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

Percentage Accurate: 99.8% → 99.8%

Time: 11.5s

Alternatives: 11

Speedup: 5.7×

• 27.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

7. Simplified 99.8%

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

Alternative 2: 99.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.6666666666666666 \cdot {x}^{2} + 0.2 \cdot {x}^{4}\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fabs x)
  (fabs
   (*
    (sqrt (/ 1.0 PI))
    (+
     2.0
     (+
      (* 0.047619047619047616 (pow x 6.0))
      (+ (* 0.6666666666666666 (pow x 2.0)) (* 0.2 (pow x 4.0)))))))))

C

double code(double x) {
	return fabs(x) * fabs((sqrt((1.0 / ((double) M_PI))) * (2.0 + ((0.047619047619047616 * pow(x, 6.0)) + ((0.6666666666666666 * pow(x, 2.0)) + (0.2 * pow(x, 4.0)))))));
}

Java

public static double code(double x) {
	return Math.abs(x) * Math.abs((Math.sqrt((1.0 / Math.PI)) * (2.0 + ((0.047619047619047616 * Math.pow(x, 6.0)) + ((0.6666666666666666 * Math.pow(x, 2.0)) + (0.2 * Math.pow(x, 4.0)))))));
}

Python

def code(x):
	return math.fabs(x) * math.fabs((math.sqrt((1.0 / math.pi)) * (2.0 + ((0.047619047619047616 * math.pow(x, 6.0)) + ((0.6666666666666666 * math.pow(x, 2.0)) + (0.2 * math.pow(x, 4.0)))))))

Julia

function code(x)
	return Float64(abs(x) * abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(2.0 + Float64(Float64(0.047619047619047616 * (x ^ 6.0)) + Float64(Float64(0.6666666666666666 * (x ^ 2.0)) + Float64(0.2 * (x ^ 4.0))))))))
end

MATLAB

function tmp = code(x)
	tmp = abs(x) * abs((sqrt((1.0 / pi)) * (2.0 + ((0.047619047619047616 * (x ^ 6.0)) + ((0.6666666666666666 * (x ^ 2.0)) + (0.2 * (x ^ 4.0)))))));
end

Wolfram

code[x_] := N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(2.0 + N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $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. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

7. Simplified 99.8%

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

8. Taylor expanded in x around 0 99.8%

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

   1. *-commutative 99.8%

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

10. Simplified 99.8%

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

    1. pow2 99.8%

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

12. Applied egg-rr 99.8%

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

    1. pow2 99.8%

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

14. Applied egg-rr 99.8%

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

15. Final simplification 99.8%

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

Alternative 4: 34.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.0001)
   (* (pow PI -0.5) (* x (+ 2.0 (* 0.6666666666666666 (pow x 2.0)))))
   (/ (* 0.047619047619047616 (pow x 7.0)) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.0001) {
		tmp = pow(((double) M_PI), -0.5) * (x * (2.0 + (0.6666666666666666 * pow(x, 2.0))));
	} else {
		tmp = (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) <= 0.0001) {
		tmp = Math.pow(Math.PI, -0.5) * (x * (2.0 + (0.6666666666666666 * Math.pow(x, 2.0))));
	} else {
		tmp = (0.047619047619047616 * Math.pow(x, 7.0)) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.0001:
		tmp = math.pow(math.pi, -0.5) * (x * (2.0 + (0.6666666666666666 * math.pow(x, 2.0))))
	else:
		tmp = (0.047619047619047616 * math.pow(x, 7.0)) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.0001)
		tmp = Float64((pi ^ -0.5) * Float64(x * Float64(2.0 + Float64(0.6666666666666666 * (x ^ 2.0)))));
	else
		tmp = Float64(Float64(0.047619047619047616 * (x ^ 7.0)) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.0001)
		tmp = (pi ^ -0.5) * (x * (2.0 + (0.6666666666666666 * (x ^ 2.0))));
	else
		tmp = (0.047619047619047616 * (x ^ 7.0)) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.0001], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * N[(2.0 + N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000005e-4

   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%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0 99.8%

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

      1. associate-*r* 99.8%

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

      2. fma-define 99.8%

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

      3. add-sqr-sqrt 52.5%

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

      4. fabs-sqr 52.5%

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

      5. add-sqr-sqrt 99.5%

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

      6. add-sqr-sqrt 52.2%

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

      7. fabs-sqr 52.2%

         \[\leadsto {\pi}^{-0.5} \cdot \mathsf{fma}\left(0.6666666666666666 \cdot {x}^{2}, x, 2 \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) \]

      8. add-sqr-sqrt 54.2%

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

   10. Applied egg-rr 54.2%

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

       1. fma-undefine 54.2%

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

       2. distribute-rgt-out 54.2%

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

   12. Simplified 54.2%

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

   if 1.00000000000000005e-4 < (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 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around inf 97.6%

      \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)} \]
   9. Step-by-step derivation

      1. metadata-eval 97.6%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left({x}^{\color{blue}{\left(2 \cdot 3\right)}} \cdot \left|x\right|\right)\right) \]

      2. pow-sqr 97.6%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{\left({x}^{3} \cdot {x}^{3}\right)} \cdot \left|x\right|\right)\right) \]

      3. cube-prod 97.5%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{{\left(x \cdot x\right)}^{3}} \cdot \left|x\right|\right)\right) \]

      4. sqr-abs 97.5%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left({\color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)}}^{3} \cdot \left|x\right|\right)\right) \]

      5. cube-prod 97.6%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{\left({\left(\left|x\right|\right)}^{3} \cdot {\left(\left|x\right|\right)}^{3}\right)} \cdot \left|x\right|\right)\right) \]

      6. pow-sqr 97.6%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{{\left(\left|x\right|\right)}^{\left(2 \cdot 3\right)}} \cdot \left|x\right|\right)\right) \]

      7. metadata-eval 97.6%

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

      8. pow-plus 97.6%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(6 + 1\right)}}\right) \]

      9. metadata-eval 97.6%

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

   10. Simplified 97.6%

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

       1. add-sqr-sqrt 97.6%

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

       2. sqrt-unprod 89.0%

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

       3. swap-sqr 89.0%

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

       4. pow-prod-up 89.0%

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

       5. metadata-eval 89.0%

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

       6. *-commutative 89.0%

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

       7. *-commutative 89.0%

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

       8. swap-sqr 89.0%

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

       9. pow-prod-up 89.0%

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

       10. add-sqr-sqrt 0.0%

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

       11. fabs-sqr 0.0%

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

       12. add-sqr-sqrt 89.0%

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

       13. metadata-eval 89.0%

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

       14. metadata-eval 89.0%

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

   12. Applied egg-rr 89.0%

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

       1. unpow-1 89.0%

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

   14. Simplified 89.0%

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

       1. associate-*l/ 89.0%

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

       2. *-un-lft-identity 89.0%

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

       3. sqrt-div 89.0%

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

       4. sqrt-prod 89.0%

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

       5. sqrt-pow1 0.1%

          \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{14}{2}\right)}} \cdot \sqrt{0.0022675736961451248}}{\sqrt{\pi}} \]

       6. metadata-eval 0.1%

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

       7. metadata-eval 0.1%

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

       8. *-commutative 0.1%

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

   16. Applied egg-rr 0.1%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 34.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.0001)
   (* (pow PI -0.5) (* x 2.0))
   (/ (* 0.047619047619047616 (pow x 7.0)) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.0001) {
		tmp = pow(((double) M_PI), -0.5) * (x * 2.0);
	} else {
		tmp = (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) <= 0.0001) {
		tmp = Math.pow(Math.PI, -0.5) * (x * 2.0);
	} else {
		tmp = (0.047619047619047616 * Math.pow(x, 7.0)) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.0001:
		tmp = math.pow(math.pi, -0.5) * (x * 2.0)
	else:
		tmp = (0.047619047619047616 * math.pow(x, 7.0)) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.0001)
		tmp = Float64((pi ^ -0.5) * Float64(x * 2.0));
	else
		tmp = Float64(Float64(0.047619047619047616 * (x ^ 7.0)) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.0001)
		tmp = (pi ^ -0.5) * (x * 2.0);
	else
		tmp = (0.047619047619047616 * (x ^ 7.0)) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.0001], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000005e-4

   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%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0 99.8%

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

   9. Taylor expanded in x around 0 99.3%

      \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(2 \cdot \left|x\right|\right)} \]
   10. Step-by-step derivation

       1. metadata-eval 99.3%

          \[\leadsto {\pi}^{-0.5} \cdot \left(\color{blue}{\left|2\right|} \cdot \left|x\right|\right) \]

       2. fabs-mul 99.3%

          \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left|2 \cdot x\right|} \]

       3. rem-square-sqrt 52.1%

          \[\leadsto {\pi}^{-0.5} \cdot \left|\color{blue}{\sqrt{2 \cdot x} \cdot \sqrt{2 \cdot x}}\right| \]

       4. fabs-sqr 52.1%

          \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(\sqrt{2 \cdot x} \cdot \sqrt{2 \cdot x}\right)} \]

       5. rem-square-sqrt 54.0%

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

   11. Simplified 54.0%

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

   if 1.00000000000000005e-4 < (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 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around inf 97.6%

      \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)} \]
   9. Step-by-step derivation

      1. metadata-eval 97.6%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left({x}^{\color{blue}{\left(2 \cdot 3\right)}} \cdot \left|x\right|\right)\right) \]

      2. pow-sqr 97.6%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{\left({x}^{3} \cdot {x}^{3}\right)} \cdot \left|x\right|\right)\right) \]

      3. cube-prod 97.5%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{{\left(x \cdot x\right)}^{3}} \cdot \left|x\right|\right)\right) \]

      4. sqr-abs 97.5%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left({\color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)}}^{3} \cdot \left|x\right|\right)\right) \]

      5. cube-prod 97.6%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{\left({\left(\left|x\right|\right)}^{3} \cdot {\left(\left|x\right|\right)}^{3}\right)} \cdot \left|x\right|\right)\right) \]

      6. pow-sqr 97.6%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{{\left(\left|x\right|\right)}^{\left(2 \cdot 3\right)}} \cdot \left|x\right|\right)\right) \]

      7. metadata-eval 97.6%

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

      8. pow-plus 97.6%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(6 + 1\right)}}\right) \]

      9. metadata-eval 97.6%

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

   10. Simplified 97.6%

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

       1. add-sqr-sqrt 97.6%

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

       2. sqrt-unprod 89.0%

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

       3. swap-sqr 89.0%

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

       4. pow-prod-up 89.0%

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

       5. metadata-eval 89.0%

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

       6. *-commutative 89.0%

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

       7. *-commutative 89.0%

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

       8. swap-sqr 89.0%

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

       9. pow-prod-up 89.0%

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

       10. add-sqr-sqrt 0.0%

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

       11. fabs-sqr 0.0%

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

       12. add-sqr-sqrt 89.0%

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

       13. metadata-eval 89.0%

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

       14. metadata-eval 89.0%

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

   12. Applied egg-rr 89.0%

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

       1. unpow-1 89.0%

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

   14. Simplified 89.0%

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

       1. associate-*l/ 89.0%

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

       2. *-un-lft-identity 89.0%

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

       3. sqrt-div 89.0%

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

       4. sqrt-prod 89.0%

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

       5. sqrt-pow1 0.1%

          \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{14}{2}\right)}} \cdot \sqrt{0.0022675736961451248}}{\sqrt{\pi}} \]

       6. metadata-eval 0.1%

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

       7. metadata-eval 0.1%

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

       8. *-commutative 0.1%

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

   16. Applied egg-rr 0.1%

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

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 34.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.0001)
   (* (pow PI -0.5) (* x 2.0))
   (* 0.047619047619047616 (/ (pow x 7.0) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.0001) {
		tmp = pow(((double) M_PI), -0.5) * (x * 2.0);
	} else {
		tmp = 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) <= 0.0001) {
		tmp = Math.pow(Math.PI, -0.5) * (x * 2.0);
	} else {
		tmp = 0.047619047619047616 * (Math.pow(x, 7.0) / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.0001:
		tmp = math.pow(math.pi, -0.5) * (x * 2.0)
	else:
		tmp = 0.047619047619047616 * (math.pow(x, 7.0) / math.sqrt(math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.0001)
		tmp = Float64((pi ^ -0.5) * Float64(x * 2.0));
	else
		tmp = 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) <= 0.0001)
		tmp = (pi ^ -0.5) * (x * 2.0);
	else
		tmp = 0.047619047619047616 * ((x ^ 7.0) / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.0001], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[x, 7.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000005e-4

   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%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0 99.8%

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

   9. Taylor expanded in x around 0 99.3%

      \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(2 \cdot \left|x\right|\right)} \]
   10. Step-by-step derivation

       1. metadata-eval 99.3%

          \[\leadsto {\pi}^{-0.5} \cdot \left(\color{blue}{\left|2\right|} \cdot \left|x\right|\right) \]

       2. fabs-mul 99.3%

          \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left|2 \cdot x\right|} \]

       3. rem-square-sqrt 52.1%

          \[\leadsto {\pi}^{-0.5} \cdot \left|\color{blue}{\sqrt{2 \cdot x} \cdot \sqrt{2 \cdot x}}\right| \]

       4. fabs-sqr 52.1%

          \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(\sqrt{2 \cdot x} \cdot \sqrt{2 \cdot x}\right)} \]

       5. rem-square-sqrt 54.0%

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

   11. Simplified 54.0%

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

   if 1.00000000000000005e-4 < (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 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around inf 97.6%

      \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)} \]
   9. Step-by-step derivation

      1. metadata-eval 97.6%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left({x}^{\color{blue}{\left(2 \cdot 3\right)}} \cdot \left|x\right|\right)\right) \]

      2. pow-sqr 97.6%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{\left({x}^{3} \cdot {x}^{3}\right)} \cdot \left|x\right|\right)\right) \]

      3. cube-prod 97.5%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{{\left(x \cdot x\right)}^{3}} \cdot \left|x\right|\right)\right) \]

      4. sqr-abs 97.5%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left({\color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)}}^{3} \cdot \left|x\right|\right)\right) \]

      5. cube-prod 97.6%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{\left({\left(\left|x\right|\right)}^{3} \cdot {\left(\left|x\right|\right)}^{3}\right)} \cdot \left|x\right|\right)\right) \]

      6. pow-sqr 97.6%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{{\left(\left|x\right|\right)}^{\left(2 \cdot 3\right)}} \cdot \left|x\right|\right)\right) \]

      7. metadata-eval 97.6%

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

      8. pow-plus 97.6%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(6 + 1\right)}}\right) \]

      9. metadata-eval 97.6%

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

   10. Simplified 97.6%

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

       1. add-sqr-sqrt 97.6%

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

       2. sqrt-unprod 89.0%

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

       3. swap-sqr 89.0%

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

       4. pow-prod-up 89.0%

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

       5. metadata-eval 89.0%

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

       6. *-commutative 89.0%

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

       7. *-commutative 89.0%

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

       8. swap-sqr 89.0%

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

       9. pow-prod-up 89.0%

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

       10. add-sqr-sqrt 0.0%

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

       11. fabs-sqr 0.0%

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

       12. add-sqr-sqrt 89.0%

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

       13. metadata-eval 89.0%

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

       14. metadata-eval 89.0%

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

   12. Applied egg-rr 89.0%

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

       1. unpow-1 89.0%

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

   14. Simplified 89.0%

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

       1. sqrt-prod 89.0%

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

       2. inv-pow 89.0%

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

       3. sqrt-pow1 89.0%

          \[\leadsto \color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \sqrt{{x}^{14} \cdot 0.0022675736961451248} \]

       4. metadata-eval 89.0%

          \[\leadsto {\pi}^{\color{blue}{-0.5}} \cdot \sqrt{{x}^{14} \cdot 0.0022675736961451248} \]

       5. sqrt-prod 89.0%

          \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(\sqrt{{x}^{14}} \cdot \sqrt{0.0022675736961451248}\right)} \]

       6. sqrt-pow1 0.1%

          \[\leadsto {\pi}^{-0.5} \cdot \left(\color{blue}{{x}^{\left(\frac{14}{2}\right)}} \cdot \sqrt{0.0022675736961451248}\right) \]

       7. metadata-eval 0.1%

          \[\leadsto {\pi}^{-0.5} \cdot \left({x}^{\color{blue}{7}} \cdot \sqrt{0.0022675736961451248}\right) \]

       8. metadata-eval 0.1%

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

       9. *-commutative 0.1%

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

       10. expm1-log1p-u 0.0%

           \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right)\right)} \]

       11. expm1-undefine 0.0%

           \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right)} - 1} \]

       12. sub-neg 0.0%

           \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right)} + \left(-1\right)} \]

   16. Applied egg-rr 0.1%

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

       1. +-commutative 0.1%

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

       2. associate-+r+ 0.1%

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

       3. metadata-eval 0.1%

          \[\leadsto \color{blue}{0} + \frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}} \]

       4. +-lft-identity 0.1%

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

       5. associate-/l* 0.1%

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

   18. Simplified 0.1%

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

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.2% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

7. Simplified 99.8%

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

8. Taylor expanded in x around inf 98.7%

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

   1. +-commutative 98.7%

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

   2. distribute-rgt-in 98.7%

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

   3. add-sqr-sqrt 35.0%

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

   4. fabs-sqr 35.0%

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

   5. add-sqr-sqrt 66.4%

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

   6. add-sqr-sqrt 34.7%

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

   7. fabs-sqr 34.7%

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

   8. add-sqr-sqrt 36.1%

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

10. Applied egg-rr 36.1%

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

11. Final simplification 36.1%

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

Alternative 8: 34.3% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{x}^{14} \cdot 0.0022675736961451248}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (* (pow PI -0.5) (* x 2.0))
   (sqrt (/ (* (pow x 14.0) 0.0022675736961451248) PI))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = pow(((double) M_PI), -0.5) * (x * 2.0);
	} else {
		tmp = 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.pow(Math.PI, -0.5) * (x * 2.0);
	} else {
		tmp = 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.pow(math.pi, -0.5) * (x * 2.0)
	else:
		tmp = 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 = Float64((pi ^ -0.5) * Float64(x * 2.0));
	else
		tmp = 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 = (pi ^ -0.5) * (x * 2.0);
	else
		tmp = sqrt((((x ^ 14.0) * 0.0022675736961451248) / pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[Power[x, 14.0], $MachinePrecision] * 0.0022675736961451248), $MachinePrecision] / Pi), $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%

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

   5. Taylor expanded in x around 0 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0 89.0%

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

   9. Taylor expanded in x around 0 68.3%

      \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(2 \cdot \left|x\right|\right)} \]
   10. Step-by-step derivation

       1. metadata-eval 68.3%

          \[\leadsto {\pi}^{-0.5} \cdot \left(\color{blue}{\left|2\right|} \cdot \left|x\right|\right) \]

       2. fabs-mul 68.3%

          \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left|2 \cdot x\right|} \]

       3. rem-square-sqrt 34.8%

          \[\leadsto {\pi}^{-0.5} \cdot \left|\color{blue}{\sqrt{2 \cdot x} \cdot \sqrt{2 \cdot x}}\right| \]

       4. fabs-sqr 34.8%

          \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(\sqrt{2 \cdot x} \cdot \sqrt{2 \cdot x}\right)} \]

       5. rem-square-sqrt 36.2%

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

   11. Simplified 36.2%

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

   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%

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

   5. Taylor expanded in x around 0 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around inf 36.1%

      \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)} \]
   9. Step-by-step derivation

      1. metadata-eval 36.1%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left({x}^{\color{blue}{\left(2 \cdot 3\right)}} \cdot \left|x\right|\right)\right) \]

      2. pow-sqr 36.1%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{\left({x}^{3} \cdot {x}^{3}\right)} \cdot \left|x\right|\right)\right) \]

      3. cube-prod 36.1%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{{\left(x \cdot x\right)}^{3}} \cdot \left|x\right|\right)\right) \]

      4. sqr-abs 36.1%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left({\color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)}}^{3} \cdot \left|x\right|\right)\right) \]

      5. cube-prod 36.1%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{\left({\left(\left|x\right|\right)}^{3} \cdot {\left(\left|x\right|\right)}^{3}\right)} \cdot \left|x\right|\right)\right) \]

      6. pow-sqr 36.1%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{{\left(\left|x\right|\right)}^{\left(2 \cdot 3\right)}} \cdot \left|x\right|\right)\right) \]

      7. metadata-eval 36.1%

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

      8. pow-plus 36.1%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(6 + 1\right)}}\right) \]

      9. metadata-eval 36.1%

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

   10. Simplified 36.1%

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

       1. add-sqr-sqrt 36.1%

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

       2. sqrt-unprod 33.2%

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

       3. swap-sqr 33.2%

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

       4. pow-prod-up 33.2%

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

       5. metadata-eval 33.2%

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

       6. *-commutative 33.2%

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

       7. *-commutative 33.2%

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

       8. swap-sqr 33.2%

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

       9. pow-prod-up 33.2%

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

       10. add-sqr-sqrt 2.0%

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

       11. fabs-sqr 2.0%

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

       12. add-sqr-sqrt 33.2%

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

       13. metadata-eval 33.2%

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

       14. metadata-eval 33.2%

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

   12. Applied egg-rr 33.2%

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

       1. unpow-1 33.2%

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

   14. Simplified 33.2%

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

       1. associate-*l/ 33.2%

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

       2. *-un-lft-identity 33.2%

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

   16. Applied egg-rr 33.2%

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

4. Final simplification 36.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{x}^{14} \cdot 0.0022675736961451248}{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.3% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.0022675736961451248 \cdot \frac{{x}^{14}}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (* (pow PI -0.5) (* x 2.0))
   (sqrt (* 0.0022675736961451248 (/ (pow x 14.0) PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = pow(((double) M_PI), -0.5) * (x * 2.0);
	} else {
		tmp = sqrt((0.0022675736961451248 * (pow(x, 14.0) / ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = Math.pow(Math.PI, -0.5) * (x * 2.0);
	} else {
		tmp = Math.sqrt((0.0022675736961451248 * (Math.pow(x, 14.0) / Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = math.pow(math.pi, -0.5) * (x * 2.0)
	else:
		tmp = math.sqrt((0.0022675736961451248 * (math.pow(x, 14.0) / math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64((pi ^ -0.5) * Float64(x * 2.0));
	else
		tmp = sqrt(Float64(0.0022675736961451248 * Float64((x ^ 14.0) / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = (pi ^ -0.5) * (x * 2.0);
	else
		tmp = sqrt((0.0022675736961451248 * ((x ^ 14.0) / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.0022675736961451248 * N[(N[Power[x, 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%

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

   5. Taylor expanded in x around 0 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0 89.0%

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

   9. Taylor expanded in x around 0 68.3%

      \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(2 \cdot \left|x\right|\right)} \]
   10. Step-by-step derivation

       1. metadata-eval 68.3%

          \[\leadsto {\pi}^{-0.5} \cdot \left(\color{blue}{\left|2\right|} \cdot \left|x\right|\right) \]

       2. fabs-mul 68.3%

          \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left|2 \cdot x\right|} \]

       3. rem-square-sqrt 34.8%

          \[\leadsto {\pi}^{-0.5} \cdot \left|\color{blue}{\sqrt{2 \cdot x} \cdot \sqrt{2 \cdot x}}\right| \]

       4. fabs-sqr 34.8%

          \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(\sqrt{2 \cdot x} \cdot \sqrt{2 \cdot x}\right)} \]

       5. rem-square-sqrt 36.2%

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

   11. Simplified 36.2%

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

   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%

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

   5. Taylor expanded in x around 0 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around inf 36.1%

      \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)} \]
   9. Step-by-step derivation

      1. metadata-eval 36.1%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left({x}^{\color{blue}{\left(2 \cdot 3\right)}} \cdot \left|x\right|\right)\right) \]

      2. pow-sqr 36.1%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{\left({x}^{3} \cdot {x}^{3}\right)} \cdot \left|x\right|\right)\right) \]

      3. cube-prod 36.1%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{{\left(x \cdot x\right)}^{3}} \cdot \left|x\right|\right)\right) \]

      4. sqr-abs 36.1%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left({\color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)}}^{3} \cdot \left|x\right|\right)\right) \]

      5. cube-prod 36.1%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{\left({\left(\left|x\right|\right)}^{3} \cdot {\left(\left|x\right|\right)}^{3}\right)} \cdot \left|x\right|\right)\right) \]

      6. pow-sqr 36.1%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{{\left(\left|x\right|\right)}^{\left(2 \cdot 3\right)}} \cdot \left|x\right|\right)\right) \]

      7. metadata-eval 36.1%

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

      8. pow-plus 36.1%

         \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(6 + 1\right)}}\right) \]

      9. metadata-eval 36.1%

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

   10. Simplified 36.1%

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

       1. add-sqr-sqrt 36.1%

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

       2. sqrt-unprod 33.2%

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

       3. swap-sqr 33.2%

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

       4. pow-prod-up 33.2%

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

       5. metadata-eval 33.2%

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

       6. *-commutative 33.2%

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

       7. *-commutative 33.2%

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

       8. swap-sqr 33.2%

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

       9. pow-prod-up 33.2%

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

       10. add-sqr-sqrt 2.0%

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

       11. fabs-sqr 2.0%

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

       12. add-sqr-sqrt 33.2%

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

       13. metadata-eval 33.2%

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

       14. metadata-eval 33.2%

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

   12. Applied egg-rr 33.2%

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

       1. associate-*r* 33.2%

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

       2. *-commutative 33.2%

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

       3. unpow-1 33.2%

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

       4. associate-*l/ 33.2%

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

       5. *-lft-identity 33.2%

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

   14. Simplified 33.2%

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

4. Final simplification 36.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.0022675736961451248 \cdot \frac{{x}^{14}}{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 34.3% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ {\pi}^{-0.5} \cdot \left(x \cdot 2\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (pow PI -0.5) (* x 2.0)))

C

double code(double x) {
	return pow(((double) M_PI), -0.5) * (x * 2.0);
}

Java

public static double code(double x) {
	return Math.pow(Math.PI, -0.5) * (x * 2.0);
}

Python

def code(x):
	return math.pow(math.pi, -0.5) * (x * 2.0)

Julia

function code(x)
	return Float64((pi ^ -0.5) * Float64(x * 2.0))
end

MATLAB

function tmp = code(x)
	tmp = (pi ^ -0.5) * (x * 2.0);
end

Wolfram

code[x_] := N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * 2.0), $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%

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

7. Simplified 99.8%

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

8. Taylor expanded in x around 0 89.0%

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

9. Taylor expanded in x around 0 68.3%

   \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(2 \cdot \left|x\right|\right)} \]
10. Step-by-step derivation

    1. metadata-eval 68.3%

       \[\leadsto {\pi}^{-0.5} \cdot \left(\color{blue}{\left|2\right|} \cdot \left|x\right|\right) \]

    2. fabs-mul 68.3%

       \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left|2 \cdot x\right|} \]

    3. rem-square-sqrt 34.8%

       \[\leadsto {\pi}^{-0.5} \cdot \left|\color{blue}{\sqrt{2 \cdot x} \cdot \sqrt{2 \cdot x}}\right| \]

    4. fabs-sqr 34.8%

       \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(\sqrt{2 \cdot x} \cdot \sqrt{2 \cdot x}\right)} \]

    5. rem-square-sqrt 36.2%

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

11. Simplified 36.2%

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

12. Final simplification 36.2%

    \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot 2\right) \]
13. Add Preprocessing

Alternative 11: 4.1% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(0\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (expm1 0.0))

C

double code(double x) {
	return expm1(0.0);
}

Java

public static double code(double x) {
	return Math.expm1(0.0);
}

Python

def code(x):
	return math.expm1(0.0)

Julia

function code(x)
	return expm1(0.0)
end

Wolfram

code[x_] := N[(Exp[0.0] - 1), $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%

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

7. Simplified 99.8%

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

8. Taylor expanded in x around inf 36.1%

   \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)} \]
9. Step-by-step derivation

   1. metadata-eval 36.1%

      \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left({x}^{\color{blue}{\left(2 \cdot 3\right)}} \cdot \left|x\right|\right)\right) \]

   2. pow-sqr 36.1%

      \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{\left({x}^{3} \cdot {x}^{3}\right)} \cdot \left|x\right|\right)\right) \]

   3. cube-prod 36.1%

      \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{{\left(x \cdot x\right)}^{3}} \cdot \left|x\right|\right)\right) \]

   4. sqr-abs 36.1%

      \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left({\color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)}}^{3} \cdot \left|x\right|\right)\right) \]

   5. cube-prod 36.1%

      \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{\left({\left(\left|x\right|\right)}^{3} \cdot {\left(\left|x\right|\right)}^{3}\right)} \cdot \left|x\right|\right)\right) \]

   6. pow-sqr 36.1%

      \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(\color{blue}{{\left(\left|x\right|\right)}^{\left(2 \cdot 3\right)}} \cdot \left|x\right|\right)\right) \]

   7. metadata-eval 36.1%

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

   8. pow-plus 36.1%

      \[\leadsto {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(6 + 1\right)}}\right) \]

   9. metadata-eval 36.1%

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

10. Simplified 36.1%

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

    1. expm1-log1p-u 35.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)} \]

    2. expm1-undefine 35.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)} - 1} \]

    3. add-sqr-sqrt 1.9%

       \[\leadsto e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{7}\right)\right)} - 1 \]

    4. fabs-sqr 1.9%

       \[\leadsto e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{7}\right)\right)} - 1 \]

    5. add-sqr-sqrt 3.5%

       \[\leadsto e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {\color{blue}{x}}^{7}\right)\right)} - 1 \]

12. Applied egg-rr 3.5%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-define 3.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right)\right)} \]

14. Simplified 3.6%

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right)\right)} \]

15. Taylor expanded in x around 0 4.1%

    \[\leadsto \mathsf{expm1}\left(\color{blue}{0}\right) \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024150 
(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)))))))