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

Percentage Accurate: 99.8% → 99.9%

Time: 12.2s

Alternatives: 10

Speedup: 3.0×

• 29.3% 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 99.8%

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

3. Simplified 99.8%

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

6. Applied egg-rr 35.8%

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

   1. associate-*r/ 36.0%

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

   2. +-commutative 36.0%

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

   3. fma-undefine 36.0%

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

   4. associate-+r+ 36.0%

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

   5. fma-define 36.0%

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

   6. +-commutative 36.0%

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

   7. associate-+r+ 36.0%

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

   8. +-commutative 36.0%

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

   9. fma-define 36.0%

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

   10. fma-define 36.0%

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

8. Simplified 36.0%

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

9. Final simplification 36.0%

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

Alternative 2: 99.2% accurate, 3.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (fabs
  (*
   (sqrt (/ 1.0 PI))
   (fma
    x_m
    2.0
    (fma
     0.047619047619047616
     (pow x_m 7.0)
     (* 0.6666666666666666 (pow x_m 3.0)))))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 99.1%

   \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7} + \left(0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)}\right| \]
6. Step-by-step derivation

   1. associate-+r+ 99.1%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(\left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7} + 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right) + 2 \cdot \left|x\right|\right)}\right| \]

   2. +-commutative 99.1%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(2 \cdot \left|x\right| + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7} + 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)\right)}\right| \]

   3. *-commutative 99.1%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\left|x\right| \cdot 2} + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7} + 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)\right)\right| \]

   4. fma-define 99.1%

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

   5. rem-square-sqrt 34.3%

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

   6. fabs-sqr 34.3%

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

   7. rem-square-sqrt 99.1%

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

   8. fma-define 99.1%

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

   9. rem-square-sqrt 34.6%

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

   10. fabs-sqr 34.6%

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

   11. rem-square-sqrt 78.1%

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

   12. cube-mult 78.1%

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

   13. sqr-abs 78.1%

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

   14. unpow2 78.1%

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

   15. associate-*r* 78.1%

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

   16. rem-square-sqrt 34.6%

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

   17. fabs-sqr 34.6%

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

   18. rem-square-sqrt 99.1%

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

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

Alternative 3: 99.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.047619047619047616, {x\_m}^{6}, 2\right), x\_m, 0.2 \cdot {x\_m}^{5}\right)\right| \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (fabs
  (*
   (sqrt (/ 1.0 PI))
   (fma
    (fma 0.047619047619047616 (pow x_m 6.0) 2.0)
    x_m
    (* 0.2 (pow x_m 5.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	return fabs((sqrt((1.0 / ((double) M_PI))) * fma(fma(0.047619047619047616, pow(x_m, 6.0), 2.0), x_m, (0.2 * pow(x_m, 5.0)))));
}

Julia

x_m = abs(x)
function code(x_m)
	return abs(Float64(sqrt(Float64(1.0 / pi)) * fma(fma(0.047619047619047616, (x_m ^ 6.0), 2.0), x_m, Float64(0.2 * (x_m ^ 5.0)))))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $MachinePrecision] + 2.0), $MachinePrecision] * x$95$m + N[(0.2 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 99.3%

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

   1. associate-+r+ 99.3%

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

   2. distribute-rgt-in 99.3%

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

   3. fma-define 99.3%

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

   4. +-commutative 99.3%

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

   5. fma-define 99.3%

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

   6. rem-square-sqrt 34.3%

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

   7. fabs-sqr 34.3%

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

   8. rem-square-sqrt 99.3%

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

   9. rem-square-sqrt 34.1%

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

   10. fabs-sqr 34.1%

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

   11. rem-square-sqrt 74.3%

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

   12. rem-square-sqrt 34.3%

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

   13. fabs-sqr 34.3%

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

   14. rem-square-sqrt 74.3%

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

   15. rem-square-sqrt 34.3%

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

   16. fabs-sqr 34.3%

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

   17. rem-square-sqrt 99.3%

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

7. Simplified 99.3%

   \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right), x, 0.2 \cdot {x}^{5}\right)}\right| \]

8. Final simplification 99.3%

   \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right), x, 0.2 \cdot {x}^{5}\right)\right| \]
9. Add Preprocessing

Alternative 4: 99.2% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around inf 99.1%

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

6. Final simplification 99.1%

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

Alternative 5: 96.6% accurate, 8.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.86:\\ \;\;\;\;x\_m \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{x\_m}^{14} \cdot 0.0022675736961451248}{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.86)
   (* x_m (/ 2.0 (sqrt PI)))
   (sqrt (/ (* (pow x_m 14.0) 0.0022675736961451248) PI))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.86) {
		tmp = x_m * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = sqrt(((pow(x_m, 14.0) * 0.0022675736961451248) / ((double) M_PI)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.86) {
		tmp = x_m * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.sqrt(((Math.pow(x_m, 14.0) * 0.0022675736961451248) / Math.PI));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.86:
		tmp = x_m * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.sqrt(((math.pow(x_m, 14.0) * 0.0022675736961451248) / math.pi))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.86)
		tmp = Float64(x_m * Float64(2.0 / sqrt(pi)));
	else
		tmp = sqrt(Float64(Float64((x_m ^ 14.0) * 0.0022675736961451248) / pi));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.86)
		tmp = x_m * (2.0 / sqrt(pi));
	else
		tmp = sqrt((((x_m ^ 14.0) * 0.0022675736961451248) / pi));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.86], N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[Power[x$95$m, 14.0], $MachinePrecision] * 0.0022675736961451248), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8600000000000001

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 35.8%

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

      1. associate-*r/ 36.0%

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

      2. +-commutative 36.0%

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

      3. fma-undefine 36.0%

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

      4. associate-+r+ 36.0%

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

      5. fma-define 36.0%

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

      6. +-commutative 36.0%

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

      7. associate-+r+ 36.0%

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

      8. +-commutative 36.0%

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

      9. fma-define 36.0%

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

      10. fma-define 36.0%

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

   8. Simplified 36.0%

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

   9. Taylor expanded in x around 0 35.7%

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

       1. associate-*r* 35.7%

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

   11. Simplified 35.7%

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

       1. associate-*l* 35.7%

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

       2. sqrt-div 35.7%

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

       3. metadata-eval 35.7%

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

       4. div-inv 35.5%

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

       5. clear-num 35.5%

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

       6. un-div-inv 35.5%

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

   13. Applied egg-rr 35.5%

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

       1. associate-/r/ 35.7%

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

   15. Simplified 35.7%

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

   if 1.8600000000000001 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 35.8%

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

      1. associate-*r/ 36.0%

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

      2. +-commutative 36.0%

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

      3. fma-undefine 36.0%

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

      4. associate-+r+ 36.0%

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

      5. fma-define 36.0%

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

      6. +-commutative 36.0%

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

      7. associate-+r+ 36.0%

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

      8. +-commutative 36.0%

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

      9. fma-define 36.0%

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

      10. fma-define 36.0%

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

   8. Simplified 36.0%

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

   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%

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

       1. add-sqr-sqrt 3.6%

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

       2. sqrt-unprod 33.4%

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

       3. swap-sqr 33.4%

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

       4. add-sqr-sqrt 33.4%

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

       5. *-commutative 33.4%

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

       6. *-commutative 33.4%

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

       7. swap-sqr 33.4%

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

       8. pow-prod-up 33.4%

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

       9. metadata-eval 33.4%

          \[\leadsto \sqrt{\frac{1}{\pi} \cdot \left({x}^{\color{blue}{14}} \cdot \left(0.047619047619047616 \cdot 0.047619047619047616\right)\right)} \]

       10. metadata-eval 33.4%

           \[\leadsto \sqrt{\frac{1}{\pi} \cdot \left({x}^{14} \cdot \color{blue}{0.0022675736961451248}\right)} \]

   13. Applied egg-rr 33.4%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi} \cdot \left({x}^{14} \cdot 0.0022675736961451248\right)}} \]
   14. Step-by-step derivation

       1. associate-*l/ 33.4%

          \[\leadsto \sqrt{\color{blue}{\frac{1 \cdot \left({x}^{14} \cdot 0.0022675736961451248\right)}{\pi}}} \]

       2. *-lft-identity 33.4%

          \[\leadsto \sqrt{\frac{\color{blue}{{x}^{14} \cdot 0.0022675736961451248}}{\pi}} \]

   15. Simplified 33.4%

       \[\leadsto \color{blue}{\sqrt{\frac{{x}^{14} \cdot 0.0022675736961451248}{\pi}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.86:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{x}^{14} \cdot 0.0022675736961451248}{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 8.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.86:\\ \;\;\;\;x\_m \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{{x\_m}^{7}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.86)
   (* 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 <= 1.86) {
		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 <= 1.86) {
		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 <= 1.86:
		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 <= 1.86)
		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 <= 1.86)
		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, 1.86], 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 < 1.8600000000000001

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 35.8%

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

      1. associate-*r/ 36.0%

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

      2. +-commutative 36.0%

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

      3. fma-undefine 36.0%

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

      4. associate-+r+ 36.0%

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

      5. fma-define 36.0%

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

      6. +-commutative 36.0%

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

      7. associate-+r+ 36.0%

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

      8. +-commutative 36.0%

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

      9. fma-define 36.0%

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

      10. fma-define 36.0%

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

   8. Simplified 36.0%

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

   9. Taylor expanded in x around 0 35.7%

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

       1. associate-*r* 35.7%

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

   11. Simplified 35.7%

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

       1. associate-*l* 35.7%

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

       2. sqrt-div 35.7%

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

       3. metadata-eval 35.7%

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

       4. div-inv 35.5%

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

       5. clear-num 35.5%

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

       6. un-div-inv 35.5%

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

   13. Applied egg-rr 35.5%

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

       1. associate-/r/ 35.7%

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

   15. Simplified 35.7%

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

   if 1.8600000000000001 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 35.8%

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

      1. associate-*r/ 36.0%

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

      2. +-commutative 36.0%

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

      3. fma-undefine 36.0%

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

      4. associate-+r+ 36.0%

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

      5. fma-define 36.0%

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

      6. +-commutative 36.0%

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

      7. associate-+r+ 36.0%

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

      8. +-commutative 36.0%

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

      9. fma-define 36.0%

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

      10. fma-define 36.0%

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

   8. Simplified 36.0%

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

   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%

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

       1. *-commutative 3.7%

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

       2. sqrt-div 3.7%

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

       3. metadata-eval 3.7%

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

       4. un-div-inv 3.7%

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

   13. Applied egg-rr 3.7%

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

       1. associate-/l* 3.7%

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

   15. Simplified 3.7%

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

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.86:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.9% accurate, 8.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.86:\\ \;\;\;\;x\_m \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616}{\frac{\sqrt{\pi}}{{x\_m}^{7}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.86)
   (* x_m (/ 2.0 (sqrt PI)))
   (/ 0.047619047619047616 (/ (sqrt PI) (pow x_m 7.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.86) {
		tmp = x_m * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = 0.047619047619047616 / (sqrt(((double) M_PI)) / 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 <= 1.86) {
		tmp = x_m * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = 0.047619047619047616 / (Math.sqrt(Math.PI) / Math.pow(x_m, 7.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.86:
		tmp = x_m * (2.0 / math.sqrt(math.pi))
	else:
		tmp = 0.047619047619047616 / (math.sqrt(math.pi) / math.pow(x_m, 7.0))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.86)
		tmp = Float64(x_m * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(0.047619047619047616 / Float64(sqrt(pi) / (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 <= 1.86)
		tmp = x_m * (2.0 / sqrt(pi));
	else
		tmp = 0.047619047619047616 / (sqrt(pi) / (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, 1.86], N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 / N[(N[Sqrt[Pi], $MachinePrecision] / N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8600000000000001

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 35.8%

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

      1. associate-*r/ 36.0%

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

      2. +-commutative 36.0%

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

      3. fma-undefine 36.0%

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

      4. associate-+r+ 36.0%

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

      5. fma-define 36.0%

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

      6. +-commutative 36.0%

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

      7. associate-+r+ 36.0%

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

      8. +-commutative 36.0%

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

      9. fma-define 36.0%

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

      10. fma-define 36.0%

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

   8. Simplified 36.0%

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

   9. Taylor expanded in x around 0 35.7%

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

       1. associate-*r* 35.7%

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

   11. Simplified 35.7%

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

       1. associate-*l* 35.7%

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

       2. sqrt-div 35.7%

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

       3. metadata-eval 35.7%

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

       4. div-inv 35.5%

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

       5. clear-num 35.5%

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

       6. un-div-inv 35.5%

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

   13. Applied egg-rr 35.5%

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

       1. associate-/r/ 35.7%

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

   15. Simplified 35.7%

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

   if 1.8600000000000001 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 35.8%

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

      1. associate-*r/ 36.0%

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

      2. +-commutative 36.0%

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

      3. fma-undefine 36.0%

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

      4. associate-+r+ 36.0%

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

      5. fma-define 36.0%

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

      6. +-commutative 36.0%

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

      7. associate-+r+ 36.0%

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

      8. +-commutative 36.0%

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

      9. fma-define 36.0%

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

      10. fma-define 36.0%

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

   8. Simplified 36.0%

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

   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%

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

       1. *-commutative 3.7%

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

       2. sqrt-div 3.7%

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

       3. metadata-eval 3.7%

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

       4. un-div-inv 3.7%

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

   13. Applied egg-rr 3.7%

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

       1. associate-/l* 3.7%

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

   15. Simplified 3.7%

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

       1. clear-num 3.7%

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

       2. un-div-inv 3.7%

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

   17. Applied egg-rr 3.7%

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

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.86:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616}{\frac{\sqrt{\pi}}{{x}^{7}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.9% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = x_m * ((2.0 + (0.047619047619047616 * (x_m ^ 6.0))) / 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]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

6. Applied egg-rr 35.8%

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

   1. associate-*r/ 36.0%

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

   2. +-commutative 36.0%

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

   3. fma-undefine 36.0%

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

   4. associate-+r+ 36.0%

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

   5. fma-define 36.0%

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

   6. +-commutative 36.0%

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

   7. associate-+r+ 36.0%

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

   8. +-commutative 36.0%

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

   9. fma-define 36.0%

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

   10. fma-define 36.0%

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

8. Simplified 36.0%

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

9. Taylor expanded in x around inf 35.6%

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

10. Final simplification 35.6%

    \[\leadsto x \cdot \frac{2 + 0.047619047619047616 \cdot {x}^{6}}{\sqrt{\pi}} \]
11. Add Preprocessing

Alternative 9: 67.1% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

6. Applied egg-rr 35.8%

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

   1. associate-*r/ 36.0%

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

   2. +-commutative 36.0%

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

   3. fma-undefine 36.0%

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

   4. associate-+r+ 36.0%

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

   5. fma-define 36.0%

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

   6. +-commutative 36.0%

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

   7. associate-+r+ 36.0%

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

   8. +-commutative 36.0%

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

   9. fma-define 36.0%

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

   10. fma-define 36.0%

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

8. Simplified 36.0%

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

9. Taylor expanded in x around 0 35.7%

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

    1. associate-*r* 35.7%

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

11. Simplified 35.7%

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

    1. associate-*l* 35.7%

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

    2. sqrt-div 35.7%

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

    3. metadata-eval 35.7%

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

    4. div-inv 35.5%

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

    5. clear-num 35.5%

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

    6. un-div-inv 35.5%

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

13. Applied egg-rr 35.5%

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

    1. associate-/r/ 35.7%

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

15. Simplified 35.7%

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

16. Final simplification 35.7%

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

Alternative 10: 4.1% accurate, 1849.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

6. Applied egg-rr 35.8%

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

   1. associate-*r/ 36.0%

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

   2. +-commutative 36.0%

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

   3. fma-undefine 36.0%

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

   4. associate-+r+ 36.0%

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

   5. fma-define 36.0%

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

   6. +-commutative 36.0%

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

   7. associate-+r+ 36.0%

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

   8. +-commutative 36.0%

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

   9. fma-define 36.0%

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

   10. fma-define 36.0%

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

8. Simplified 36.0%

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

9. Taylor expanded in x around 0 35.7%

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

    1. associate-*r* 35.7%

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

11. Simplified 35.7%

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

    1. expm1-log1p-u 35.6%

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

    2. expm1-undefine 4.0%

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

    3. associate-*l* 4.0%

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

    4. sqrt-div 4.0%

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

    5. metadata-eval 4.0%

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

    6. div-inv 4.0%

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

    7. *-commutative 4.0%

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

13. Applied egg-rr 4.0%

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

    1. sub-neg 4.0%

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

    2. metadata-eval 4.0%

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

    3. +-commutative 4.0%

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

    4. log1p-undefine 4.0%

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

    5. rem-exp-log 4.2%

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

    6. +-commutative 4.2%

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

    7. associate-*l/ 4.2%

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

    8. associate-/l* 4.2%

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

    9. fma-define 4.2%

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

15. Simplified 4.2%

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

16. Taylor expanded in x around 0 4.1%

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

17. Final simplification 4.1%

    \[\leadsto 0 \]
18. Add Preprocessing

Reproduce

herbie shell --seed 2024068 
(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)))))))