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

Percentage Accurate: 99.8% → 99.8%

Time: 10.4s

Alternatives: 10

Speedup: 1.5×

• 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, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot \left(x \cdot x\right)\\ t_1 := \left|x\right| \cdot \left(\left|x\right| \cdot t\_0\right)\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + 0.6666666666666666 \cdot t\_0\right) + 0.2 \cdot t\_1\right) + 0.047619047619047616 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot t\_1\right)\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (* x x))) (t_1 (* (fabs x) (* (fabs x) t_0))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* 0.6666666666666666 t_0)) (* 0.2 t_1))
      (* 0.047619047619047616 (* (fabs x) (* (fabs x) t_1))))))))

C

double code(double x) {
	double t_0 = fabs(x) * (x * x);
	double t_1 = fabs(x) * (fabs(x) * t_0);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (fabs(x) * (fabs(x) * t_1))))));
}

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * (x * x);
	double t_1 = Math.abs(x) * (Math.abs(x) * t_0);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (Math.abs(x) * (Math.abs(x) * t_1))))));
}

Python

def code(x):
	t_0 = math.fabs(x) * (x * x)
	t_1 = math.fabs(x) * (math.fabs(x) * t_0)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (math.fabs(x) * (math.fabs(x) * t_1))))))

Julia

function code(x)
	t_0 = Float64(abs(x) * Float64(x * x))
	t_1 = Float64(abs(x) * Float64(abs(x) * t_0))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(0.6666666666666666 * t_0)) + Float64(0.2 * t_1)) + Float64(0.047619047619047616 * Float64(abs(x) * Float64(abs(x) * t_1))))))
end

MATLAB

function tmp = code(x)
	t_0 = abs(x) * (x * x);
	t_1 = abs(x) * (abs(x) * t_0);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (abs(x) * (abs(x) * t_1))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$0), $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[(0.6666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.2 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 34.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. add-sqr-sqrt 34.9%

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

   2. fabs-sqr 34.9%

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

   3. add-sqr-sqrt 36.3%

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

   4. *-un-lft-identity 36.3%

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

6. Applied egg-rr 36.3%

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

   1. *-lft-identity 36.3%

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

8. Simplified 36.3%

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

Alternative 3: 77.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot t\_0\right) + 0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot t\_0\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (* x x)) (* x x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 x) (* 0.6666666666666666 (pow x 3.0))) (* 0.2 t_0))
      (* 0.047619047619047616 (* (* x x) t_0)))))))

C

double code(double x) {
	double t_0 = (fabs(x) * (x * x)) * (x * x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * x) + (0.6666666666666666 * pow(x, 3.0))) + (0.2 * t_0)) + (0.047619047619047616 * ((x * x) * t_0)))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * (x * x)) * (x * x);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * x) + (0.6666666666666666 * Math.pow(x, 3.0))) + (0.2 * t_0)) + (0.047619047619047616 * ((x * x) * t_0)))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * (x * x)) * (x * x)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * x) + (0.6666666666666666 * math.pow(x, 3.0))) + (0.2 * t_0)) + (0.047619047619047616 * ((x * x) * t_0)))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * Float64(x * x)) * Float64(x * x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * x) + Float64(0.6666666666666666 * (x ^ 3.0))) + Float64(0.2 * t_0)) + Float64(0.047619047619047616 * Float64(Float64(x * x) * t_0)))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * (x * x)) * (x * x);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * x) + (0.6666666666666666 * (x ^ 3.0))) + (0.2 * t_0)) + (0.047619047619047616 * ((x * x) * t_0)))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * x), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. fma-undefine 99.8%

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

   2. add-sqr-sqrt 34.9%

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

   3. fabs-sqr 34.9%

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

   4. add-sqr-sqrt 99.5%

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

   5. add-sqr-sqrt 35.1%

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

   6. fabs-sqr 35.1%

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

   7. add-sqr-sqrt 78.1%

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

   8. cube-mult 78.1%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot \color{blue}{{x}^{3}}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

6. Applied egg-rr 78.1%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\color{blue}{\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)} + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

7. Final simplification 78.1%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right| \]
8. Add Preprocessing

Alternative 4: 98.9% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[Abs[N[(x * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(2.0 + N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[Power[x, 2.0], $MachinePrecision] * 0.047619047619047616 + 0.2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 99.3%

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

   1. associate-*r* 99.2%

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

   2. *-commutative 99.2%

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

   3. rem-square-sqrt 34.7%

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

   4. fabs-sqr 34.7%

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

   5. rem-square-sqrt 99.2%

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

   6. associate-*l* 99.3%

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

   7. fma-define 99.3%

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

   8. rem-square-sqrt 35.0%

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

   9. fabs-sqr 35.0%

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

   10. rem-square-sqrt 99.3%

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

   11. rem-square-sqrt 35.0%

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

   12. fabs-sqr 35.0%

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

   13. rem-square-sqrt 99.3%

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

7. Simplified 99.3%

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

Alternative 5: 98.8% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 99.3%

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

   1. associate-*r* 99.2%

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

   2. *-commutative 99.2%

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

   3. rem-square-sqrt 34.7%

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

   4. fabs-sqr 34.7%

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

   5. rem-square-sqrt 99.2%

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

   6. associate-*l* 99.3%

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

   7. fma-define 99.3%

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

   8. rem-square-sqrt 35.0%

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

   9. fabs-sqr 35.0%

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

   10. rem-square-sqrt 99.3%

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

   11. rem-square-sqrt 35.0%

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

   12. fabs-sqr 35.0%

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

   13. rem-square-sqrt 99.3%

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

7. Simplified 99.3%

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

8. Taylor expanded in x around inf 98.8%

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

   1. *-un-lft-identity 98.8%

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

   2. inv-pow 98.8%

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

   3. sqrt-pow1 98.8%

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

   4. metadata-eval 98.8%

      \[\leadsto \left|x \cdot \left(\left(1 \cdot {\pi}^{\color{blue}{-0.5}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

10. Applied egg-rr 98.8%

    \[\leadsto \left|x \cdot \left(\color{blue}{\left(1 \cdot {\pi}^{-0.5}\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]
11. Step-by-step derivation

    1. *-lft-identity 98.8%

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

12. Simplified 98.8%

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

Alternative 6: 34.3% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. associate-*r* 99.2%

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

      2. *-commutative 99.2%

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

      3. rem-square-sqrt 34.7%

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

      4. fabs-sqr 34.7%

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

      5. rem-square-sqrt 99.2%

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

      6. associate-*l* 99.3%

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

      7. fma-define 99.3%

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

      8. rem-square-sqrt 35.0%

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

      9. fabs-sqr 35.0%

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

      10. rem-square-sqrt 99.3%

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

      11. rem-square-sqrt 35.0%

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

      12. fabs-sqr 35.0%

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

      13. rem-square-sqrt 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around 0 68.3%

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

      1. *-commutative 68.3%

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

      2. *-commutative 68.3%

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

      3. associate-*l* 68.3%

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

      4. unpow-1 68.3%

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

      5. metadata-eval 68.3%

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

      6. pow-sqr 68.3%

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

      7. rem-sqrt-square 68.3%

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

      8. rem-square-sqrt 68.3%

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

      9. fabs-sqr 68.3%

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

      10. rem-square-sqrt 68.3%

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

   10. Simplified 68.3%

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

       1. add-sqr-sqrt 34.7%

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

       2. fabs-sqr 34.7%

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

       3. add-sqr-sqrt 36.2%

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

       4. *-commutative 36.2%

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

       5. associate-*l* 36.2%

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

       6. metadata-eval 36.2%

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

       7. pow-flip 36.2%

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

       8. pow1/2 36.2%

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

       9. un-div-inv 36.2%

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

   12. Applied egg-rr 36.2%

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

   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 99.4%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. associate-*r* 99.2%

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

      2. *-commutative 99.2%

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

      3. rem-square-sqrt 34.7%

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

      4. fabs-sqr 34.7%

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

      5. rem-square-sqrt 99.2%

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

      6. associate-*l* 99.3%

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

      7. fma-define 99.3%

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

      8. rem-square-sqrt 35.0%

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

      9. fabs-sqr 35.0%

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

      10. rem-square-sqrt 99.3%

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

      11. rem-square-sqrt 35.0%

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

      12. fabs-sqr 35.0%

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

      13. rem-square-sqrt 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around inf 36.1%

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

      1. *-commutative 36.1%

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

      2. unpow-1 36.1%

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

      3. metadata-eval 36.1%

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

      4. pow-sqr 36.1%

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

      5. rem-sqrt-square 36.1%

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

      6. rem-square-sqrt 36.1%

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

      7. fabs-sqr 36.1%

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

      8. rem-square-sqrt 36.1%

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

   10. Simplified 36.1%

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

       1. add-sqr-sqrt 3.4%

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

       2. fabs-sqr 3.4%

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

       3. add-sqr-sqrt 3.6%

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

       4. *-commutative 3.6%

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

       5. associate-*l* 3.6%

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

   12. Applied egg-rr 3.6%

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

4. Final simplification 36.2%

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

Alternative 7: 34.3% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. associate-*r* 99.2%

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

      2. *-commutative 99.2%

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

      3. rem-square-sqrt 34.7%

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

      4. fabs-sqr 34.7%

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

      5. rem-square-sqrt 99.2%

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

      6. associate-*l* 99.3%

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

      7. fma-define 99.3%

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

      8. rem-square-sqrt 35.0%

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

      9. fabs-sqr 35.0%

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

      10. rem-square-sqrt 99.3%

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

      11. rem-square-sqrt 35.0%

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

      12. fabs-sqr 35.0%

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

      13. rem-square-sqrt 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around 0 68.3%

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

      1. *-commutative 68.3%

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

      2. *-commutative 68.3%

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

      3. associate-*l* 68.3%

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

      4. unpow-1 68.3%

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

      5. metadata-eval 68.3%

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

      6. pow-sqr 68.3%

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

      7. rem-sqrt-square 68.3%

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

      8. rem-square-sqrt 68.3%

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

      9. fabs-sqr 68.3%

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

      10. rem-square-sqrt 68.3%

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

   10. Simplified 68.3%

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

       1. add-sqr-sqrt 34.7%

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

       2. fabs-sqr 34.7%

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

       3. add-sqr-sqrt 36.2%

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

       4. *-commutative 36.2%

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

       5. associate-*l* 36.2%

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

       6. metadata-eval 36.2%

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

       7. pow-flip 36.2%

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

       8. pow1/2 36.2%

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

       9. un-div-inv 36.2%

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

   12. Applied egg-rr 36.2%

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

   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 99.4%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. associate-*r* 99.2%

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

      2. *-commutative 99.2%

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

      3. rem-square-sqrt 34.7%

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

      4. fabs-sqr 34.7%

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

      5. rem-square-sqrt 99.2%

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

      6. associate-*l* 99.3%

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

      7. fma-define 99.3%

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

      8. rem-square-sqrt 35.0%

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

      9. fabs-sqr 35.0%

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

      10. rem-square-sqrt 99.3%

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

      11. rem-square-sqrt 35.0%

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

      12. fabs-sqr 35.0%

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

      13. rem-square-sqrt 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around inf 36.1%

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

      1. *-commutative 36.1%

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

      2. unpow-1 36.1%

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

      3. metadata-eval 36.1%

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

      4. pow-sqr 36.1%

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

      5. rem-sqrt-square 36.1%

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

      6. rem-square-sqrt 36.1%

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

      7. fabs-sqr 36.1%

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

      8. rem-square-sqrt 36.1%

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

   10. Simplified 36.1%

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

       1. add-sqr-sqrt 3.4%

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

       2. fabs-sqr 3.4%

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

       3. add-sqr-sqrt 3.6%

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

       4. *-commutative 3.6%

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

   12. Applied egg-rr 3.6%

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

4. Final simplification 36.2%

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

Alternative 8: 34.3% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[Power[x, 14.0], $MachinePrecision] / Pi), $MachinePrecision] * 0.0022675736961451248), $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 99.4%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. associate-*r* 99.2%

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

      2. *-commutative 99.2%

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

      3. rem-square-sqrt 34.7%

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

      4. fabs-sqr 34.7%

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

      5. rem-square-sqrt 99.2%

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

      6. associate-*l* 99.3%

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

      7. fma-define 99.3%

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

      8. rem-square-sqrt 35.0%

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

      9. fabs-sqr 35.0%

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

      10. rem-square-sqrt 99.3%

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

      11. rem-square-sqrt 35.0%

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

      12. fabs-sqr 35.0%

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

      13. rem-square-sqrt 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around 0 68.3%

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

      1. *-commutative 68.3%

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

      2. *-commutative 68.3%

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

      3. associate-*l* 68.3%

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

      4. unpow-1 68.3%

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

      5. metadata-eval 68.3%

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

      6. pow-sqr 68.3%

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

      7. rem-sqrt-square 68.3%

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

      8. rem-square-sqrt 68.3%

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

      9. fabs-sqr 68.3%

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

      10. rem-square-sqrt 68.3%

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

   10. Simplified 68.3%

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

       1. add-sqr-sqrt 34.7%

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

       2. fabs-sqr 34.7%

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

       3. add-sqr-sqrt 36.2%

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

       4. *-commutative 36.2%

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

       5. associate-*l* 36.2%

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

       6. metadata-eval 36.2%

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

       7. pow-flip 36.2%

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

       8. pow1/2 36.2%

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

       9. un-div-inv 36.2%

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

   12. Applied egg-rr 36.2%

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

   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 99.4%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. associate-*r* 99.2%

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

      2. *-commutative 99.2%

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

      3. rem-square-sqrt 34.7%

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

      4. fabs-sqr 34.7%

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

      5. rem-square-sqrt 99.2%

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

      6. associate-*l* 99.3%

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

      7. fma-define 99.3%

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

      8. rem-square-sqrt 35.0%

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

      9. fabs-sqr 35.0%

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

      10. rem-square-sqrt 99.3%

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

      11. rem-square-sqrt 35.0%

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

      12. fabs-sqr 35.0%

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

      13. rem-square-sqrt 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around inf 36.1%

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

      1. *-commutative 36.1%

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

      2. unpow-1 36.1%

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

      3. metadata-eval 36.1%

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

      4. pow-sqr 36.1%

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

      5. rem-sqrt-square 36.1%

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

      6. rem-square-sqrt 36.1%

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

      7. fabs-sqr 36.1%

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

      8. rem-square-sqrt 36.1%

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

   10. Simplified 36.1%

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

       1. add-sqr-sqrt 3.4%

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

       2. fabs-sqr 3.4%

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

       3. sqrt-unprod 33.2%

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

       4. *-commutative 33.2%

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

       5. *-commutative 33.2%

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

       6. swap-sqr 33.2%

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

       7. swap-sqr 33.2%

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

       8. pow-prod-up 33.2%

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

       9. metadata-eval 33.2%

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

       10. inv-pow 33.2%

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

       11. pow-prod-up 33.2%

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

       12. metadata-eval 33.2%

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

       13. metadata-eval 33.2%

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

   12. Applied egg-rr 33.2%

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

       1. associate-*l/ 33.2%

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

       2. *-lft-identity 33.2%

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

   14. Simplified 33.2%

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

Alternative 9: 34.3% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.4444444444444444 \cdot \frac{{x}^{6}}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.75)
   (* x (/ 2.0 (sqrt PI)))
   (sqrt (* 0.4444444444444444 (/ (pow x 6.0) PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.75) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = sqrt((0.4444444444444444 * (pow(x, 6.0) / ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.75) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.sqrt((0.4444444444444444 * (Math.pow(x, 6.0) / Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.75:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.sqrt((0.4444444444444444 * (math.pow(x, 6.0) / math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.75)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = sqrt(Float64(0.4444444444444444 * Float64((x ^ 6.0) / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.75)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = sqrt((0.4444444444444444 * ((x ^ 6.0) / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.75], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.4444444444444444 * N[(N[Power[x, 6.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.75

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. associate-*r* 99.2%

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

      2. *-commutative 99.2%

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

      3. rem-square-sqrt 34.7%

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

      4. fabs-sqr 34.7%

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

      5. rem-square-sqrt 99.2%

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

      6. associate-*l* 99.3%

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

      7. fma-define 99.3%

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

      8. rem-square-sqrt 35.0%

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

      9. fabs-sqr 35.0%

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

      10. rem-square-sqrt 99.3%

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

      11. rem-square-sqrt 35.0%

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

      12. fabs-sqr 35.0%

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

      13. rem-square-sqrt 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around 0 68.3%

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

      1. *-commutative 68.3%

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

      2. *-commutative 68.3%

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

      3. associate-*l* 68.3%

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

      4. unpow-1 68.3%

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

      5. metadata-eval 68.3%

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

      6. pow-sqr 68.3%

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

      7. rem-sqrt-square 68.3%

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

      8. rem-square-sqrt 68.3%

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

      9. fabs-sqr 68.3%

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

      10. rem-square-sqrt 68.3%

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

   10. Simplified 68.3%

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

       1. add-sqr-sqrt 34.7%

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

       2. fabs-sqr 34.7%

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

       3. add-sqr-sqrt 36.2%

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

       4. *-commutative 36.2%

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

       5. associate-*l* 36.2%

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

       6. metadata-eval 36.2%

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

       7. pow-flip 36.2%

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

       8. pow1/2 36.2%

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

       9. un-div-inv 36.2%

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

   12. Applied egg-rr 36.2%

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

   if 1.75 < 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 99.4%

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

   5. Taylor expanded in x around inf 26.4%

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

      1. *-commutative 26.4%

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

      2. *-commutative 26.4%

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

      3. associate-*l* 26.4%

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

      4. rem-square-sqrt 2.2%

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

      5. fabs-sqr 2.2%

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

      6. rem-square-sqrt 26.4%

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

      7. unpow2 26.4%

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

      8. unpow3 26.4%

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

      9. *-commutative 26.4%

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

   7. Simplified 26.4%

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

      1. add-sqr-sqrt 3.4%

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

      2. fabs-sqr 3.4%

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

      3. sqrt-unprod 30.6%

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

      4. swap-sqr 30.6%

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

      5. add-sqr-sqrt 30.6%

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

      6. *-commutative 30.6%

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

      7. *-commutative 30.6%

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

      8. swap-sqr 30.6%

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

      9. pow-prod-up 30.6%

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

      10. metadata-eval 30.6%

          \[\leadsto \sqrt{\frac{1}{\pi} \cdot \left({x}^{\color{blue}{6}} \cdot \left(0.6666666666666666 \cdot 0.6666666666666666\right)\right)} \]

      11. metadata-eval 30.6%

          \[\leadsto \sqrt{\frac{1}{\pi} \cdot \left({x}^{6} \cdot \color{blue}{0.4444444444444444}\right)} \]

   9. Applied egg-rr 30.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi} \cdot \left({x}^{6} \cdot 0.4444444444444444\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* 30.6%

          \[\leadsto \sqrt{\color{blue}{\left(\frac{1}{\pi} \cdot {x}^{6}\right) \cdot 0.4444444444444444}} \]

       2. *-commutative 30.6%

          \[\leadsto \sqrt{\color{blue}{0.4444444444444444 \cdot \left(\frac{1}{\pi} \cdot {x}^{6}\right)}} \]

       3. associate-*l/ 30.6%

          \[\leadsto \sqrt{0.4444444444444444 \cdot \color{blue}{\frac{1 \cdot {x}^{6}}{\pi}}} \]

       4. *-lft-identity 30.6%

          \[\leadsto \sqrt{0.4444444444444444 \cdot \frac{\color{blue}{{x}^{6}}}{\pi}} \]

   11. Simplified 30.6%

       \[\leadsto \color{blue}{\sqrt{0.4444444444444444 \cdot \frac{{x}^{6}}{\pi}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 34.3% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 99.3%

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

   1. associate-*r* 99.2%

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

   2. *-commutative 99.2%

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

   3. rem-square-sqrt 34.7%

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

   4. fabs-sqr 34.7%

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

   5. rem-square-sqrt 99.2%

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

   6. associate-*l* 99.3%

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

   7. fma-define 99.3%

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

   8. rem-square-sqrt 35.0%

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

   9. fabs-sqr 35.0%

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

   10. rem-square-sqrt 99.3%

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

   11. rem-square-sqrt 35.0%

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

   12. fabs-sqr 35.0%

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

   13. rem-square-sqrt 99.3%

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

7. Simplified 99.3%

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

8. Taylor expanded in x around 0 68.3%

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

   1. *-commutative 68.3%

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

   2. *-commutative 68.3%

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

   3. associate-*l* 68.3%

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

   4. unpow-1 68.3%

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

   5. metadata-eval 68.3%

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

   6. pow-sqr 68.3%

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

   7. rem-sqrt-square 68.3%

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

   8. rem-square-sqrt 68.3%

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

   9. fabs-sqr 68.3%

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

   10. rem-square-sqrt 68.3%

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

10. Simplified 68.3%

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

    1. add-sqr-sqrt 34.7%

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

    2. fabs-sqr 34.7%

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

    3. add-sqr-sqrt 36.2%

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

    4. *-commutative 36.2%

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

    5. associate-*l* 36.2%

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

    6. metadata-eval 36.2%

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

    7. pow-flip 36.2%

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

    8. pow1/2 36.2%

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

    9. un-div-inv 36.2%

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

12. Applied egg-rr 36.2%

    \[\leadsto \color{blue}{x \cdot \frac{2}{\sqrt{\pi}}} \]
13. 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)))))))