
		tmp = t_0 * (0.047619047619047616 * math.pow(x_m, 7.0))
	return tmp

Julia

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

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = sqrt((1.0 / pi));
	tmp = 0.0;
	if (x_m <= 2.2)
		tmp = t_0 * ((x_m * (0.6666666666666666 * (x_m ^ 2.0))) + (2.0 * x_m));
	else
		tmp = t_0 * (0.047619047619047616 * (x_m ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 33.7%

         \[\leadsto \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

      2. fabs-sqr 33.7%

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

      3. add-sqr-sqrt 33.4%

         \[\leadsto \left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left|\color{blue}{\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}}\right| \]

      4. fabs-sqr 33.4%

         \[\leadsto \left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}\right)} \]

      5. add-sqr-sqrt 34.8%

         \[\leadsto \color{blue}{x} \cdot \left(\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}\right) \]

      6. add-sqr-sqrt 35.4%

         \[\leadsto x \cdot \color{blue}{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \]

      7. *-commutative 35.4%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}} \cdot x} \]

      8. associate-*l/ 35.1%

         \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right) \cdot x}{\sqrt{\pi}}} \]

   6. Applied egg-rr 35.1%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot x}{\sqrt{\pi}}} \]

   7. Taylor expanded in x around 0 35.3%

      \[\leadsto \color{blue}{x \cdot \left(0.6666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   8. Step-by-step derivation

      1. distribute-rgt-in 35.3%

         \[\leadsto \color{blue}{\left(0.6666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right) \cdot x + \left(2 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot x} \]

      2. *-commutative 35.3%

         \[\leadsto \color{blue}{x \cdot \left(0.6666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} + \left(2 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot x \]

      3. associate-*r* 35.3%

         \[\leadsto x \cdot \color{blue}{\left(\left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \sqrt{\frac{1}{\pi}}\right)} + \left(2 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot x \]

      4. associate-*r* 35.3%

         \[\leadsto \color{blue}{\left(x \cdot \left(0.6666666666666666 \cdot {x}^{2}\right)\right) \cdot \sqrt{\frac{1}{\pi}}} + \left(2 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot x \]

      5. associate-*r* 35.3%

         \[\leadsto \left(x \cdot \left(0.6666666666666666 \cdot {x}^{2}\right)\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot x\right)} \]

      6. *-commutative 35.3%

         \[\leadsto \left(x \cdot \left(0.6666666666666666 \cdot {x}^{2}\right)\right) \cdot \sqrt{\frac{1}{\pi}} + 2 \cdot \color{blue}{\left(x \cdot \sqrt{\frac{1}{\pi}}\right)} \]

      7. associate-*r* 35.3%

         \[\leadsto \left(x \cdot \left(0.6666666666666666 \cdot {x}^{2}\right)\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} \]

      8. distribute-rgt-out 35.3%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(0.6666666666666666 \cdot {x}^{2}\right) + 2 \cdot x\right)} \]

      9. *-commutative 35.3%

         \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(0.6666666666666666 \cdot {x}^{2}\right) + \color{blue}{x \cdot 2}\right) \]

      10. distribute-lft-in 35.3%

          \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(x \cdot \left(0.6666666666666666 \cdot {x}^{2} + 2\right)\right)} \]

      11. fma-define 35.3%

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

   9. Simplified 35.3%

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

       1. fma-undefine 35.3%

          \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \color{blue}{\left(0.6666666666666666 \cdot {x}^{2} + 2\right)}\right) \]

       2. distribute-rgt-in 35.3%

          \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(\left(0.6666666666666666 \cdot {x}^{2}\right) \cdot x + 2 \cdot x\right)} \]

       3. *-commutative 35.3%

          \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left(0.6666666666666666 \cdot {x}^{2}\right) \cdot x + \color{blue}{x \cdot 2}\right) \]

   11. Applied egg-rr 35.3%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(\left(0.6666666666666666 \cdot {x}^{2}\right) \cdot x + x \cdot 2\right)} \]

   if 2.2000000000000002 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 33.7%

         \[\leadsto \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

      2. fabs-sqr 33.7%

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

      3. add-sqr-sqrt 33.4%

         \[\leadsto \left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left|\color{blue}{\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}}\right| \]

      4. fabs-sqr 33.4%

         \[\leadsto \left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}\right)} \]

      5. add-sqr-sqrt 34.8%

         \[\leadsto \color{blue}{x} \cdot \left(\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}\right) \]

      6. add-sqr-sqrt 35.4%

         \[\leadsto x \cdot \color{blue}{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \]

      7. *-commutative 35.4%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}} \cdot x} \]

      8. associate-*l/ 35.1%

         \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right) \cdot x}{\sqrt{\pi}}} \]

   6. Applied egg-rr 35.1%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot x}{\sqrt{\pi}}} \]

   7. Taylor expanded in x around inf 3.8%

      \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 3.8%

         \[\leadsto \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}} \]

   9. Simplified 3.8%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(0.6666666666666666 \cdot {x}^{2}\right) + 2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 8.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.2:\\ \;\;\;\;x\_m \cdot \left({\pi}^{-0.5} \cdot \left(2 + 0.6666666666666666 \cdot {x\_m}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x\_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.2)
   (* x_m (* (pow PI -0.5) (+ 2.0 (* 0.6666666666666666 (pow x_m 2.0)))))
   (* (sqrt (/ 1.0 PI)) (* 0.047619047619047616 (pow x_m 7.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.2) {
		tmp = x_m * (pow(((double) M_PI), -0.5) * (2.0 + (0.6666666666666666 * pow(x_m, 2.0))));
	} else {
		tmp = sqrt((1.0 / ((double) M_PI))) * (0.047619047619047616 * pow(x_m, 7.0));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.2) {
		tmp = x_m * (Math.pow(Math.PI, -0.5) * (2.0 + (0.6666666666666666 * Math.pow(x_m, 2.0))));
	} else {
		tmp = Math.sqrt((1.0 / Math.PI)) * (0.047619047619047616 * Math.pow(x_m, 7.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.2:
		tmp = x_m * (math.pow(math.pi, -0.5) * (2.0 + (0.6666666666666666 * math.pow(x_m, 2.0))))
	else:
		tmp = math.sqrt((1.0 / math.pi)) * (0.047619047619047616 * math.pow(x_m, 7.0))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.2)
		tmp = Float64(x_m * Float64((pi ^ -0.5) * Float64(2.0 + Float64(0.6666666666666666 * (x_m ^ 2.0)))));
	else
		tmp = Float64(sqrt(Float64(1.0 / pi)) * Float64(0.047619047619047616 * (x_m ^ 7.0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.2)
		tmp = x_m * ((pi ^ -0.5) * (2.0 + (0.6666666666666666 * (x_m ^ 2.0))));
	else
		tmp = sqrt((1.0 / pi)) * (0.047619047619047616 * (x_m ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.2], N[(x$95$m * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(2.0 + N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(0.047619047619047616 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 33.7%

         \[\leadsto \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

      2. fabs-sqr 33.7%

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

      3. add-sqr-sqrt 33.4%

         \[\leadsto \left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left|\color{blue}{\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}}\right| \]

      4. fabs-sqr 33.4%

         \[\leadsto \left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}\right)} \]

      5. add-sqr-sqrt 34.8%

         \[\leadsto \color{blue}{x} \cdot \left(\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}\right) \]

      6. add-sqr-sqrt 35.4%

         \[\leadsto x \cdot \color{blue}{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \]

      7. *-commutative 35.4%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}} \cdot x} \]

      8. associate-*l/ 35.1%

         \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right) \cdot x}{\sqrt{\pi}}} \]

   6. Applied egg-rr 35.1%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot x}{\sqrt{\pi}}} \]

   7. Taylor expanded in x around 0 35.3%

      \[\leadsto \color{blue}{x \cdot \left(0.6666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 35.3%

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot \sqrt{\frac{1}{\pi}} + 0.6666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]

      2. associate-*r* 35.3%

         \[\leadsto x \cdot \left(2 \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \sqrt{\frac{1}{\pi}}}\right) \]

      3. distribute-rgt-out 35.3%

         \[\leadsto x \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right)} \]

      4. inv-pow 35.3%

         \[\leadsto x \cdot \left(\sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right) \]

      5. sqrt-pow1 35.3%

         \[\leadsto x \cdot \left(\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right) \]

      6. metadata-eval 35.3%

         \[\leadsto x \cdot \left({\pi}^{\color{blue}{-0.5}} \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right) \]

   9. Applied egg-rr 35.3%

      \[\leadsto x \cdot \color{blue}{\left({\pi}^{-0.5} \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right)} \]

   if 2.2000000000000002 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 33.7%

         \[\leadsto \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

      2. fabs-sqr 33.7%

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

      3. add-sqr-sqrt 33.4%

         \[\leadsto \left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left|\color{blue}{\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}}\right| \]

      4. fabs-sqr 33.4%

         \[\leadsto \left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}\right)} \]

      5. add-sqr-sqrt 34.8%

         \[\leadsto \color{blue}{x} \cdot \left(\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}\right) \]

      6. add-sqr-sqrt 35.4%

         \[\leadsto x \cdot \color{blue}{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \]

      7. *-commutative 35.4%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}} \cdot x} \]

      8. associate-*l/ 35.1%

         \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right) \cdot x}{\sqrt{\pi}}} \]

   6. Applied egg-rr 35.1%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot x}{\sqrt{\pi}}} \]

   7. Taylor expanded in x around inf 3.8%

      \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 3.8%

         \[\leadsto \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}} \]

   9. Simplified 3.8%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;x \cdot \left({\pi}^{-0.5} \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.85:\\ \;\;\;\;x\_m \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x\_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.85)
   (* x_m (/ 2.0 (sqrt PI)))
   (* (sqrt (/ 1.0 PI)) (* 0.047619047619047616 (pow x_m 7.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = sqrt((1.0 / ((double) M_PI))) * (0.047619047619047616 * pow(x_m, 7.0));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.sqrt((1.0 / Math.PI)) * (0.047619047619047616 * Math.pow(x_m, 7.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.85:
		tmp = x_m * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.sqrt((1.0 / math.pi)) * (0.047619047619047616 * math.pow(x_m, 7.0))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.85)
		tmp = Float64(x_m * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(sqrt(Float64(1.0 / pi)) * Float64(0.047619047619047616 * (x_m ^ 7.0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.85)
		tmp = x_m * (2.0 / sqrt(pi));
	else
		tmp = sqrt((1.0 / pi)) * (0.047619047619047616 * (x_m ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.85], N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(0.047619047619047616 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 33.7%

         \[\leadsto \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

      2. fabs-sqr 33.7%

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

      3. add-sqr-sqrt 33.4%

         \[\leadsto \left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left|\color{blue}{\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}}\right| \]

      4. fabs-sqr 33.4%

         \[\leadsto \left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}\right)} \]

      5. add-sqr-sqrt 34.8%

         \[\leadsto \color{blue}{x} \cdot \left(\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}\right) \]

      6. add-sqr-sqrt 35.4%

         \[\leadsto x \cdot \color{blue}{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \]

      7. *-commutative 35.4%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}} \cdot x} \]

      8. associate-*l/ 35.1%

         \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right) \cdot x}{\sqrt{\pi}}} \]

   6. Applied egg-rr 35.1%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot x}{\sqrt{\pi}}} \]

   7. Taylor expanded in x around 0 35.3%

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

      1. associate-*r* 35.3%

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

   9. Simplified 35.3%

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

       1. sqrt-div 35.3%

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

       2. metadata-eval 35.3%

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

       3. un-div-inv 35.0%

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

       4. *-commutative 35.0%

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

   11. Applied egg-rr 35.0%

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

       1. associate-*r/ 35.3%

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

   13. Simplified 35.3%

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

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 33.7%

         \[\leadsto \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

      2. fabs-sqr 33.7%

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

      3. add-sqr-sqrt 33.4%

         \[\leadsto \left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left|\color{blue}{\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}}\right| \]

      4. fabs-sqr 33.4%

         \[\leadsto \left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}\right)} \]

      5. add-sqr-sqrt 34.8%

         \[\leadsto \color{blue}{x} \cdot \left(\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}\right) \]

      6. add-sqr-sqrt 35.4%

         \[\leadsto x \cdot \color{blue}{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \]

      7. *-commutative 35.4%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}} \cdot x} \]

      8. associate-*l/ 35.1%

         \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right) \cdot x}{\sqrt{\pi}}} \]

   6. Applied egg-rr 35.1%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot x}{\sqrt{\pi}}} \]

   7. Taylor expanded in x around inf 3.8%

      \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 3.8%

         \[\leadsto \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}} \]

   9. Simplified 3.8%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.0% accurate, 8.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.75:\\ \;\;\;\;x\_m \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.2 \cdot {x\_m}^{5}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.75)
   (* x_m (/ 2.0 (sqrt PI)))
   (/ (* 0.2 (pow x_m 5.0)) (sqrt PI))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.75) {
		tmp = x_m * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = (0.2 * pow(x_m, 5.0)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.75) {
		tmp = x_m * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = (0.2 * Math.pow(x_m, 5.0)) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.75:
		tmp = x_m * (2.0 / math.sqrt(math.pi))
	else:
		tmp = (0.2 * math.pow(x_m, 5.0)) / math.sqrt(math.pi)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.75)
		tmp = Float64(x_m * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(Float64(0.2 * (x_m ^ 5.0)) / sqrt(pi));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.75)
		tmp = x_m * (2.0 / sqrt(pi));
	else
		tmp = (0.2 * (x_m ^ 5.0)) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.75], N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.2 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.75

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 33.7%

         \[\leadsto \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

      2. fabs-sqr 33.7%

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

      3. add-sqr-sqrt 33.4%

         \[\leadsto \left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left|\color{blue}{\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}}\right| \]

      4. fabs-sqr 33.4%

         \[\leadsto \left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}\right)} \]

      5. add-sqr-sqrt 34.8%

         \[\leadsto \color{blue}{x} \cdot \left(\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}\right) \]

      6. add-sqr-sqrt 35.4%

         \[\leadsto x \cdot \color{blue}{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \]

      7. *-commutative 35.4%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}} \cdot x} \]

      8. associate-*l/ 35.1%

         \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right) \cdot x}{\sqrt{\pi}}} \]

   6. Applied egg-rr 35.1%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot x}{\sqrt{\pi}}} \]

   7. Taylor expanded in x around 0 35.3%

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

      1. associate-*r* 35.3%

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

   9. Simplified 35.3%

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

       1. sqrt-div 35.3%

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

       2. metadata-eval 35.3%

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

       3. un-div-inv 35.0%

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

       4. *-commutative 35.0%

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

   11. Applied egg-rr 35.0%

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

       1. associate-*r/ 35.3%

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

   13. Simplified 35.3%

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

   if 1.75 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 33.7%

         \[\leadsto \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

      2. fabs-sqr 33.7%

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

      3. add-sqr-sqrt 33.4%

         \[\leadsto \left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left|\color{blue}{\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}}\right| \]

      4. fabs-sqr 33.4%

         \[\leadsto \left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}\right)} \]

      5. add-sqr-sqrt 34.8%

         \[\leadsto \color{blue}{x} \cdot \left(\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}\right) \]

      6. add-sqr-sqrt 35.4%

         \[\leadsto x \cdot \color{blue}{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \]

      7. *-commutative 35.4%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}} \cdot x} \]

      8. associate-*l/ 35.1%

         \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right) \cdot x}{\sqrt{\pi}}} \]

   6. Applied egg-rr 35.1%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot x}{\sqrt{\pi}}} \]

   7. Taylor expanded in x around 0 35.1%

      \[\leadsto \frac{\left(\color{blue}{0.2 \cdot {x}^{4}} + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot x}{\sqrt{\pi}} \]

   8. Taylor expanded in x around 0 34.9%

      \[\leadsto \frac{\left(0.2 \cdot {x}^{4} + \color{blue}{2}\right) \cdot x}{\sqrt{\pi}} \]

   9. Taylor expanded in x around inf 3.9%

      \[\leadsto \frac{\color{blue}{0.2 \cdot {x}^{5}}}{\sqrt{\pi}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 83.1% accurate, 8.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 6 \cdot 10^{-16}:\\ \;\;\;\;x\_m \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x\_m}^{2} \cdot \frac{4}{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 6e-16)
   (* x_m (/ 2.0 (sqrt PI)))
   (sqrt (* (pow x_m 2.0) (/ 4.0 PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 6e-16) {
		tmp = x_m * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = sqrt((pow(x_m, 2.0) * (4.0 / ((double) M_PI))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 6e-16) {
		tmp = x_m * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.sqrt((Math.pow(x_m, 2.0) * (4.0 / Math.PI)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 6e-16:
		tmp = x_m * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.sqrt((math.pow(x_m, 2.0) * (4.0 / math.pi)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 6e-16)
		tmp = Float64(x_m * Float64(2.0 / sqrt(pi)));
	else
		tmp = sqrt(Float64((x_m ^ 2.0) * Float64(4.0 / pi)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 6e-16)
		tmp = x_m * (2.0 / sqrt(pi));
	else
		tmp = sqrt(((x_m ^ 2.0) * (4.0 / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 6e-16], N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(4.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.99999999999999987e-16

   1. Initial program 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 33.5%

         \[\leadsto \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

      2. fabs-sqr 33.5%

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

      3. add-sqr-sqrt 33.2%

         \[\leadsto \left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left|\color{blue}{\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}}\right| \]

      4. fabs-sqr 33.2%

         \[\leadsto \left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}\right)} \]

      5. add-sqr-sqrt 34.6%

         \[\leadsto \color{blue}{x} \cdot \left(\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}\right) \]

      6. add-sqr-sqrt 35.2%

         \[\leadsto x \cdot \color{blue}{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \]

      7. *-commutative 35.2%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}} \cdot x} \]

      8. associate-*l/ 34.9%

         \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right) \cdot x}{\sqrt{\pi}}} \]

   6. Applied egg-rr 34.9%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot x}{\sqrt{\pi}}} \]

   7. Taylor expanded in x around 0 35.2%

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

      1. associate-*r* 35.2%

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

   9. Simplified 35.2%

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

       1. sqrt-div 35.2%

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

       2. metadata-eval 35.2%

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

       3. un-div-inv 35.0%

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

       4. *-commutative 35.0%

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

   11. Applied egg-rr 35.0%

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

       1. associate-*r/ 35.2%

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

   13. Simplified 35.2%

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

   if 5.99999999999999987e-16 < x

   1. Initial program 100.0%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

         \[\leadsto \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

      2. fabs-sqr 100.0%

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

      3. add-sqr-sqrt 98.4%

         \[\leadsto \left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left|\color{blue}{\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}}\right| \]

      4. fabs-sqr 98.4%

         \[\leadsto \left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}\right)} \]

      5. add-sqr-sqrt 97.5%

         \[\leadsto \color{blue}{x} \cdot \left(\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}\right) \]

      6. add-sqr-sqrt 98.4%

         \[\leadsto x \cdot \color{blue}{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \]

      7. *-commutative 98.4%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}} \cdot x} \]

      8. associate-*l/ 98.4%

         \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right) \cdot x}{\sqrt{\pi}}} \]

   6. Applied egg-rr 98.4%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot x}{\sqrt{\pi}}} \]

   7. Taylor expanded in x around 0 46.5%

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

      1. associate-*r* 46.5%

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

   9. Simplified 46.5%

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

       1. sqrt-div 46.5%

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

       2. metadata-eval 46.5%

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

       3. un-div-inv 46.5%

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

       4. *-commutative 46.5%

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

   11. Applied egg-rr 46.5%

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

       1. associate-*r/ 46.5%

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

   13. Simplified 46.5%

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

       1. add-sqr-sqrt 46.5%

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

       2. sqrt-unprod 46.5%

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

       3. swap-sqr 46.5%

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

       4. unpow2 46.5%

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

       5. frac-times 46.5%

          \[\leadsto \sqrt{{x}^{2} \cdot \color{blue}{\frac{2 \cdot 2}{\sqrt{\pi} \cdot \sqrt{\pi}}}} \]

       6. metadata-eval 46.5%

          \[\leadsto \sqrt{{x}^{2} \cdot \frac{\color{blue}{4}}{\sqrt{\pi} \cdot \sqrt{\pi}}} \]

       7. add-sqr-sqrt 46.5%

          \[\leadsto \sqrt{{x}^{2} \cdot \frac{4}{\color{blue}{\pi}}} \]

   15. Applied egg-rr 46.5%

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

Alternative 11: 67.8% accurate, 17.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* x_m (/ 2.0 (sqrt PI))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 33.7%

      \[\leadsto \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

   2. fabs-sqr 33.7%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

   3. add-sqr-sqrt 33.4%

      \[\leadsto \left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left|\color{blue}{\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}}\right| \]

   4. fabs-sqr 33.4%

      \[\leadsto \left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}\right)} \]

   5. add-sqr-sqrt 34.8%

      \[\leadsto \color{blue}{x} \cdot \left(\sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}\right) \]

   6. add-sqr-sqrt 35.4%

      \[\leadsto x \cdot \color{blue}{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \]

   7. *-commutative 35.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}} \cdot x} \]

   8. associate-*l/ 35.1%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right) \cdot x}{\sqrt{\pi}}} \]

6. Applied egg-rr 35.1%

   \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right) \cdot x}{\sqrt{\pi}}} \]

7. Taylor expanded in x around 0 35.3%

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

   1. associate-*r* 35.3%

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

9. Simplified 35.3%

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

    1. sqrt-div 35.3%

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

    2. metadata-eval 35.3%

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

    3. un-div-inv 35.0%

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

    4. *-commutative 35.0%

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

11. Applied egg-rr 35.0%

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

    1. associate-*r/ 35.3%

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

13. Simplified 35.3%

    \[\leadsto \color{blue}{x \cdot \frac{2}{\sqrt{\pi}}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024083 
(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)))))))