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

Percentage Accurate: 99.8% → 99.9%

Time: 13.1s

Alternatives: 8

Speedup: 4.4×

• 28.1% 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, 4.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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 0 99.8%

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

   1. *-commutative 99.8%

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

   2. fabs-mul 99.8%

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

   3. associate-*l* 99.8%

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

7. Simplified 99.8%

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

   1. rem-sqrt-square 99.8%

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

   2. add-sqr-sqrt 99.8%

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

   3. add-sqr-sqrt 40.2%

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

   4. fabs-sqr 40.2%

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

   5. add-sqr-sqrt 41.8%

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

   6. distribute-lft-in 41.8%

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

   7. distribute-lft-in 41.8%

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

9. Applied egg-rr 41.8%

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

    1. distribute-lft-out 41.8%

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

    2. distribute-lft-in 41.8%

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

11. Simplified 41.8%

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

    1. fma-undefine 41.8%

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

13. Applied egg-rr 41.8%

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

    1. fma-undefine 41.8%

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

15. Applied egg-rr 41.8%

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

16. Final simplification 41.8%

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

Alternative 2: 99.3% accurate, 4.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 0.5:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x\_m \cdot \left(0.2 \cdot {x\_m}^{4} + \left(2 + 0.6666666666666666 \cdot {x\_m}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{7} \cdot \left({\pi}^{-0.5} \cdot \left(0.047619047619047616 + \frac{0.2}{{x\_m}^{2}}\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.5)
   (*
    (pow PI -0.5)
    (*
     x_m
     (+ (* 0.2 (pow x_m 4.0)) (+ 2.0 (* 0.6666666666666666 (pow x_m 2.0))))))
   (*
    (pow x_m 7.0)
    (* (pow PI -0.5) (+ 0.047619047619047616 (/ 0.2 (pow x_m 2.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.5) {
		tmp = pow(((double) M_PI), -0.5) * (x_m * ((0.2 * pow(x_m, 4.0)) + (2.0 + (0.6666666666666666 * pow(x_m, 2.0)))));
	} else {
		tmp = pow(x_m, 7.0) * (pow(((double) M_PI), -0.5) * (0.047619047619047616 + (0.2 / pow(x_m, 2.0))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 0.5) {
		tmp = Math.pow(Math.PI, -0.5) * (x_m * ((0.2 * Math.pow(x_m, 4.0)) + (2.0 + (0.6666666666666666 * Math.pow(x_m, 2.0)))));
	} else {
		tmp = Math.pow(x_m, 7.0) * (Math.pow(Math.PI, -0.5) * (0.047619047619047616 + (0.2 / Math.pow(x_m, 2.0))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 0.5:
		tmp = math.pow(math.pi, -0.5) * (x_m * ((0.2 * math.pow(x_m, 4.0)) + (2.0 + (0.6666666666666666 * math.pow(x_m, 2.0)))))
	else:
		tmp = math.pow(x_m, 7.0) * (math.pow(math.pi, -0.5) * (0.047619047619047616 + (0.2 / math.pow(x_m, 2.0))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.5)
		tmp = Float64((pi ^ -0.5) * Float64(x_m * Float64(Float64(0.2 * (x_m ^ 4.0)) + Float64(2.0 + Float64(0.6666666666666666 * (x_m ^ 2.0))))));
	else
		tmp = Float64((x_m ^ 7.0) * Float64((pi ^ -0.5) * Float64(0.047619047619047616 + Float64(0.2 / (x_m ^ 2.0)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (abs(x_m) <= 0.5)
		tmp = (pi ^ -0.5) * (x_m * ((0.2 * (x_m ^ 4.0)) + (2.0 + (0.6666666666666666 * (x_m ^ 2.0)))));
	else
		tmp = (x_m ^ 7.0) * ((pi ^ -0.5) * (0.047619047619047616 + (0.2 / (x_m ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.5], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x$95$m * N[(N[(0.2 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x$95$m, 7.0], $MachinePrecision] * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(0.047619047619047616 + N[(0.2 / N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.5

   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 0 99.8%

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

      1. *-commutative 99.8%

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

      2. fabs-mul 99.8%

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

      3. associate-*l* 99.8%

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

   7. Simplified 99.8%

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

      1. rem-sqrt-square 99.8%

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

      2. add-sqr-sqrt 99.8%

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

      3. add-sqr-sqrt 57.2%

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

      4. fabs-sqr 57.2%

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

      5. add-sqr-sqrt 59.4%

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

      6. distribute-lft-in 59.4%

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

      7. distribute-lft-in 59.4%

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

   9. Applied egg-rr 59.4%

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

       1. distribute-lft-out 59.4%

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

       2. distribute-lft-in 59.4%

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

   11. Simplified 59.4%

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

       1. fma-undefine 59.4%

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

   13. Applied egg-rr 59.4%

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

   14. Taylor expanded in x around 0 59.1%

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

   if 0.5 < (fabs.f64 x)

   1. Initial program 99.7%

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

   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 0 99.8%

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

      1. *-commutative 99.8%

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

      2. fabs-mul 99.8%

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

      3. associate-*l* 99.8%

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

   7. Simplified 99.8%

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

      1. rem-sqrt-square 99.8%

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

      2. add-sqr-sqrt 99.8%

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

      3. add-sqr-sqrt 0.0%

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

      4. fabs-sqr 0.0%

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

      5. add-sqr-sqrt 0.1%

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

      6. distribute-lft-in 0.1%

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

      7. distribute-lft-in 0.1%

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

   9. Applied egg-rr 0.1%

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

       1. distribute-lft-out 0.1%

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

       2. distribute-lft-in 0.1%

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

   11. Simplified 0.1%

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

   12. Taylor expanded in x around inf 0.1%

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

       1. associate-*r* 0.1%

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

       2. distribute-rgt-out 0.1%

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

       3. rem-exp-log 0.1%

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

       4. exp-neg 0.1%

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

       5. unpow1/2 0.1%

          \[\leadsto {x}^{7} \cdot \left(\color{blue}{{\left(e^{-\log \pi}\right)}^{0.5}} \cdot \left(0.047619047619047616 + 0.2 \cdot \frac{1}{{x}^{2}}\right)\right) \]

       6. exp-prod 0.1%

          \[\leadsto {x}^{7} \cdot \left(\color{blue}{e^{\left(-\log \pi\right) \cdot 0.5}} \cdot \left(0.047619047619047616 + 0.2 \cdot \frac{1}{{x}^{2}}\right)\right) \]

       7. distribute-lft-neg-out 0.1%

          \[\leadsto {x}^{7} \cdot \left(e^{\color{blue}{-\log \pi \cdot 0.5}} \cdot \left(0.047619047619047616 + 0.2 \cdot \frac{1}{{x}^{2}}\right)\right) \]

       8. distribute-rgt-neg-in 0.1%

          \[\leadsto {x}^{7} \cdot \left(e^{\color{blue}{\log \pi \cdot \left(-0.5\right)}} \cdot \left(0.047619047619047616 + 0.2 \cdot \frac{1}{{x}^{2}}\right)\right) \]

       9. metadata-eval 0.1%

          \[\leadsto {x}^{7} \cdot \left(e^{\log \pi \cdot \color{blue}{-0.5}} \cdot \left(0.047619047619047616 + 0.2 \cdot \frac{1}{{x}^{2}}\right)\right) \]

       10. exp-to-pow 0.1%

           \[\leadsto {x}^{7} \cdot \left(\color{blue}{{\pi}^{-0.5}} \cdot \left(0.047619047619047616 + 0.2 \cdot \frac{1}{{x}^{2}}\right)\right) \]

       11. associate-*r/ 0.1%

           \[\leadsto {x}^{7} \cdot \left({\pi}^{-0.5} \cdot \left(0.047619047619047616 + \color{blue}{\frac{0.2 \cdot 1}{{x}^{2}}}\right)\right) \]

       12. metadata-eval 0.1%

           \[\leadsto {x}^{7} \cdot \left({\pi}^{-0.5} \cdot \left(0.047619047619047616 + \frac{\color{blue}{0.2}}{{x}^{2}}\right)\right) \]

   14. Simplified 0.1%

       \[\leadsto \color{blue}{{x}^{7} \cdot \left({\pi}^{-0.5} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot \left(0.2 \cdot {x}^{4} + \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot \left({\pi}^{-0.5} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 0.5:\\ \;\;\;\;x\_m \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{7} \cdot \left({\pi}^{-0.5} \cdot \left(0.047619047619047616 + \frac{0.2}{{x\_m}^{2}}\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.5)
   (* x_m (/ 2.0 (sqrt PI)))
   (*
    (pow x_m 7.0)
    (* (pow PI -0.5) (+ 0.047619047619047616 (/ 0.2 (pow x_m 2.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.5) {
		tmp = x_m * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = pow(x_m, 7.0) * (pow(((double) M_PI), -0.5) * (0.047619047619047616 + (0.2 / pow(x_m, 2.0))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 0.5) {
		tmp = x_m * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.pow(x_m, 7.0) * (Math.pow(Math.PI, -0.5) * (0.047619047619047616 + (0.2 / Math.pow(x_m, 2.0))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 0.5:
		tmp = x_m * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.pow(x_m, 7.0) * (math.pow(math.pi, -0.5) * (0.047619047619047616 + (0.2 / math.pow(x_m, 2.0))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.5)
		tmp = Float64(x_m * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64((x_m ^ 7.0) * Float64((pi ^ -0.5) * Float64(0.047619047619047616 + Float64(0.2 / (x_m ^ 2.0)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (abs(x_m) <= 0.5)
		tmp = x_m * (2.0 / sqrt(pi));
	else
		tmp = (x_m ^ 7.0) * ((pi ^ -0.5) * (0.047619047619047616 + (0.2 / (x_m ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.5], N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x$95$m, 7.0], $MachinePrecision] * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(0.047619047619047616 + N[(0.2 / N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.5

   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%

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

   5. Taylor expanded in x around 0 99.1%

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

      1. add-sqr-sqrt 98.5%

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

      2. fabs-sqr 98.5%

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

      3. add-sqr-sqrt 99.1%

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

      4. associate-*r* 99.1%

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

      5. add-sqr-sqrt 56.7%

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

      6. fabs-sqr 56.7%

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

      7. add-sqr-sqrt 58.8%

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

      8. sqrt-div 58.8%

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

      9. metadata-eval 58.8%

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

      10. un-div-inv 58.8%

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

   7. Applied egg-rr 58.8%

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

   if 0.5 < (fabs.f64 x)

   1. Initial program 99.7%

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

   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 0 99.8%

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

      1. *-commutative 99.8%

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

      2. fabs-mul 99.8%

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

      3. associate-*l* 99.8%

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

   7. Simplified 99.8%

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

      1. rem-sqrt-square 99.8%

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

      2. add-sqr-sqrt 99.8%

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

      3. add-sqr-sqrt 0.0%

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

      4. fabs-sqr 0.0%

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

      5. add-sqr-sqrt 0.1%

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

      6. distribute-lft-in 0.1%

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

      7. distribute-lft-in 0.1%

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

   9. Applied egg-rr 0.1%

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

       1. distribute-lft-out 0.1%

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

       2. distribute-lft-in 0.1%

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

   11. Simplified 0.1%

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

   12. Taylor expanded in x around inf 0.1%

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

       1. associate-*r* 0.1%

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

       2. distribute-rgt-out 0.1%

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

       3. rem-exp-log 0.1%

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

       4. exp-neg 0.1%

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

       5. unpow1/2 0.1%

          \[\leadsto {x}^{7} \cdot \left(\color{blue}{{\left(e^{-\log \pi}\right)}^{0.5}} \cdot \left(0.047619047619047616 + 0.2 \cdot \frac{1}{{x}^{2}}\right)\right) \]

       6. exp-prod 0.1%

          \[\leadsto {x}^{7} \cdot \left(\color{blue}{e^{\left(-\log \pi\right) \cdot 0.5}} \cdot \left(0.047619047619047616 + 0.2 \cdot \frac{1}{{x}^{2}}\right)\right) \]

       7. distribute-lft-neg-out 0.1%

          \[\leadsto {x}^{7} \cdot \left(e^{\color{blue}{-\log \pi \cdot 0.5}} \cdot \left(0.047619047619047616 + 0.2 \cdot \frac{1}{{x}^{2}}\right)\right) \]

       8. distribute-rgt-neg-in 0.1%

          \[\leadsto {x}^{7} \cdot \left(e^{\color{blue}{\log \pi \cdot \left(-0.5\right)}} \cdot \left(0.047619047619047616 + 0.2 \cdot \frac{1}{{x}^{2}}\right)\right) \]

       9. metadata-eval 0.1%

          \[\leadsto {x}^{7} \cdot \left(e^{\log \pi \cdot \color{blue}{-0.5}} \cdot \left(0.047619047619047616 + 0.2 \cdot \frac{1}{{x}^{2}}\right)\right) \]

       10. exp-to-pow 0.1%

           \[\leadsto {x}^{7} \cdot \left(\color{blue}{{\pi}^{-0.5}} \cdot \left(0.047619047619047616 + 0.2 \cdot \frac{1}{{x}^{2}}\right)\right) \]

       11. associate-*r/ 0.1%

           \[\leadsto {x}^{7} \cdot \left({\pi}^{-0.5} \cdot \left(0.047619047619047616 + \color{blue}{\frac{0.2 \cdot 1}{{x}^{2}}}\right)\right) \]

       12. metadata-eval 0.1%

           \[\leadsto {x}^{7} \cdot \left({\pi}^{-0.5} \cdot \left(0.047619047619047616 + \frac{\color{blue}{0.2}}{{x}^{2}}\right)\right) \]

   14. Simplified 0.1%

       \[\leadsto \color{blue}{{x}^{7} \cdot \left({\pi}^{-0.5} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot \left({\pi}^{-0.5} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 5.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 0.5:\\ \;\;\;\;x\_m \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{x\_m \cdot {x\_m}^{6}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.5)
   (* x_m (/ 2.0 (sqrt PI)))
   (* 0.047619047619047616 (/ (* x_m (pow x_m 6.0)) (sqrt PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.5) {
		tmp = x_m * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = 0.047619047619047616 * ((x_m * pow(x_m, 6.0)) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 0.5) {
		tmp = x_m * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = 0.047619047619047616 * ((x_m * Math.pow(x_m, 6.0)) / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 0.5:
		tmp = x_m * (2.0 / math.sqrt(math.pi))
	else:
		tmp = 0.047619047619047616 * ((x_m * math.pow(x_m, 6.0)) / math.sqrt(math.pi))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.5)
		tmp = Float64(x_m * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(0.047619047619047616 * Float64(Float64(x_m * (x_m ^ 6.0)) / sqrt(pi)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (abs(x_m) <= 0.5)
		tmp = x_m * (2.0 / sqrt(pi));
	else
		tmp = 0.047619047619047616 * ((x_m * (x_m ^ 6.0)) / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.5], N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[(x$95$m * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.5

   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%

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

   5. Taylor expanded in x around 0 99.1%

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

      1. add-sqr-sqrt 98.5%

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

      2. fabs-sqr 98.5%

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

      3. add-sqr-sqrt 99.1%

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

      4. associate-*r* 99.1%

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

      5. add-sqr-sqrt 56.7%

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

      6. fabs-sqr 56.7%

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

      7. add-sqr-sqrt 58.8%

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

      8. sqrt-div 58.8%

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

      9. metadata-eval 58.8%

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

      10. un-div-inv 58.8%

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

   7. Applied egg-rr 58.8%

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

   if 0.5 < (fabs.f64 x)

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 99.2%

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

      1. associate-*r* 99.2%

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

      2. *-commutative 99.2%

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

   7. Simplified 99.2%

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

      1. add-sqr-sqrt 99.1%

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

      2. fabs-sqr 99.1%

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

      3. add-sqr-sqrt 99.2%

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

      4. associate-*l* 99.2%

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

      5. sqrt-div 99.2%

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

      6. metadata-eval 99.2%

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

      7. un-div-inv 99.3%

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

      8. add-sqr-sqrt 0.0%

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

      9. fabs-sqr 0.0%

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

      10. add-sqr-sqrt 0.1%

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

   9. Applied egg-rr 0.1%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.5:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{x \cdot {x}^{6}}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 8.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.85:\\ \;\;\;\;x\_m \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x\_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.85)
   (* x_m (/ 2.0 (sqrt PI)))
   (* 0.047619047619047616 (* (pow PI -0.5) (pow x_m 7.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = 0.047619047619047616 * (pow(((double) M_PI), -0.5) * pow(x_m, 7.0));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = 0.047619047619047616 * (Math.pow(Math.PI, -0.5) * Math.pow(x_m, 7.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.85:
		tmp = x_m * (2.0 / math.sqrt(math.pi))
	else:
		tmp = 0.047619047619047616 * (math.pow(math.pi, -0.5) * math.pow(x_m, 7.0))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.85)
		tmp = Float64(x_m * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(0.047619047619047616 * Float64((pi ^ -0.5) * (x_m ^ 7.0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.85)
		tmp = x_m * (2.0 / sqrt(pi));
	else
		tmp = 0.047619047619047616 * ((pi ^ -0.5) * (x_m ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.85], N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 71.4%

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

      1. add-sqr-sqrt 70.9%

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

      2. fabs-sqr 70.9%

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

      3. add-sqr-sqrt 71.4%

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

      4. associate-*r* 71.4%

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

      5. add-sqr-sqrt 39.9%

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

      6. fabs-sqr 39.9%

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

      7. add-sqr-sqrt 41.5%

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

      8. sqrt-div 41.5%

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

      9. metadata-eval 41.5%

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

      10. un-div-inv 41.5%

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

   7. Applied egg-rr 41.5%

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

   if 1.8500000000000001 < 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

   5. Taylor expanded in x around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. fabs-mul 99.8%

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

      3. associate-*l* 99.8%

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

   7. Simplified 99.8%

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

      1. rem-sqrt-square 99.8%

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

      2. add-sqr-sqrt 99.8%

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

      3. add-sqr-sqrt 40.2%

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

      4. fabs-sqr 40.2%

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

      5. add-sqr-sqrt 41.8%

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

      6. distribute-lft-in 41.8%

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

      7. distribute-lft-in 41.8%

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

   9. Applied egg-rr 41.8%

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

       1. distribute-lft-out 41.8%

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

       2. distribute-lft-in 41.8%

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

   11. Simplified 41.8%

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

   12. Taylor expanded in x around inf 3.9%

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

       1. associate-*r* 3.9%

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

       2. rem-exp-log 3.9%

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

       3. exp-neg 3.9%

          \[\leadsto \left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\color{blue}{e^{-\log \pi}}} \]

       4. unpow1/2 3.9%

          \[\leadsto \left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \color{blue}{{\left(e^{-\log \pi}\right)}^{0.5}} \]

       5. exp-prod 3.9%

          \[\leadsto \left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \color{blue}{e^{\left(-\log \pi\right) \cdot 0.5}} \]

       6. distribute-lft-neg-out 3.9%

          \[\leadsto \left(0.047619047619047616 \cdot {x}^{7}\right) \cdot e^{\color{blue}{-\log \pi \cdot 0.5}} \]

       7. distribute-rgt-neg-in 3.9%

          \[\leadsto \left(0.047619047619047616 \cdot {x}^{7}\right) \cdot e^{\color{blue}{\log \pi \cdot \left(-0.5\right)}} \]

       8. metadata-eval 3.9%

          \[\leadsto \left(0.047619047619047616 \cdot {x}^{7}\right) \cdot e^{\log \pi \cdot \color{blue}{-0.5}} \]

       9. exp-to-pow 3.9%

          \[\leadsto \left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \color{blue}{{\pi}^{-0.5}} \]

       10. *-commutative 3.9%

           \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)} \]

   14. Simplified 3.9%

       \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)} \]
   15. Step-by-step derivation

       1. pow1 3.9%

          \[\leadsto \color{blue}{{\left({\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right)}^{1}} \]

   16. Applied egg-rr 3.9%

       \[\leadsto \color{blue}{{\left({\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right)}^{1}} \]
   17. Step-by-step derivation

       1. unpow1 3.9%

          \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)} \]

       2. *-commutative 3.9%

          \[\leadsto \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot {\pi}^{-0.5}} \]

       3. associate-*l* 3.9%

          \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)} \]

   18. Simplified 3.9%

       \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.3% accurate, 8.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.85:\\ \;\;\;\;x\_m \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.0022675736961451248 \cdot \frac{{x\_m}^{14}}{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.85)
   (* x_m (/ 2.0 (sqrt PI)))
   (sqrt (* 0.0022675736961451248 (/ (pow x_m 14.0) PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = sqrt((0.0022675736961451248 * (pow(x_m, 14.0) / ((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 = x_m * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.sqrt((0.0022675736961451248 * (Math.pow(x_m, 14.0) / Math.PI)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.85:
		tmp = x_m * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.sqrt((0.0022675736961451248 * (math.pow(x_m, 14.0) / math.pi)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.85)
		tmp = Float64(x_m * Float64(2.0 / sqrt(pi)));
	else
		tmp = sqrt(Float64(0.0022675736961451248 * Float64((x_m ^ 14.0) / 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 = x_m * (2.0 / sqrt(pi));
	else
		tmp = sqrt((0.0022675736961451248 * ((x_m ^ 14.0) / 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[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.0022675736961451248 * N[(N[Power[x$95$m, 14.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   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%

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

   5. Taylor expanded in x around 0 71.4%

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

      1. add-sqr-sqrt 70.9%

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

      2. fabs-sqr 70.9%

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

      3. add-sqr-sqrt 71.4%

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

      4. associate-*r* 71.4%

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

      5. add-sqr-sqrt 39.9%

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

      6. fabs-sqr 39.9%

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

      7. add-sqr-sqrt 41.5%

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

      8. sqrt-div 41.5%

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

      9. metadata-eval 41.5%

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

      10. un-div-inv 41.5%

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

   7. Applied egg-rr 41.5%

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

   if 1.8500000000000001 < 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

   5. Taylor expanded in x around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. fabs-mul 99.8%

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

      3. associate-*l* 99.8%

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

   7. Simplified 99.8%

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

      1. rem-sqrt-square 99.8%

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

      2. add-sqr-sqrt 99.8%

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

      3. add-sqr-sqrt 40.2%

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

      4. fabs-sqr 40.2%

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

      5. add-sqr-sqrt 41.8%

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

      6. distribute-lft-in 41.8%

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

      7. distribute-lft-in 41.8%

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

   9. Applied egg-rr 41.8%

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

       1. distribute-lft-out 41.8%

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

       2. distribute-lft-in 41.8%

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

   11. Simplified 41.8%

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

   12. Taylor expanded in x around inf 3.9%

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

       1. associate-*r* 3.9%

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

       2. rem-exp-log 3.9%

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

       3. exp-neg 3.9%

          \[\leadsto \left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\color{blue}{e^{-\log \pi}}} \]

       4. unpow1/2 3.9%

          \[\leadsto \left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \color{blue}{{\left(e^{-\log \pi}\right)}^{0.5}} \]

       5. exp-prod 3.9%

          \[\leadsto \left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \color{blue}{e^{\left(-\log \pi\right) \cdot 0.5}} \]

       6. distribute-lft-neg-out 3.9%

          \[\leadsto \left(0.047619047619047616 \cdot {x}^{7}\right) \cdot e^{\color{blue}{-\log \pi \cdot 0.5}} \]

       7. distribute-rgt-neg-in 3.9%

          \[\leadsto \left(0.047619047619047616 \cdot {x}^{7}\right) \cdot e^{\color{blue}{\log \pi \cdot \left(-0.5\right)}} \]

       8. metadata-eval 3.9%

          \[\leadsto \left(0.047619047619047616 \cdot {x}^{7}\right) \cdot e^{\log \pi \cdot \color{blue}{-0.5}} \]

       9. exp-to-pow 3.9%

          \[\leadsto \left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \color{blue}{{\pi}^{-0.5}} \]

       10. *-commutative 3.9%

           \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)} \]

   14. Simplified 3.9%

       \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)} \]
   15. Step-by-step derivation

       1. add-sqr-sqrt 3.7%

          \[\leadsto \color{blue}{\sqrt{{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)} \cdot \sqrt{{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)}} \]

       2. sqrt-unprod 30.5%

          \[\leadsto \color{blue}{\sqrt{\left({\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right) \cdot \left({\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right)}} \]

       3. swap-sqr 30.5%

          \[\leadsto \sqrt{\color{blue}{\left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right) \cdot \left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right)}} \]

       4. pow-prod-up 30.5%

          \[\leadsto \sqrt{\color{blue}{{\pi}^{\left(-0.5 + -0.5\right)}} \cdot \left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right)} \]

       5. metadata-eval 30.5%

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

       6. *-commutative 30.5%

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

       7. *-commutative 30.5%

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

       8. swap-sqr 30.6%

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

       9. pow-prod-up 30.6%

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

       10. metadata-eval 30.6%

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

       11. metadata-eval 30.6%

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

   16. Applied egg-rr 30.6%

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

       1. associate-*r* 30.6%

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

       2. *-commutative 30.6%

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

       3. unpow-1 30.6%

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

       4. metadata-eval 30.6%

          \[\leadsto \sqrt{0.0022675736961451248 \cdot \left(\frac{1}{\pi} \cdot {x}^{\color{blue}{\left(2 \cdot 7\right)}}\right)} \]

       5. pow-sqr 30.5%

          \[\leadsto \sqrt{0.0022675736961451248 \cdot \left(\frac{1}{\pi} \cdot \color{blue}{\left({x}^{7} \cdot {x}^{7}\right)}\right)} \]

       6. associate-*l/ 30.5%

          \[\leadsto \sqrt{0.0022675736961451248 \cdot \color{blue}{\frac{1 \cdot \left({x}^{7} \cdot {x}^{7}\right)}{\pi}}} \]

       7. *-lft-identity 30.5%

          \[\leadsto \sqrt{0.0022675736961451248 \cdot \frac{\color{blue}{{x}^{7} \cdot {x}^{7}}}{\pi}} \]

       8. pow-sqr 30.6%

          \[\leadsto \sqrt{0.0022675736961451248 \cdot \frac{\color{blue}{{x}^{\left(2 \cdot 7\right)}}}{\pi}} \]

       9. metadata-eval 30.6%

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

   18. Simplified 30.6%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.0022675736961451248 \cdot \frac{{x}^{14}}{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.8% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 71.4%

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

   1. add-sqr-sqrt 70.9%

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

   2. fabs-sqr 70.9%

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

   3. add-sqr-sqrt 71.4%

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

   4. associate-*r* 71.4%

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

   5. add-sqr-sqrt 39.9%

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

   6. fabs-sqr 39.9%

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

   7. add-sqr-sqrt 41.5%

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

   8. sqrt-div 41.5%

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

   9. metadata-eval 41.5%

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

   10. un-div-inv 41.5%

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

7. Applied egg-rr 41.5%

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

8. Final simplification 41.5%

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

Alternative 8: 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%

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

5. Taylor expanded in x around 0 71.4%

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

   1. expm1-log1p-u 71.4%

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

   2. expm1-undefine 6.9%

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

7. Applied egg-rr 4.6%

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

8. Taylor expanded in x around 0 4.3%

   \[\leadsto \color{blue}{1} - 1 \]
9. Step-by-step derivation

   1. metadata-eval 4.3%

      \[\leadsto \color{blue}{0} \]

10. Applied egg-rr 4.3%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024112 
(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)))))))