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

Percentage Accurate: 99.8% → 99.9%

Time: 11.0s

Alternatives: 7

Speedup: 2.6×

• 28.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fabs 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 fabs(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(abs(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[(N[Abs[x], $MachinePrecision] * 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 99.9%

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

3. Simplified 99.9%

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

Alternative 2: 34.0% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 99.9%

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

   1. add-sqr-sqrt 33.3%

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

   2. fabs-sqr 33.3%

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

   3. add-sqr-sqrt 34.9%

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

   4. add-sqr-sqrt 34.3%

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

   5. fabs-sqr 34.3%

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

   6. add-sqr-sqrt 34.9%

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

   7. distribute-lft-in 34.9%

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

7. Applied egg-rr 34.9%

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

   1. distribute-lft-out 34.9%

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

   2. distribute-lft-out 34.9%

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

   3. fma-undefine 34.9%

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

   4. *-commutative 34.9%

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

   5. fma-define 34.9%

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

9. Simplified 34.9%

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

    1. fma-undefine 34.9%

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

    2. *-commutative 34.9%

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

    3. fma-undefine 34.9%

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

    4. associate-+r+ 34.9%

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

    5. *-commutative 34.9%

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

11. Applied egg-rr 34.9%

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

12. Final simplification 34.9%

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

Alternative 3: 33.9% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 99.9%

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

   1. add-sqr-sqrt 33.3%

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

   2. fabs-sqr 33.3%

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

   3. add-sqr-sqrt 34.9%

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

   4. add-sqr-sqrt 34.3%

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

   5. fabs-sqr 34.3%

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

   6. add-sqr-sqrt 34.9%

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

   7. distribute-lft-in 34.9%

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

7. Applied egg-rr 34.9%

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

   1. distribute-lft-out 34.9%

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

   2. distribute-lft-out 34.9%

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

   3. fma-undefine 34.9%

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

   4. *-commutative 34.9%

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

   5. fma-define 34.9%

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

9. Simplified 34.9%

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

10. Taylor expanded in x around 0 34.9%

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

Alternative 4: 33.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (* (pow PI -0.5) (* x 2.0))
   (* (sqrt (/ 1.0 PI)) (* 0.047619047619047616 (pow x 7.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = pow(((double) M_PI), -0.5) * (x * 2.0);
	} else {
		tmp = sqrt((1.0 / ((double) M_PI))) * (0.047619047619047616 * pow(x, 7.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = Math.pow(Math.PI, -0.5) * (x * 2.0);
	} else {
		tmp = Math.sqrt((1.0 / Math.PI)) * (0.047619047619047616 * Math.pow(x, 7.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = math.pow(math.pi, -0.5) * (x * 2.0)
	else:
		tmp = math.sqrt((1.0 / math.pi)) * (0.047619047619047616 * math.pow(x, 7.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64((pi ^ -0.5) * Float64(x * 2.0));
	else
		tmp = Float64(sqrt(Float64(1.0 / pi)) * Float64(0.047619047619047616 * (x ^ 7.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = (pi ^ -0.5) * (x * 2.0);
	else
		tmp = sqrt((1.0 / pi)) * (0.047619047619047616 * (x ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $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 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

      1. add-sqr-sqrt 33.3%

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

      2. fabs-sqr 33.3%

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

      3. add-sqr-sqrt 34.9%

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

      4. add-sqr-sqrt 34.3%

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

      5. fabs-sqr 34.3%

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

      6. add-sqr-sqrt 34.9%

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

      7. distribute-lft-in 34.9%

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

   7. Applied egg-rr 34.9%

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

      1. distribute-lft-out 34.9%

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

      2. distribute-lft-out 34.9%

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

      3. fma-undefine 34.9%

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

      4. *-commutative 34.9%

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

      5. fma-define 34.9%

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

   9. Simplified 34.9%

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

   10. Taylor expanded in x around 0 34.9%

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

       1. associate-*r* 34.9%

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

   12. Simplified 34.9%

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

       1. pow1 34.9%

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

       2. *-commutative 34.9%

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

       3. inv-pow 34.9%

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

       4. sqrt-pow1 34.9%

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

       5. metadata-eval 34.9%

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

       6. *-commutative 34.9%

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

   14. Applied egg-rr 34.9%

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

       1. unpow1 34.9%

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

       2. *-commutative 34.9%

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

   16. Simplified 34.9%

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

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

      1. add-sqr-sqrt 33.3%

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

      2. fabs-sqr 33.3%

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

      3. add-sqr-sqrt 34.9%

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

      4. add-sqr-sqrt 34.3%

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

      5. fabs-sqr 34.3%

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

      6. add-sqr-sqrt 34.9%

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

      7. distribute-lft-in 34.9%

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

   7. Applied egg-rr 34.9%

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

      1. distribute-lft-out 34.9%

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

      2. distribute-lft-out 34.9%

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

      3. fma-undefine 34.9%

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

      4. *-commutative 34.9%

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

      5. fma-define 34.9%

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

   9. Simplified 34.9%

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

   10. Taylor expanded in x around inf 3.6%

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

       1. associate-*r* 3.6%

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

       2. *-commutative 3.6%

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

   12. Simplified 3.6%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 33.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{7}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (* (pow PI -0.5) (* x 2.0))
   (* 0.047619047619047616 (* (sqrt (/ 1.0 PI)) (pow x 7.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = pow(((double) M_PI), -0.5) * (x * 2.0);
	} else {
		tmp = 0.047619047619047616 * (sqrt((1.0 / ((double) M_PI))) * pow(x, 7.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = Math.pow(Math.PI, -0.5) * (x * 2.0);
	} else {
		tmp = 0.047619047619047616 * (Math.sqrt((1.0 / Math.PI)) * Math.pow(x, 7.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = math.pow(math.pi, -0.5) * (x * 2.0)
	else:
		tmp = 0.047619047619047616 * (math.sqrt((1.0 / math.pi)) * math.pow(x, 7.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64((pi ^ -0.5) * Float64(x * 2.0));
	else
		tmp = Float64(0.047619047619047616 * Float64(sqrt(Float64(1.0 / pi)) * (x ^ 7.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = (pi ^ -0.5) * (x * 2.0);
	else
		tmp = 0.047619047619047616 * (sqrt((1.0 / pi)) * (x ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

      1. add-sqr-sqrt 33.3%

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

      2. fabs-sqr 33.3%

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

      3. add-sqr-sqrt 34.9%

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

      4. add-sqr-sqrt 34.3%

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

      5. fabs-sqr 34.3%

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

      6. add-sqr-sqrt 34.9%

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

      7. distribute-lft-in 34.9%

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

   7. Applied egg-rr 34.9%

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

      1. distribute-lft-out 34.9%

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

      2. distribute-lft-out 34.9%

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

      3. fma-undefine 34.9%

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

      4. *-commutative 34.9%

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

      5. fma-define 34.9%

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

   9. Simplified 34.9%

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

   10. Taylor expanded in x around 0 34.9%

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

       1. associate-*r* 34.9%

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

   12. Simplified 34.9%

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

       1. pow1 34.9%

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

       2. *-commutative 34.9%

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

       3. inv-pow 34.9%

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

       4. sqrt-pow1 34.9%

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

       5. metadata-eval 34.9%

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

       6. *-commutative 34.9%

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

   14. Applied egg-rr 34.9%

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

       1. unpow1 34.9%

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

       2. *-commutative 34.9%

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

   16. Simplified 34.9%

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

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

      1. add-sqr-sqrt 33.3%

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

      2. fabs-sqr 33.3%

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

      3. add-sqr-sqrt 34.9%

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

      4. add-sqr-sqrt 34.3%

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

      5. fabs-sqr 34.3%

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

      6. add-sqr-sqrt 34.9%

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

      7. distribute-lft-in 34.9%

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

   7. Applied egg-rr 34.9%

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

      1. distribute-lft-out 34.9%

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

      2. distribute-lft-out 34.9%

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

      3. fma-undefine 34.9%

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

      4. *-commutative 34.9%

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

      5. fma-define 34.9%

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

   9. Simplified 34.9%

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

   10. Taylor expanded in x around inf 3.6%

       \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 33.8% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 99.9%

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

   1. add-sqr-sqrt 33.3%

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

   2. fabs-sqr 33.3%

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

   3. add-sqr-sqrt 34.9%

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

   4. add-sqr-sqrt 34.3%

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

   5. fabs-sqr 34.3%

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

   6. add-sqr-sqrt 34.9%

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

   7. distribute-lft-in 34.9%

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

7. Applied egg-rr 34.9%

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

   1. distribute-lft-out 34.9%

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

   2. distribute-lft-out 34.9%

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

   3. fma-undefine 34.9%

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

   4. *-commutative 34.9%

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

   5. fma-define 34.9%

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

9. Simplified 34.9%

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

10. Taylor expanded in x around inf 34.8%

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

11. Final simplification 34.8%

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

Alternative 7: 33.8% accurate, 17.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 99.9%

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

   1. add-sqr-sqrt 33.3%

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

   2. fabs-sqr 33.3%

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

   3. add-sqr-sqrt 34.9%

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

   4. add-sqr-sqrt 34.3%

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

   5. fabs-sqr 34.3%

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

   6. add-sqr-sqrt 34.9%

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

   7. distribute-lft-in 34.9%

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

7. Applied egg-rr 34.9%

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

   1. distribute-lft-out 34.9%

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

   2. distribute-lft-out 34.9%

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

   3. fma-undefine 34.9%

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

   4. *-commutative 34.9%

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

   5. fma-define 34.9%

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

9. Simplified 34.9%

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

10. Taylor expanded in x around 0 34.9%

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

    1. associate-*r* 34.9%

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

12. Simplified 34.9%

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

    1. pow1 34.9%

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

    2. *-commutative 34.9%

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

    3. inv-pow 34.9%

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

    4. sqrt-pow1 34.9%

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

    5. metadata-eval 34.9%

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

    6. *-commutative 34.9%

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

14. Applied egg-rr 34.9%

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

    1. unpow1 34.9%

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

    2. *-commutative 34.9%

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

16. Simplified 34.9%

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

17. Final simplification 34.9%

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

Reproduce

herbie shell --seed 2024101 
(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)))))))