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

Percentage Accurate: 99.8% → 99.9%

Time: 11.9s

Alternatives: 10

Speedup: 3.0×

• 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, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

   1. div-inv 99.8%

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

6. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l* 99.8%

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

   3. fma-define 99.8%

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

   4. +-commutative 99.8%

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

   5. associate-+l+ 99.8%

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

   6. fma-define 99.8%

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

   7. associate-+l+ 99.8%

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

   8. +-commutative 99.8%

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

   9. +-commutative 99.8%

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

   10. fma-define 99.8%

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

   11. fma-define 99.8%

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

8. Simplified 99.8%

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

   1. fma-undefine 99.8%

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

   2. fma-undefine 99.8%

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

   3. associate-+r+ 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 99.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 99.4%

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

6. Final simplification 99.4%

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

Alternative 3: 99.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.4)
   (fabs
    (*
     (sqrt (/ 1.0 PI))
     (* (fabs x) (fma 0.6666666666666666 (pow x 2.0) 2.0))))
   (fabs (* 0.047619047619047616 (/ (pow x 7.0) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.4) {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * (fabs(x) * fma(0.6666666666666666, pow(x, 2.0), 2.0))));
	} else {
		tmp = fabs((0.047619047619047616 * (pow(x, 7.0) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.4)
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(abs(x) * fma(0.6666666666666666, (x ^ 2.0), 2.0))));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64((x ^ 7.0) / sqrt(pi))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.4], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[Power[x, 7.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.40000000000000002

   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 99.2%

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

      1. associate-*r* 99.2%

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

      2. unpow2 99.2%

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

      3. sqr-abs 99.2%

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

      4. unpow3 99.2%

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

      5. *-commutative 99.2%

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

      6. associate-*r* 99.2%

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

      7. distribute-rgt-in 99.2%

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

      8. *-commutative 99.2%

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

   7. Simplified 99.2%

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

   if 0.40000000000000002 < (fabs.f64 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 99.7%

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

   5. Taylor expanded in x around inf 99.6%

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

      1. associate-*r* 99.5%

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

      2. sqrt-div 99.5%

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

      3. metadata-eval 99.5%

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

      4. un-div-inv 99.6%

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

      5. add-sqr-sqrt 0.0%

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

      6. fabs-sqr 0.0%

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

      7. add-sqr-sqrt 99.6%

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

      8. associate-*r* 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. associate-*l* 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-/l* 99.6%

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

      4. *-commutative 99.6%

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

      5. pow-plus 99.6%

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

      6. metadata-eval 99.6%

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

   9. Simplified 99.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x):
	return math.fabs(((2.0 + (((0.047619047619047616 * math.pow(x, 6.0)) + (0.2 * math.pow(x, 4.0))) + (0.6666666666666666 * math.pow(x, 2.0)))) * (x / math.sqrt(math.pi))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

   1. div-inv 99.8%

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

6. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l* 99.8%

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

   3. fma-define 99.8%

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

   4. +-commutative 99.8%

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

   5. associate-+l+ 99.8%

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

   6. fma-define 99.8%

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

   7. associate-+l+ 99.8%

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

   8. +-commutative 99.8%

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

   9. +-commutative 99.8%

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

   10. fma-define 99.8%

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

   11. fma-define 99.8%

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

8. Simplified 99.8%

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

   1. fma-undefine 99.8%

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

   2. fma-undefine 99.8%

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

   3. associate-+r+ 99.8%

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

10. Applied egg-rr 99.8%

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

    1. metadata-eval 99.8%

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

    2. sqrt-pow1 99.8%

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

    3. inv-pow 99.8%

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

    4. add-sqr-sqrt 29.0%

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

    5. fabs-sqr 29.0%

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

    6. add-sqr-sqrt 99.8%

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

    7. sqrt-div 99.8%

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

    8. metadata-eval 99.8%

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

    9. un-div-inv 99.4%

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

12. Applied egg-rr 99.4%

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

13. Final simplification 99.4%

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

Alternative 5: 98.7% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around inf 98.9%

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

6. Final simplification 98.9%

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

Alternative 6: 99.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around inf 99.3%

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

6. Final simplification 99.3%

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

Alternative 7: 99.1% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.4)
   (fabs
    (* (sqrt (/ 1.0 PI)) (+ (* 0.6666666666666666 (pow x 3.0)) (* 2.0 x))))
   (fabs (* 0.047619047619047616 (/ (pow x 7.0) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.4) {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * ((0.6666666666666666 * pow(x, 3.0)) + (2.0 * x))));
	} else {
		tmp = fabs((0.047619047619047616 * (pow(x, 7.0) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.4) {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * ((0.6666666666666666 * Math.pow(x, 3.0)) + (2.0 * x))));
	} else {
		tmp = Math.abs((0.047619047619047616 * (Math.pow(x, 7.0) / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.4:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * ((0.6666666666666666 * math.pow(x, 3.0)) + (2.0 * x))))
	else:
		tmp = math.fabs((0.047619047619047616 * (math.pow(x, 7.0) / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.4)
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(0.6666666666666666 * (x ^ 3.0)) + Float64(2.0 * x))));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64((x ^ 7.0) / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.4)
		tmp = abs((sqrt((1.0 / pi)) * ((0.6666666666666666 * (x ^ 3.0)) + (2.0 * x))));
	else
		tmp = abs((0.047619047619047616 * ((x ^ 7.0) / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.4], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[Power[x, 7.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.40000000000000002

   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 99.2%

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

      1. associate-*r* 99.2%

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

      2. unpow2 99.2%

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

      3. sqr-abs 99.2%

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

      4. unpow3 99.2%

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

      5. *-commutative 99.2%

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

      6. associate-*r* 99.2%

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

      7. distribute-rgt-in 99.2%

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

      8. *-commutative 99.2%

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

   7. Simplified 99.2%

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

      1. add-sqr-sqrt 43.0%

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

      2. fabs-sqr 43.0%

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

      3. add-sqr-sqrt 99.2%

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

      4. fma-undefine 99.2%

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

      5. distribute-rgt-in 99.2%

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

   9. Applied egg-rr 99.2%

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

   10. Taylor expanded in x around 0 99.2%

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

   if 0.40000000000000002 < (fabs.f64 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 99.7%

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

   5. Taylor expanded in x around inf 99.6%

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

      1. associate-*r* 99.5%

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

      2. sqrt-div 99.5%

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

      3. metadata-eval 99.5%

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

      4. un-div-inv 99.6%

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

      5. add-sqr-sqrt 0.0%

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

      6. fabs-sqr 0.0%

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

      7. add-sqr-sqrt 99.6%

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

      8. associate-*r* 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. associate-*l* 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-/l* 99.6%

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

      4. *-commutative 99.6%

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

      5. pow-plus 99.6%

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

      6. metadata-eval 99.6%

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

   9. Simplified 99.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.7% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.4)
   (fabs (* x (/ 2.0 (sqrt PI))))
   (fabs (* 0.047619047619047616 (/ (pow x 7.0) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.4) {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((0.047619047619047616 * (pow(x, 7.0) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.4) {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs((0.047619047619047616 * (Math.pow(x, 7.0) / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.4:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs((0.047619047619047616 * (math.pow(x, 7.0) / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.4)
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64((x ^ 7.0) / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.4)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = abs((0.047619047619047616 * ((x ^ 7.0) / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.4], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[Power[x, 7.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.40000000000000002

   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 98.7%

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

      1. associate-*r* 98.7%

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

   7. Simplified 98.7%

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

      1. pow1 98.7%

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

      2. *-commutative 98.7%

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

      3. add-sqr-sqrt 42.8%

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

      4. fabs-sqr 42.8%

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

      5. add-sqr-sqrt 98.7%

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

      6. sqrt-div 98.7%

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

      7. metadata-eval 98.7%

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

      8. un-div-inv 98.7%

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

   9. Applied egg-rr 98.7%

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

       1. unpow1 98.7%

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

       2. associate-*r/ 98.0%

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

   11. Simplified 98.0%

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

       1. associate-*r/ 98.7%

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

       2. *-commutative 98.7%

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

   13. Applied egg-rr 98.7%

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

   if 0.40000000000000002 < (fabs.f64 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 99.7%

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

   5. Taylor expanded in x around inf 99.6%

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

      1. associate-*r* 99.5%

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

      2. sqrt-div 99.5%

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

      3. metadata-eval 99.5%

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

      4. un-div-inv 99.6%

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

      5. add-sqr-sqrt 0.0%

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

      6. fabs-sqr 0.0%

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

      7. add-sqr-sqrt 99.6%

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

      8. associate-*r* 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. associate-*l* 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-/l* 99.6%

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

      4. *-commutative 99.6%

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

      5. pow-plus 99.6%

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

      6. metadata-eval 99.6%

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

   9. Simplified 99.6%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.1% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(sqrt(Float64(Float64((x ^ 14.0) * 0.0022675736961451248) / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = abs(sqrt((((x ^ 14.0) * 0.0022675736961451248) / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[Sqrt[N[(N[(N[Power[x, 14.0], $MachinePrecision] * 0.0022675736961451248), $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 68.0%

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

      1. associate-*r* 68.0%

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

   7. Simplified 68.0%

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

      1. pow1 68.0%

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

      2. *-commutative 68.0%

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

      3. add-sqr-sqrt 28.7%

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

      4. fabs-sqr 28.7%

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

      5. add-sqr-sqrt 68.0%

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

      6. sqrt-div 68.0%

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

      7. metadata-eval 68.0%

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

      8. un-div-inv 68.0%

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

   9. Applied egg-rr 68.0%

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

       1. unpow1 68.0%

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

       2. associate-*r/ 67.5%

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

   11. Simplified 67.5%

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

       1. associate-*r/ 68.0%

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

       2. *-commutative 68.0%

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

   13. Applied egg-rr 68.0%

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

   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%

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

   5. Taylor expanded in x around inf 36.4%

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

      1. associate-*r* 36.4%

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

      2. sqrt-div 36.4%

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

      3. metadata-eval 36.4%

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

      4. un-div-inv 36.5%

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

      5. add-sqr-sqrt 1.7%

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

      6. fabs-sqr 1.7%

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

      7. add-sqr-sqrt 36.5%

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

      8. associate-*r* 36.5%

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

   7. Applied egg-rr 36.5%

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

      1. *-commutative 36.5%

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

      2. *-commutative 36.5%

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

      3. associate-*r* 36.5%

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

      4. *-commutative 36.5%

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

      5. pow-plus 36.4%

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

      6. metadata-eval 36.4%

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

      7. *-commutative 36.4%

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

      8. add-sqr-sqrt 3.4%

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

      9. sqrt-unprod 33.5%

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

      10. frac-times 33.5%

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

      11. *-commutative 33.5%

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

      12. *-commutative 33.5%

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

      13. swap-sqr 33.5%

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

      14. pow-prod-up 33.5%

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

      15. metadata-eval 33.5%

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

      16. metadata-eval 33.5%

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

      17. add-sqr-sqrt 33.5%

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

   9. Applied egg-rr 33.5%

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

4. Final simplification 68.0%

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

Alternative 10: 68.1% accurate, 9.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (fabs (* x (/ 2.0 (sqrt PI)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 68.0%

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

   1. associate-*r* 68.0%

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

7. Simplified 68.0%

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

   1. pow1 68.0%

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

   2. *-commutative 68.0%

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

   3. add-sqr-sqrt 28.7%

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

   4. fabs-sqr 28.7%

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

   5. add-sqr-sqrt 68.0%

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

   6. sqrt-div 68.0%

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

   7. metadata-eval 68.0%

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

   8. un-div-inv 68.0%

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

9. Applied egg-rr 68.0%

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

    1. unpow1 68.0%

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

    2. associate-*r/ 67.5%

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

11. Simplified 67.5%

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

    1. associate-*r/ 68.0%

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

    2. *-commutative 68.0%

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

13. Applied egg-rr 68.0%

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

14. Final simplification 68.0%

    \[\leadsto \left|x \cdot \frac{2}{\sqrt{\pi}}\right| \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024054 
(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)))))))