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

Percentage Accurate: 99.8% → 99.9%

Time: 12.0s

Alternatives: 10

Speedup: 3.0×

• 28.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 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 99.9%

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

3. Simplified 99.9%

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

6. Applied egg-rr 38.0%

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

   1. associate-*r/ 38.2%

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

   2. fma-define 38.2%

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

   3. +-commutative 38.2%

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

   4. associate-+l+ 38.2%

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

   5. fma-undefine 38.2%

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

   6. associate-+l+ 38.2%

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

   7. +-commutative 38.2%

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

   8. +-commutative 38.2%

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

   9. fma-define 38.2%

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

   10. fma-define 38.2%

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

8. Simplified 38.2%

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

Alternative 2: 99.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 99.2%

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

6. Final simplification 99.2%

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

Alternative 3: 98.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 98.7%

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

6. Final simplification 98.7%

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

Alternative 4: 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 99.9%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 98.7%

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

6. Taylor expanded in x around inf 98.6%

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

7. Final simplification 98.6%

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

Alternative 5: 99.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.85:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x\_m}^{3} + x\_m \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{7} \cdot \frac{0.047619047619047616 + 0.2 \cdot {x\_m}^{-2}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.85)
   (* (sqrt (/ 1.0 PI)) (+ (* 0.6666666666666666 (pow x_m 3.0)) (* x_m 2.0)))
   (*
    (pow x_m 7.0)
    (/ (+ 0.047619047619047616 (* 0.2 (pow x_m -2.0))) (sqrt PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = sqrt((1.0 / ((double) M_PI))) * ((0.6666666666666666 * pow(x_m, 3.0)) + (x_m * 2.0));
	} else {
		tmp = pow(x_m, 7.0) * ((0.047619047619047616 + (0.2 * pow(x_m, -2.0))) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = Math.sqrt((1.0 / Math.PI)) * ((0.6666666666666666 * Math.pow(x_m, 3.0)) + (x_m * 2.0));
	} else {
		tmp = Math.pow(x_m, 7.0) * ((0.047619047619047616 + (0.2 * Math.pow(x_m, -2.0))) / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.85:
		tmp = math.sqrt((1.0 / math.pi)) * ((0.6666666666666666 * math.pow(x_m, 3.0)) + (x_m * 2.0))
	else:
		tmp = math.pow(x_m, 7.0) * ((0.047619047619047616 + (0.2 * math.pow(x_m, -2.0))) / math.sqrt(math.pi))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.85)
		tmp = Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(0.6666666666666666 * (x_m ^ 3.0)) + Float64(x_m * 2.0)));
	else
		tmp = Float64((x_m ^ 7.0) * Float64(Float64(0.047619047619047616 + Float64(0.2 * (x_m ^ -2.0))) / sqrt(pi)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.85)
		tmp = sqrt((1.0 / pi)) * ((0.6666666666666666 * (x_m ^ 3.0)) + (x_m * 2.0));
	else
		tmp = (x_m ^ 7.0) * ((0.047619047619047616 + (0.2 * (x_m ^ -2.0))) / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.85], N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(0.6666666666666666 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x$95$m, 7.0], $MachinePrecision] * N[(N[(0.047619047619047616 + N[(0.2 * N[Power[x$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 38.0%

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

      1. associate-*r/ 38.2%

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

      2. fma-define 38.2%

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

      3. +-commutative 38.2%

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

      4. associate-+l+ 38.2%

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

      5. fma-undefine 38.2%

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

      6. associate-+l+ 38.2%

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

      7. +-commutative 38.2%

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

      8. +-commutative 38.2%

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

      9. fma-define 38.2%

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

      10. fma-define 38.2%

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

   8. Simplified 38.2%

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

   9. Taylor expanded in x around 0 38.1%

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

       1. distribute-lft-in 38.1%

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

       2. fma-define 38.1%

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

       3. *-commutative 38.1%

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

       4. associate-*r* 38.1%

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

       5. *-commutative 38.1%

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

       6. fma-undefine 38.1%

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

   11. Simplified 38.1%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right)} \]

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 38.0%

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

      1. associate-*r/ 38.2%

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

      2. fma-define 38.2%

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

      3. +-commutative 38.2%

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

      4. associate-+l+ 38.2%

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

      5. fma-undefine 38.2%

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

      6. associate-+l+ 38.2%

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

      7. +-commutative 38.2%

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

      8. +-commutative 38.2%

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

      9. fma-define 38.2%

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

      10. fma-define 38.2%

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

   8. Simplified 38.2%

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

   9. Taylor expanded in x around inf 1.4%

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

       1. associate-*r* 1.4%

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

       2. distribute-rgt-out 1.4%

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

       3. associate-*r/ 1.4%

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

       4. metadata-eval 1.4%

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

   11. Simplified 1.4%

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

       1. distribute-rgt-in 1.4%

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

       2. distribute-lft-in 1.4%

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

       3. sqrt-div 1.4%

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

       4. metadata-eval 1.4%

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

       5. un-div-inv 1.4%

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

       6. sqrt-div 1.4%

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

       7. metadata-eval 1.4%

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

       8. un-div-inv 1.4%

          \[\leadsto {x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}} + {x}^{7} \cdot \color{blue}{\frac{\frac{0.2}{{x}^{2}}}{\sqrt{\pi}}} \]

       9. div-inv 1.4%

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

       10. pow-flip 1.4%

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

       11. metadata-eval 1.4%

           \[\leadsto {x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}} + {x}^{7} \cdot \frac{0.2 \cdot {x}^{\color{blue}{-2}}}{\sqrt{\pi}} \]

   13. Applied egg-rr 1.4%

       \[\leadsto \color{blue}{{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}} + {x}^{7} \cdot \frac{0.2 \cdot {x}^{-2}}{\sqrt{\pi}}} \]
   14. Step-by-step derivation

       1. associate-*r/ 1.4%

          \[\leadsto \color{blue}{\frac{{x}^{7} \cdot 0.047619047619047616}{\sqrt{\pi}}} + {x}^{7} \cdot \frac{0.2 \cdot {x}^{-2}}{\sqrt{\pi}} \]

       2. associate-*l/ 1.4%

          \[\leadsto \color{blue}{\frac{{x}^{7}}{\sqrt{\pi}} \cdot 0.047619047619047616} + {x}^{7} \cdot \frac{0.2 \cdot {x}^{-2}}{\sqrt{\pi}} \]

       3. associate-*r/ 1.4%

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

       4. associate-*l/ 1.4%

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

       5. +-commutative 1.4%

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

       6. distribute-lft-out 1.4%

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

       7. fma-undefine 1.4%

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

       8. associate-*l/ 1.4%

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

       9. associate-*r/ 1.4%

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

   15. Simplified 1.4%

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

       1. fma-undefine 1.4%

          \[\leadsto {x}^{7} \cdot \frac{\color{blue}{0.2 \cdot {x}^{-2} + 0.047619047619047616}}{\sqrt{\pi}} \]

   17. Applied egg-rr 1.4%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3} + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot \frac{0.047619047619047616 + 0.2 \cdot {x}^{-2}}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 8.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.1:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x\_m}^{3} + x\_m \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot {x\_m}^{6}\right) \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.1)
   (* (sqrt (/ 1.0 PI)) (+ (* 0.6666666666666666 (pow x_m 3.0)) (* x_m 2.0)))
   (* (* x_m (pow x_m 6.0)) (/ 0.047619047619047616 (sqrt PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.1) {
		tmp = sqrt((1.0 / ((double) M_PI))) * ((0.6666666666666666 * pow(x_m, 3.0)) + (x_m * 2.0));
	} else {
		tmp = (x_m * pow(x_m, 6.0)) * (0.047619047619047616 / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.1) {
		tmp = Math.sqrt((1.0 / Math.PI)) * ((0.6666666666666666 * Math.pow(x_m, 3.0)) + (x_m * 2.0));
	} else {
		tmp = (x_m * Math.pow(x_m, 6.0)) * (0.047619047619047616 / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.1:
		tmp = math.sqrt((1.0 / math.pi)) * ((0.6666666666666666 * math.pow(x_m, 3.0)) + (x_m * 2.0))
	else:
		tmp = (x_m * math.pow(x_m, 6.0)) * (0.047619047619047616 / math.sqrt(math.pi))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.1)
		tmp = Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(0.6666666666666666 * (x_m ^ 3.0)) + Float64(x_m * 2.0)));
	else
		tmp = Float64(Float64(x_m * (x_m ^ 6.0)) * Float64(0.047619047619047616 / sqrt(pi)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.1)
		tmp = sqrt((1.0 / pi)) * ((0.6666666666666666 * (x_m ^ 3.0)) + (x_m * 2.0));
	else
		tmp = (x_m * (x_m ^ 6.0)) * (0.047619047619047616 / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.1], N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(0.6666666666666666 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision] * N[(0.047619047619047616 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.10000000000000009

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 38.0%

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

      1. associate-*r/ 38.2%

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

      2. fma-define 38.2%

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

      3. +-commutative 38.2%

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

      4. associate-+l+ 38.2%

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

      5. fma-undefine 38.2%

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

      6. associate-+l+ 38.2%

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

      7. +-commutative 38.2%

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

      8. +-commutative 38.2%

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

      9. fma-define 38.2%

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

      10. fma-define 38.2%

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

   8. Simplified 38.2%

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

   9. Taylor expanded in x around 0 38.1%

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

       1. distribute-lft-in 38.1%

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

       2. fma-define 38.1%

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

       3. *-commutative 38.1%

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

       4. associate-*r* 38.1%

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

       5. *-commutative 38.1%

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

       6. fma-undefine 38.1%

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

   11. Simplified 38.1%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right)} \]

   if 2.10000000000000009 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 38.0%

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

      1. associate-*r/ 38.2%

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

      2. fma-define 38.2%

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

      3. +-commutative 38.2%

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

      4. associate-+l+ 38.2%

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

      5. fma-undefine 38.2%

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

      6. associate-+l+ 38.2%

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

      7. +-commutative 38.2%

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

      8. +-commutative 38.2%

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

      9. fma-define 38.2%

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

      10. fma-define 38.2%

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

   8. Simplified 38.2%

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

   9. Taylor expanded in x around inf 3.7%

      \[\leadsto x \cdot \color{blue}{\left(0.047619047619047616 \cdot \left({x}^{6} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]
   10. Step-by-step derivation

       1. pow1 3.7%

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

       2. associate-*r* 3.7%

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

       3. *-commutative 3.7%

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

       4. associate-*l* 3.7%

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

       5. sqrt-div 3.7%

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

       6. metadata-eval 3.7%

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

       7. un-div-inv 3.7%

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

   11. Applied egg-rr 3.7%

       \[\leadsto \color{blue}{{\left(x \cdot \left({x}^{6} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right)\right)}^{1}} \]
   12. Step-by-step derivation

       1. unpow1 3.7%

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

       2. associate-*r* 3.7%

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

   13. Simplified 3.7%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3} + x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot {x}^{6}\right) \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.9:\\ \;\;\;\;x\_m \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot {x\_m}^{6}\right) \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.9)
   (* x_m (/ 2.0 (sqrt PI)))
   (* (* x_m (pow x_m 6.0)) (/ 0.047619047619047616 (sqrt PI)))))

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 = (x_m * pow(x_m, 6.0)) * (0.047619047619047616 / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.9) {
		tmp = x_m * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = (x_m * Math.pow(x_m, 6.0)) * (0.047619047619047616 / Math.sqrt(Math.PI));
	}
	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 = (x_m * math.pow(x_m, 6.0)) * (0.047619047619047616 / math.sqrt(math.pi))
	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(Float64(x_m * (x_m ^ 6.0)) * Float64(0.047619047619047616 / sqrt(pi)));
	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 = (x_m * (x_m ^ 6.0)) * (0.047619047619047616 / sqrt(pi));
	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[(x$95$m * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision] * N[(0.047619047619047616 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 38.0%

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

      1. associate-*r/ 38.2%

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

      2. fma-define 38.2%

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

      3. +-commutative 38.2%

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

      4. associate-+l+ 38.2%

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

      5. fma-undefine 38.2%

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

      6. associate-+l+ 38.2%

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

      7. +-commutative 38.2%

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

      8. +-commutative 38.2%

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

      9. fma-define 38.2%

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

      10. fma-define 38.2%

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

   8. Simplified 38.2%

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

   9. Taylor expanded in x around 0 38.0%

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

       1. associate-*r* 38.0%

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

   11. Simplified 38.0%

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

       1. sqrt-div 38.0%

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

       2. metadata-eval 38.0%

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

       3. un-div-inv 37.8%

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

       4. *-commutative 37.8%

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

   13. Applied egg-rr 37.8%

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

       1. associate-/l* 38.0%

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

   15. Simplified 38.0%

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

   if 1.8999999999999999 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 38.0%

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

      1. associate-*r/ 38.2%

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

      2. fma-define 38.2%

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

      3. +-commutative 38.2%

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

      4. associate-+l+ 38.2%

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

      5. fma-undefine 38.2%

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

      6. associate-+l+ 38.2%

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

      7. +-commutative 38.2%

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

      8. +-commutative 38.2%

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

      9. fma-define 38.2%

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

      10. fma-define 38.2%

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

   8. Simplified 38.2%

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

   9. Taylor expanded in x around inf 3.7%

      \[\leadsto x \cdot \color{blue}{\left(0.047619047619047616 \cdot \left({x}^{6} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]
   10. Step-by-step derivation

       1. pow1 3.7%

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

       2. associate-*r* 3.7%

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

       3. *-commutative 3.7%

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

       4. associate-*l* 3.7%

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

       5. sqrt-div 3.7%

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

       6. metadata-eval 3.7%

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

       7. un-div-inv 3.7%

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

   11. Applied egg-rr 3.7%

       \[\leadsto \color{blue}{{\left(x \cdot \left({x}^{6} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right)\right)}^{1}} \]
   12. Step-by-step derivation

       1. unpow1 3.7%

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

       2. associate-*r* 3.7%

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

   13. Simplified 3.7%

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

Alternative 8: 98.8% accurate, 8.7× speedup

Math

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

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 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 38.0%

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

      1. associate-*r/ 38.2%

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

      2. fma-define 38.2%

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

      3. +-commutative 38.2%

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

      4. associate-+l+ 38.2%

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

      5. fma-undefine 38.2%

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

      6. associate-+l+ 38.2%

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

      7. +-commutative 38.2%

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

      8. +-commutative 38.2%

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

      9. fma-define 38.2%

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

      10. fma-define 38.2%

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

   8. Simplified 38.2%

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

   9. Taylor expanded in x around 0 38.0%

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

       1. associate-*r* 38.0%

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

   11. Simplified 38.0%

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

       1. sqrt-div 38.0%

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

       2. metadata-eval 38.0%

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

       3. un-div-inv 37.8%

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

       4. *-commutative 37.8%

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

   13. Applied egg-rr 37.8%

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

       1. associate-/l* 38.0%

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

   15. Simplified 38.0%

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

   if 1.8999999999999999 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 38.0%

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

      1. associate-*r/ 38.2%

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

      2. fma-define 38.2%

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

      3. +-commutative 38.2%

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

      4. associate-+l+ 38.2%

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

      5. fma-undefine 38.2%

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

      6. associate-+l+ 38.2%

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

      7. +-commutative 38.2%

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

      8. +-commutative 38.2%

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

      9. fma-define 38.2%

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

      10. fma-define 38.2%

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

   8. Simplified 38.2%

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

   9. Taylor expanded in x around 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. Add Preprocessing

Alternative 9: 98.8% accurate, 8.8× speedup

Math

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

FPCore

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

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 = 0.047619047619047616 * (pow(x_m, 7.0) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.9) {
		tmp = x_m * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = 0.047619047619047616 * (Math.pow(x_m, 7.0) / Math.sqrt(Math.PI));
	}
	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 = 0.047619047619047616 * (math.pow(x_m, 7.0) / math.sqrt(math.pi))
	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(0.047619047619047616 * Float64((x_m ^ 7.0) / sqrt(pi)));
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 38.0%

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

      1. associate-*r/ 38.2%

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

      2. fma-define 38.2%

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

      3. +-commutative 38.2%

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

      4. associate-+l+ 38.2%

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

      5. fma-undefine 38.2%

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

      6. associate-+l+ 38.2%

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

      7. +-commutative 38.2%

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

      8. +-commutative 38.2%

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

      9. fma-define 38.2%

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

      10. fma-define 38.2%

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

   8. Simplified 38.2%

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

   9. Taylor expanded in x around 0 38.0%

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

       1. associate-*r* 38.0%

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

   11. Simplified 38.0%

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

       1. sqrt-div 38.0%

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

       2. metadata-eval 38.0%

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

       3. un-div-inv 37.8%

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

       4. *-commutative 37.8%

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

   13. Applied egg-rr 37.8%

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

       1. associate-/l* 38.0%

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

   15. Simplified 38.0%

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

   if 1.8999999999999999 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 38.0%

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

      1. associate-*r/ 38.2%

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

      2. fma-define 38.2%

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

      3. +-commutative 38.2%

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

      4. associate-+l+ 38.2%

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

      5. fma-undefine 38.2%

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

      6. associate-+l+ 38.2%

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

      7. +-commutative 38.2%

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

      8. +-commutative 38.2%

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

      9. fma-define 38.2%

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

      10. fma-define 38.2%

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

   8. Simplified 38.2%

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

   9. Taylor expanded in x around inf 3.7%

      \[\leadsto x \cdot \color{blue}{\left(0.047619047619047616 \cdot \left({x}^{6} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]
   10. Step-by-step derivation

       1. pow1 3.7%

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

       2. associate-*r* 3.7%

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

       3. *-commutative 3.7%

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

       4. associate-*l* 3.7%

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

       5. sqrt-div 3.7%

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

       6. metadata-eval 3.7%

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

       7. un-div-inv 3.7%

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

   11. Applied egg-rr 3.7%

       \[\leadsto \color{blue}{{\left(x \cdot \left({x}^{6} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right)\right)}^{1}} \]
   12. Step-by-step derivation

       1. unpow1 3.7%

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

       2. associate-*r* 3.7%

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

       3. associate-*r/ 3.7%

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

       4. *-commutative 3.7%

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

       5. pow-plus 3.7%

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

       6. metadata-eval 3.7%

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

       7. *-commutative 3.7%

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

       8. associate-*r/ 3.7%

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

   13. Simplified 3.7%

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

Alternative 10: 67.7% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

6. Applied egg-rr 38.0%

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

   1. associate-*r/ 38.2%

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

   2. fma-define 38.2%

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

   3. +-commutative 38.2%

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

   4. associate-+l+ 38.2%

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

   5. fma-undefine 38.2%

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

   6. associate-+l+ 38.2%

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

   7. +-commutative 38.2%

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

   8. +-commutative 38.2%

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

   9. fma-define 38.2%

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

   10. fma-define 38.2%

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

8. Simplified 38.2%

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

9. Taylor expanded in x around 0 38.0%

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

    1. associate-*r* 38.0%

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

11. Simplified 38.0%

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

    1. sqrt-div 38.0%

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

    2. metadata-eval 38.0%

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

    3. un-div-inv 37.8%

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

    4. *-commutative 37.8%

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

13. Applied egg-rr 37.8%

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

    1. associate-/l* 38.0%

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

15. Simplified 38.0%

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

Reproduce

herbie shell --seed 2024085 
(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)))))))