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

Percentage Accurate: 99.9% → 99.9%

Time: 11.7s

Alternatives: 10

Speedup: 3.0×

• 28.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x\_m \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x\_m}^{6}, \mathsf{fma}\left(0.2, {x\_m}^{4}, 0.6666666666666666 \cdot {x\_m}^{2}\right)\right)}{\sqrt{\pi}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  x_m
  (/
   (+
    2.0
    (fma
     0.047619047619047616
     (pow x_m 6.0)
     (fma 0.2 (pow x_m 4.0) (* 0.6666666666666666 (pow x_m 2.0)))))
   (sqrt PI))))

C

x_m = fabs(x);
double code(double x_m) {
	return x_m * ((2.0 + fma(0.047619047619047616, pow(x_m, 6.0), fma(0.2, pow(x_m, 4.0), (0.6666666666666666 * pow(x_m, 2.0))))) / sqrt(((double) M_PI)));
}

Julia

x_m = abs(x)
function code(x_m)
	return Float64(x_m * Float64(Float64(2.0 + fma(0.047619047619047616, (x_m ^ 6.0), fma(0.2, (x_m ^ 4.0), Float64(0.6666666666666666 * (x_m ^ 2.0))))) / sqrt(pi)))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * N[(N[(2.0 + N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $MachinePrecision] + N[(0.2 * N[Power[x$95$m, 4.0], $MachinePrecision] + N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

6. Applied egg-rr 34.3%

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

   1. associate-*r/ 34.5%

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

   2. +-commutative 34.5%

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

   3. fma-undefine 34.5%

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

   4. associate-+r+ 34.5%

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

   5. fma-define 34.5%

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

   6. +-commutative 34.5%

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

   7. associate-+r+ 34.5%

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

   8. +-commutative 34.5%

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

   9. fma-define 34.5%

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

   10. fma-define 34.5%

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

8. Simplified 34.5%

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

Alternative 2: 99.9% accurate, 4.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \left(\left(2 + 0.6666666666666666 \cdot {x\_m}^{2}\right) + \left(0.2 \cdot {x\_m}^{4} + 0.047619047619047616 \cdot {x\_m}^{6}\right)\right) \cdot \left(x\_m \cdot {\pi}^{-0.5}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (+
   (+ 2.0 (* 0.6666666666666666 (pow x_m 2.0)))
   (+ (* 0.2 (pow x_m 4.0)) (* 0.047619047619047616 (pow x_m 6.0))))
  (* x_m (pow PI -0.5))))

C

x_m = fabs(x);
double code(double x_m) {
	return ((2.0 + (0.6666666666666666 * pow(x_m, 2.0))) + ((0.2 * pow(x_m, 4.0)) + (0.047619047619047616 * pow(x_m, 6.0)))) * (x_m * pow(((double) M_PI), -0.5));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return ((2.0 + (0.6666666666666666 * Math.pow(x_m, 2.0))) + ((0.2 * Math.pow(x_m, 4.0)) + (0.047619047619047616 * Math.pow(x_m, 6.0)))) * (x_m * Math.pow(Math.PI, -0.5));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return ((2.0 + (0.6666666666666666 * math.pow(x_m, 2.0))) + ((0.2 * math.pow(x_m, 4.0)) + (0.047619047619047616 * math.pow(x_m, 6.0)))) * (x_m * math.pow(math.pi, -0.5))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(Float64(2.0 + Float64(0.6666666666666666 * (x_m ^ 2.0))) + Float64(Float64(0.2 * (x_m ^ 4.0)) + Float64(0.047619047619047616 * (x_m ^ 6.0)))) * Float64(x_m * (pi ^ -0.5)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = ((2.0 + (0.6666666666666666 * (x_m ^ 2.0))) + ((0.2 * (x_m ^ 4.0)) + (0.047619047619047616 * (x_m ^ 6.0)))) * (x_m * (pi ^ -0.5));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(N[(2.0 + N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.2 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

6. Applied egg-rr 34.3%

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

   1. div-inv 34.5%

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

   2. *-commutative 34.5%

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

   3. metadata-eval 34.5%

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

   4. sqrt-div 34.5%

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

   5. associate-*l* 34.5%

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

   6. inv-pow 34.5%

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

   7. sqrt-pow1 34.5%

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

   8. metadata-eval 34.5%

      \[\leadsto \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right) \cdot \left(x \cdot {\pi}^{\color{blue}{-0.5}}\right) \]

8. Applied egg-rr 34.5%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right) \cdot \left(x \cdot {\pi}^{-0.5}\right)} \]
9. Step-by-step derivation

   1. fma-undefine 34.5%

      \[\leadsto \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \color{blue}{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)}\right) \cdot \left(x \cdot {\pi}^{-0.5}\right) \]

10. Applied egg-rr 34.5%

    \[\leadsto \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \color{blue}{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)}\right) \cdot \left(x \cdot {\pi}^{-0.5}\right) \]
11. Step-by-step derivation

    1. fma-undefine 34.5%

       \[\leadsto \left(\color{blue}{\left(0.6666666666666666 \cdot {x}^{2} + 2\right)} + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right) \cdot \left(x \cdot {\pi}^{-0.5}\right) \]

12. Applied egg-rr 34.5%

    \[\leadsto \left(\color{blue}{\left(0.6666666666666666 \cdot {x}^{2} + 2\right)} + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right) \cdot \left(x \cdot {\pi}^{-0.5}\right) \]

13. Final simplification 34.5%

    \[\leadsto \left(\left(2 + 0.6666666666666666 \cdot {x}^{2}\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right) \cdot \left(x \cdot {\pi}^{-0.5}\right) \]
14. Add Preprocessing

Alternative 3: 98.4% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.abs(((Math.abs(x_m) / Math.sqrt(Math.PI)) * (2.0 + (0.047619047619047616 * Math.pow(x_m, 6.0)))));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.fabs(((math.fabs(x_m) / math.sqrt(math.pi)) * (2.0 + (0.047619047619047616 * math.pow(x_m, 6.0)))))

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = abs(((abs(x_m) / sqrt(pi)) * (2.0 + (0.047619047619047616 * (x_m ^ 6.0)))));
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around inf 98.3%

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

6. Taylor expanded in x around 0 97.7%

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

7. Final simplification 97.7%

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

Alternative 4: 99.3% accurate, 5.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \mathbf{if}\;x\_m \leq 1.9:\\ \;\;\;\;x\_m \cdot \left(0.6666666666666666 \cdot \left({x\_m}^{2} \cdot t\_0\right) + 2 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{7} \cdot \left(t\_0 \cdot \left(0.047619047619047616 + \frac{0.2}{{x\_m}^{2}}\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (if (<= x_m 1.9)
     (* x_m (+ (* 0.6666666666666666 (* (pow x_m 2.0) t_0)) (* 2.0 t_0)))
     (*
      (pow x_m 7.0)
      (* t_0 (+ 0.047619047619047616 (/ 0.2 (pow x_m 2.0))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double tmp;
	if (x_m <= 1.9) {
		tmp = x_m * ((0.6666666666666666 * (pow(x_m, 2.0) * t_0)) + (2.0 * t_0));
	} else {
		tmp = pow(x_m, 7.0) * (t_0 * (0.047619047619047616 + (0.2 / pow(x_m, 2.0))));
	}
	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 <= 1.9) {
		tmp = x_m * ((0.6666666666666666 * (Math.pow(x_m, 2.0) * t_0)) + (2.0 * t_0));
	} else {
		tmp = Math.pow(x_m, 7.0) * (t_0 * (0.047619047619047616 + (0.2 / Math.pow(x_m, 2.0))));
	}
	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 <= 1.9:
		tmp = x_m * ((0.6666666666666666 * (math.pow(x_m, 2.0) * t_0)) + (2.0 * t_0))
	else:
		tmp = math.pow(x_m, 7.0) * (t_0 * (0.047619047619047616 + (0.2 / math.pow(x_m, 2.0))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = sqrt(Float64(1.0 / pi))
	tmp = 0.0
	if (x_m <= 1.9)
		tmp = Float64(x_m * Float64(Float64(0.6666666666666666 * Float64((x_m ^ 2.0) * t_0)) + Float64(2.0 * t_0)));
	else
		tmp = Float64((x_m ^ 7.0) * Float64(t_0 * Float64(0.047619047619047616 + Float64(0.2 / (x_m ^ 2.0)))));
	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 <= 1.9)
		tmp = x_m * ((0.6666666666666666 * ((x_m ^ 2.0) * t_0)) + (2.0 * t_0));
	else
		tmp = (x_m ^ 7.0) * (t_0 * (0.047619047619047616 + (0.2 / (x_m ^ 2.0))));
	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, 1.9], N[(x$95$m * N[(N[(0.6666666666666666 * N[(N[Power[x$95$m, 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x$95$m, 7.0], $MachinePrecision] * N[(t$95$0 * N[(0.047619047619047616 + N[(0.2 / N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 34.3%

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

      1. associate-*r/ 34.5%

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

      2. +-commutative 34.5%

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

      3. fma-undefine 34.5%

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

      4. associate-+r+ 34.5%

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

      5. fma-define 34.5%

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

      6. +-commutative 34.5%

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

      7. associate-+r+ 34.5%

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

      8. +-commutative 34.5%

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

      9. fma-define 34.5%

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

      10. fma-define 34.5%

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

   8. Simplified 34.5%

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

   9. Taylor expanded in x around 0 34.6%

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

   if 1.8999999999999999 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 34.3%

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

      1. associate-*r/ 34.5%

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

      2. +-commutative 34.5%

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

      3. fma-undefine 34.5%

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

      4. associate-+r+ 34.5%

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

      5. fma-define 34.5%

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

      6. +-commutative 34.5%

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

      7. associate-+r+ 34.5%

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

      8. +-commutative 34.5%

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

      9. fma-define 34.5%

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

      10. fma-define 34.5%

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

   8. Simplified 34.5%

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

   9. Taylor expanded in x around inf 1.6%

      \[\leadsto \color{blue}{{x}^{7} \cdot \left(0.047619047619047616 \cdot \sqrt{\frac{1}{\pi}} + 0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 1.6%

          \[\leadsto {x}^{7} \cdot \left(0.047619047619047616 \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \sqrt{\frac{1}{\pi}}}\right) \]

       2. distribute-rgt-out 1.6%

          \[\leadsto {x}^{7} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + 0.2 \cdot \frac{1}{{x}^{2}}\right)\right)} \]

       3. associate-*r/ 1.6%

          \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \color{blue}{\frac{0.2 \cdot 1}{{x}^{2}}}\right)\right) \]

       4. metadata-eval 1.6%

          \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \frac{\color{blue}{0.2}}{{x}^{2}}\right)\right) \]

   11. Simplified 1.6%

       \[\leadsto \color{blue}{{x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.3% accurate, 5.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.9:\\ \;\;\;\;x\_m \cdot \frac{2 + 0.6666666666666666 \cdot {x\_m}^{2}}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \frac{0.2}{{x\_m}^{2}}\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.9)
   (* x_m (/ (+ 2.0 (* 0.6666666666666666 (pow x_m 2.0))) (sqrt PI)))
   (*
    (pow x_m 7.0)
    (* (sqrt (/ 1.0 PI)) (+ 0.047619047619047616 (/ 0.2 (pow x_m 2.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.9) {
		tmp = x_m * ((2.0 + (0.6666666666666666 * pow(x_m, 2.0))) / sqrt(((double) M_PI)));
	} else {
		tmp = pow(x_m, 7.0) * (sqrt((1.0 / ((double) M_PI))) * (0.047619047619047616 + (0.2 / pow(x_m, 2.0))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.9) {
		tmp = x_m * ((2.0 + (0.6666666666666666 * Math.pow(x_m, 2.0))) / Math.sqrt(Math.PI));
	} else {
		tmp = Math.pow(x_m, 7.0) * (Math.sqrt((1.0 / Math.PI)) * (0.047619047619047616 + (0.2 / Math.pow(x_m, 2.0))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.9:
		tmp = x_m * ((2.0 + (0.6666666666666666 * math.pow(x_m, 2.0))) / math.sqrt(math.pi))
	else:
		tmp = math.pow(x_m, 7.0) * (math.sqrt((1.0 / math.pi)) * (0.047619047619047616 + (0.2 / math.pow(x_m, 2.0))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.9)
		tmp = Float64(x_m * Float64(Float64(2.0 + Float64(0.6666666666666666 * (x_m ^ 2.0))) / sqrt(pi)));
	else
		tmp = Float64((x_m ^ 7.0) * Float64(sqrt(Float64(1.0 / pi)) * Float64(0.047619047619047616 + Float64(0.2 / (x_m ^ 2.0)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.9)
		tmp = x_m * ((2.0 + (0.6666666666666666 * (x_m ^ 2.0))) / sqrt(pi));
	else
		tmp = (x_m ^ 7.0) * (sqrt((1.0 / pi)) * (0.047619047619047616 + (0.2 / (x_m ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.9], N[(x$95$m * N[(N[(2.0 + N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x$95$m, 7.0], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(0.047619047619047616 + N[(0.2 / N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 34.3%

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

      1. associate-*r/ 34.5%

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

      2. +-commutative 34.5%

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

      3. fma-undefine 34.5%

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

      4. associate-+r+ 34.5%

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

      5. fma-define 34.5%

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

      6. +-commutative 34.5%

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

      7. associate-+r+ 34.5%

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

      8. +-commutative 34.5%

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

      9. fma-define 34.5%

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

      10. fma-define 34.5%

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

   8. Simplified 34.5%

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

   9. Taylor expanded in x around 0 34.6%

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

       1. associate-*r* 34.6%

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

       2. distribute-rgt-out 34.6%

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

       3. fma-define 34.6%

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

   11. Simplified 34.6%

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

       1. *-commutative 34.6%

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

       2. sqrt-div 34.6%

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

       3. metadata-eval 34.6%

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

       4. un-div-inv 34.6%

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

   13. Applied egg-rr 34.6%

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

       1. fma-undefine 34.5%

          \[\leadsto \left(\color{blue}{\left(0.6666666666666666 \cdot {x}^{2} + 2\right)} + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right) \cdot \left(x \cdot {\pi}^{-0.5}\right) \]

   15. Applied egg-rr 34.6%

       \[\leadsto x \cdot \frac{\color{blue}{0.6666666666666666 \cdot {x}^{2} + 2}}{\sqrt{\pi}} \]

   if 1.8999999999999999 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 34.3%

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

      1. associate-*r/ 34.5%

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

      2. +-commutative 34.5%

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

      3. fma-undefine 34.5%

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

      4. associate-+r+ 34.5%

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

      5. fma-define 34.5%

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

      6. +-commutative 34.5%

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

      7. associate-+r+ 34.5%

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

      8. +-commutative 34.5%

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

      9. fma-define 34.5%

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

      10. fma-define 34.5%

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

   8. Simplified 34.5%

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

   9. Taylor expanded in x around inf 1.6%

      \[\leadsto \color{blue}{{x}^{7} \cdot \left(0.047619047619047616 \cdot \sqrt{\frac{1}{\pi}} + 0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 1.6%

          \[\leadsto {x}^{7} \cdot \left(0.047619047619047616 \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \sqrt{\frac{1}{\pi}}}\right) \]

       2. distribute-rgt-out 1.6%

          \[\leadsto {x}^{7} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + 0.2 \cdot \frac{1}{{x}^{2}}\right)\right)} \]

       3. associate-*r/ 1.6%

          \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \color{blue}{\frac{0.2 \cdot 1}{{x}^{2}}}\right)\right) \]

       4. metadata-eval 1.6%

          \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \frac{\color{blue}{0.2}}{{x}^{2}}\right)\right) \]

   11. Simplified 1.6%

       \[\leadsto \color{blue}{{x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;x \cdot \frac{2 + 0.6666666666666666 \cdot {x}^{2}}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 8.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.2)
   (* x_m (/ (+ 2.0 (* 0.6666666666666666 (pow 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 <= 2.2) {
		tmp = x_m * ((2.0 + (0.6666666666666666 * pow(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 <= 2.2) {
		tmp = x_m * ((2.0 + (0.6666666666666666 * Math.pow(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 <= 2.2:
		tmp = x_m * ((2.0 + (0.6666666666666666 * math.pow(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 <= 2.2)
		tmp = Float64(x_m * Float64(Float64(2.0 + Float64(0.6666666666666666 * (x_m ^ 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 <= 2.2)
		tmp = x_m * ((2.0 + (0.6666666666666666 * (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, 2.2], N[(x$95$m * N[(N[(2.0 + N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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 < 2.2000000000000002

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 34.3%

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

      1. associate-*r/ 34.5%

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

      2. +-commutative 34.5%

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

      3. fma-undefine 34.5%

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

      4. associate-+r+ 34.5%

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

      5. fma-define 34.5%

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

      6. +-commutative 34.5%

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

      7. associate-+r+ 34.5%

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

      8. +-commutative 34.5%

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

      9. fma-define 34.5%

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

      10. fma-define 34.5%

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

   8. Simplified 34.5%

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

   9. Taylor expanded in x around 0 34.6%

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

       1. associate-*r* 34.6%

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

       2. distribute-rgt-out 34.6%

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

       3. fma-define 34.6%

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

   11. Simplified 34.6%

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

       1. *-commutative 34.6%

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

       2. sqrt-div 34.6%

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

       3. metadata-eval 34.6%

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

       4. un-div-inv 34.6%

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

   13. Applied egg-rr 34.6%

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

       1. fma-undefine 34.5%

          \[\leadsto \left(\color{blue}{\left(0.6666666666666666 \cdot {x}^{2} + 2\right)} + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right) \cdot \left(x \cdot {\pi}^{-0.5}\right) \]

   15. Applied egg-rr 34.6%

       \[\leadsto x \cdot \frac{\color{blue}{0.6666666666666666 \cdot {x}^{2} + 2}}{\sqrt{\pi}} \]

   if 2.2000000000000002 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 34.3%

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

      1. associate-*r/ 34.5%

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

      2. +-commutative 34.5%

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

      3. fma-undefine 34.5%

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

      4. associate-+r+ 34.5%

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

      5. fma-define 34.5%

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

      6. +-commutative 34.5%

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

      7. associate-+r+ 34.5%

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

      8. +-commutative 34.5%

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

      9. fma-define 34.5%

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

      10. fma-define 34.5%

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

   8. Simplified 34.5%

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

   9. Taylor expanded in x around inf 3.7%

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

       1. associate-*r* 3.7%

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

       2. *-commutative 3.7%

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

   11. Simplified 3.7%

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

4. Final simplification 34.6%

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

Alternative 7: 98.9% accurate, 8.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.9:\\ \;\;\;\;x\_m \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x\_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.9)
   (* 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 <= 1.9) {
		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 <= 1.9) {
		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 <= 1.9:
		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 <= 1.9)
		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 <= 1.9)
		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, 1.9], 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 < 1.8999999999999999

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 34.3%

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

      1. associate-*r/ 34.5%

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

      2. +-commutative 34.5%

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

      3. fma-undefine 34.5%

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

      4. associate-+r+ 34.5%

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

      5. fma-define 34.5%

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

      6. +-commutative 34.5%

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

      7. associate-+r+ 34.5%

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

      8. +-commutative 34.5%

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

      9. fma-define 34.5%

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

      10. fma-define 34.5%

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

   8. Simplified 34.5%

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

   9. Taylor expanded in x around 0 34.6%

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

       1. associate-*r* 34.6%

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

   11. Simplified 34.6%

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

       1. sqrt-div 34.6%

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

       2. metadata-eval 34.6%

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

       3. un-div-inv 34.4%

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

       4. associate-*r/ 34.4%

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

       5. add-sqr-sqrt 32.9%

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

       6. fabs-sqr 32.9%

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

       7. add-sqr-sqrt 70.7%

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

       8. clear-num 70.7%

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

       9. un-div-inv 70.7%

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

       10. add-sqr-sqrt 32.8%

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

       11. fabs-sqr 32.8%

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

       12. add-sqr-sqrt 34.4%

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

   13. Applied egg-rr 34.4%

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

       1. associate-/r/ 34.6%

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

   15. Simplified 34.6%

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

   if 1.8999999999999999 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 34.3%

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

      1. associate-*r/ 34.5%

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

      2. +-commutative 34.5%

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

      3. fma-undefine 34.5%

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

      4. associate-+r+ 34.5%

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

      5. fma-define 34.5%

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

      6. +-commutative 34.5%

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

      7. associate-+r+ 34.5%

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

      8. +-commutative 34.5%

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

      9. fma-define 34.5%

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

      10. fma-define 34.5%

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

   8. Simplified 34.5%

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

   9. Taylor expanded in x around inf 3.7%

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

       1. associate-*r* 3.7%

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

       2. *-commutative 3.7%

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

   11. Simplified 3.7%

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

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.7% accurate, 8.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-99}:\\ \;\;\;\;x\_m \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{4 \cdot \frac{{x\_m}^{2}}{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 5e-99)
   (* x_m (/ 2.0 (sqrt PI)))
   (sqrt (* 4.0 (/ (pow x_m 2.0) PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 5e-99) {
		tmp = x_m * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = sqrt((4.0 * (pow(x_m, 2.0) / ((double) M_PI))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 5e-99) {
		tmp = x_m * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.sqrt((4.0 * (Math.pow(x_m, 2.0) / Math.PI)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 5e-99:
		tmp = x_m * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.sqrt((4.0 * (math.pow(x_m, 2.0) / math.pi)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 5e-99)
		tmp = Float64(x_m * Float64(2.0 / sqrt(pi)));
	else
		tmp = sqrt(Float64(4.0 * Float64((x_m ^ 2.0) / pi)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 5e-99)
		tmp = x_m * (2.0 / sqrt(pi));
	else
		tmp = sqrt((4.0 * ((x_m ^ 2.0) / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 5e-99], N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(4.0 * N[(N[Power[x$95$m, 2.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999999999969e-99

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 26.0%

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

      1. associate-*r/ 26.2%

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

      2. +-commutative 26.2%

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

      3. fma-undefine 26.2%

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

      4. associate-+r+ 26.2%

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

      5. fma-define 26.2%

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

      6. +-commutative 26.2%

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

      7. associate-+r+ 26.2%

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

      8. +-commutative 26.2%

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

      9. fma-define 26.2%

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

      10. fma-define 26.2%

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

   8. Simplified 26.2%

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

   9. Taylor expanded in x around 0 26.3%

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

       1. associate-*r* 26.3%

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

   11. Simplified 26.3%

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

       1. sqrt-div 26.3%

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

       2. metadata-eval 26.3%

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

       3. un-div-inv 26.1%

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

       4. associate-*r/ 26.1%

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

       5. add-sqr-sqrt 24.4%

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

       6. fabs-sqr 24.4%

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

       7. add-sqr-sqrt 67.0%

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

       8. clear-num 67.0%

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

       9. un-div-inv 67.0%

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

       10. add-sqr-sqrt 24.4%

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

       11. fabs-sqr 24.4%

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

       12. add-sqr-sqrt 26.1%

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

   13. Applied egg-rr 26.1%

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

       1. associate-/r/ 26.3%

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

   15. Simplified 26.3%

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

   if 4.99999999999999969e-99 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 99.1%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. fma-undefine 99.8%

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

      4. associate-+r+ 99.8%

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

      5. fma-define 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+r+ 99.8%

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

      8. +-commutative 99.8%

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

      9. fma-define 99.8%

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

      10. fma-define 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around 0 99.8%

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

       1. associate-*r* 99.8%

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

   11. Simplified 99.8%

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

       1. add-sqr-sqrt 99.0%

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

       2. sqrt-unprod 99.8%

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

       3. sqrt-div 99.8%

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

       4. metadata-eval 99.8%

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

       5. un-div-inv 99.7%

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

       6. associate-*r/ 99.7%

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

       7. add-sqr-sqrt 99.6%

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

       8. fabs-sqr 99.6%

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

       9. add-sqr-sqrt 99.7%

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

   13. Applied egg-rr 99.9%

       \[\leadsto \color{blue}{\sqrt{4 \cdot \frac{{x}^{2}}{\pi}}} \]
   14. Step-by-step derivation

       1. *-commutative 99.9%

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

   15. Simplified 99.9%

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

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-99}:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{4 \cdot \frac{{x}^{2}}{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.6% 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 99.8%

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

3. Simplified 99.8%

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

6. Applied egg-rr 34.3%

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

   1. associate-*r/ 34.5%

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

   2. +-commutative 34.5%

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

   3. fma-undefine 34.5%

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

   4. associate-+r+ 34.5%

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

   5. fma-define 34.5%

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

   6. +-commutative 34.5%

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

   7. associate-+r+ 34.5%

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

   8. +-commutative 34.5%

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

   9. fma-define 34.5%

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

   10. fma-define 34.5%

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

8. Simplified 34.5%

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

9. Taylor expanded in x around 0 34.6%

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

    1. associate-*r* 34.6%

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

11. Simplified 34.6%

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

    1. sqrt-div 34.6%

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

    2. metadata-eval 34.6%

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

    3. un-div-inv 34.4%

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

    4. associate-*r/ 34.4%

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

    5. add-sqr-sqrt 32.9%

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

    6. fabs-sqr 32.9%

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

    7. add-sqr-sqrt 70.7%

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

    8. clear-num 70.7%

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

    9. un-div-inv 70.7%

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

    10. add-sqr-sqrt 32.8%

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

    11. fabs-sqr 32.8%

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

    12. add-sqr-sqrt 34.4%

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

13. Applied egg-rr 34.4%

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

    1. associate-/r/ 34.6%

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

15. Simplified 34.6%

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

16. Final simplification 34.6%

    \[\leadsto x \cdot \frac{2}{\sqrt{\pi}} \]
17. Add Preprocessing

Alternative 10: 4.1% accurate, 1849.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.0;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 0.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.0

Julia

x_m = abs(x)
function code(x_m)
	return 0.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 0.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.0

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

6. Applied egg-rr 34.3%

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

   1. associate-*r/ 34.5%

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

   2. +-commutative 34.5%

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

   3. fma-undefine 34.5%

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

   4. associate-+r+ 34.5%

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

   5. fma-define 34.5%

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

   6. +-commutative 34.5%

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

   7. associate-+r+ 34.5%

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

   8. +-commutative 34.5%

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

   9. fma-define 34.5%

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

   10. fma-define 34.5%

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

8. Simplified 34.5%

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

9. Taylor expanded in x around 0 34.6%

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

    1. associate-*r* 34.6%

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

11. Simplified 34.6%

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

    1. expm1-log1p-u 34.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]

    2. expm1-undefine 3.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}\right)} - 1} \]

    3. sqrt-div 3.9%

       \[\leadsto e^{\mathsf{log1p}\left(\left(2 \cdot x\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)} - 1 \]

    4. metadata-eval 3.9%

       \[\leadsto e^{\mathsf{log1p}\left(\left(2 \cdot x\right) \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)} - 1 \]

    5. un-div-inv 3.9%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{2 \cdot x}{\sqrt{\pi}}}\right)} - 1 \]

    6. *-commutative 3.9%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{x \cdot 2}}{\sqrt{\pi}}\right)} - 1 \]

13. Applied egg-rr 3.9%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot 2}{\sqrt{\pi}}\right)} - 1} \]
14. Step-by-step derivation

    1. sub-neg 3.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x \cdot 2}{\sqrt{\pi}}\right)} + \left(-1\right)} \]

    2. metadata-eval 3.9%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{x \cdot 2}{\sqrt{\pi}}\right)} + \color{blue}{-1} \]

    3. +-commutative 3.9%

       \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(\frac{x \cdot 2}{\sqrt{\pi}}\right)}} \]

    4. log1p-undefine 3.9%

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

    5. rem-exp-log 4.0%

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

    6. +-commutative 4.0%

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

    7. associate-/l* 4.0%

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

    8. fma-define 4.0%

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

15. Simplified 4.0%

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

16. Taylor expanded in x around 0 4.1%

    \[\leadsto -1 + \color{blue}{1} \]

17. Final simplification 4.1%

    \[\leadsto 0 \]
18. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(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)))))))