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

Percentage Accurate: 99.8% → 99.9%

Time: 12.5s

Alternatives: 8

Speedup: 3.6×

• 29.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[Abs[N[(N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $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 99.4%

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

   1. div-inv 99.9%

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

   2. pow1/2 99.9%

      \[\leadsto \left|\left(\left|x\right| \cdot \frac{1}{\color{blue}{{\pi}^{0.5}}}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   3. pow-flip 99.9%

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

   4. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

   1. pow1 99.9%

      \[\leadsto \left|\color{blue}{{\left(\left|x\right| \cdot {\pi}^{-0.5}\right)}^{1}} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   2. add-sqr-sqrt 27.5%

      \[\leadsto \left|{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot {\pi}^{-0.5}\right)}^{1} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   3. fabs-sqr 27.5%

      \[\leadsto \left|{\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {\pi}^{-0.5}\right)}^{1} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   4. add-sqr-sqrt 99.9%

      \[\leadsto \left|{\left(\color{blue}{x} \cdot {\pi}^{-0.5}\right)}^{1} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

8. Applied egg-rr 99.9%

   \[\leadsto \left|\color{blue}{{\left(x \cdot {\pi}^{-0.5}\right)}^{1}} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]
9. Step-by-step derivation

   1. unpow1 99.9%

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

10. Simplified 99.9%

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

    1. fma-undefine 99.9%

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

12. Applied egg-rr 99.9%

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

13. Final simplification 99.9%

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

Alternative 2: 99.4% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[Abs[N[(N[(N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(x / 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 99.4%

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

   1. div-inv 99.9%

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

   2. pow1/2 99.9%

      \[\leadsto \left|\left(\left|x\right| \cdot \frac{1}{\color{blue}{{\pi}^{0.5}}}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   3. pow-flip 99.9%

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

   4. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

   1. pow1 99.9%

      \[\leadsto \left|\color{blue}{{\left(\left|x\right| \cdot {\pi}^{-0.5}\right)}^{1}} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   2. add-sqr-sqrt 27.5%

      \[\leadsto \left|{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot {\pi}^{-0.5}\right)}^{1} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   3. fabs-sqr 27.5%

      \[\leadsto \left|{\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {\pi}^{-0.5}\right)}^{1} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   4. add-sqr-sqrt 99.9%

      \[\leadsto \left|{\left(\color{blue}{x} \cdot {\pi}^{-0.5}\right)}^{1} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

8. Applied egg-rr 99.9%

   \[\leadsto \left|\color{blue}{{\left(x \cdot {\pi}^{-0.5}\right)}^{1}} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]
9. Step-by-step derivation

   1. unpow1 99.9%

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

10. Simplified 99.9%

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

    1. fma-undefine 99.9%

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

12. Applied egg-rr 99.9%

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

    1. metadata-eval 99.9%

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

    2. pow-flip 99.9%

       \[\leadsto \left|\left(x \cdot \color{blue}{\frac{1}{{\pi}^{0.5}}}\right) \cdot \left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

    3. pow1/2 99.9%

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

    4. div-inv 99.4%

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

14. Applied egg-rr 99.4%

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

15. Final simplification 99.4%

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

Alternative 3: 34.2% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(x * N[(N[(2.0 + N[(0.2 * N[Power[x, 4.0], $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $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 99.4%

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

5. Taylor expanded in x around 0 98.0%

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

   1. div-inv 98.5%

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

   2. add-sqr-sqrt 26.6%

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

   3. fabs-sqr 26.6%

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

   4. add-sqr-sqrt 28.3%

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

   5. add-sqr-sqrt 28.3%

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

   6. fabs-sqr 28.3%

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

   7. add-sqr-sqrt 28.3%

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

   8. clear-num 28.3%

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

7. Applied egg-rr 28.3%

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

8. Final simplification 28.3%

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

Alternative 4: 34.2% accurate, 8.7× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = 0.047619047619047616 * (pow(x, 7.0) * sqrt((1.0 / ((double) M_PI))));
	}
	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(x, 7.0) * Math.sqrt((1.0 / Math.PI)));
	}
	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(x, 7.0) * math.sqrt((1.0 / math.pi)))
	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((x ^ 7.0) * sqrt(Float64(1.0 / pi))));
	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 * ((x ^ 7.0) * sqrt((1.0 / pi)));
	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[x, 7.0], $MachinePrecision] * N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $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 99.4%

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

   5. Taylor expanded in x around 0 65.1%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{0.5 \cdot \sqrt{\pi}}\right|} \]
   6. Step-by-step derivation

      1. div-inv 65.6%

         \[\leadsto \color{blue}{\left|x\right| \cdot \frac{1}{\left|0.5 \cdot \sqrt{\pi}\right|}} \]

      2. add-sqr-sqrt 26.6%

         \[\leadsto \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \frac{1}{\left|0.5 \cdot \sqrt{\pi}\right|} \]

      3. fabs-sqr 26.6%

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

      4. add-sqr-sqrt 28.4%

         \[\leadsto \color{blue}{x} \cdot \frac{1}{\left|0.5 \cdot \sqrt{\pi}\right|} \]

      5. add-sqr-sqrt 28.4%

         \[\leadsto x \cdot \frac{1}{\left|\color{blue}{\sqrt{0.5 \cdot \sqrt{\pi}} \cdot \sqrt{0.5 \cdot \sqrt{\pi}}}\right|} \]

      6. fabs-sqr 28.4%

         \[\leadsto x \cdot \frac{1}{\color{blue}{\sqrt{0.5 \cdot \sqrt{\pi}} \cdot \sqrt{0.5 \cdot \sqrt{\pi}}}} \]

      7. add-sqr-sqrt 28.4%

         \[\leadsto x \cdot \frac{1}{\color{blue}{0.5 \cdot \sqrt{\pi}}} \]

   7. Applied egg-rr 28.4%

      \[\leadsto \color{blue}{x \cdot \frac{1}{0.5 \cdot \sqrt{\pi}}} \]
   8. Step-by-step derivation

      1. associate-/r* 28.4%

         \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{0.5}}{\sqrt{\pi}}} \]

      2. metadata-eval 28.4%

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

   9. Simplified 28.4%

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

   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 99.4%

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

   5. Taylor expanded in x around 0 98.0%

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

      1. div-inv 98.5%

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

      2. add-sqr-sqrt 26.6%

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

      3. fabs-sqr 26.6%

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

      4. add-sqr-sqrt 28.3%

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

      5. add-sqr-sqrt 28.3%

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

      6. fabs-sqr 28.3%

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

      7. add-sqr-sqrt 28.3%

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

      8. clear-num 28.3%

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

   7. Applied egg-rr 28.3%

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

   8. Taylor expanded in x around inf 3.7%

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

4. Final simplification 28.4%

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

Alternative 5: 34.2% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-34}:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{x}^{2}}{\pi \cdot 0.25}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e-34) (* x (/ 2.0 (sqrt PI))) (sqrt (/ (pow x 2.0) (* PI 0.25)))))

C

double code(double x) {
	double tmp;
	if (x <= 2e-34) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = sqrt((pow(x, 2.0) / (((double) M_PI) * 0.25)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2e-34) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.sqrt((Math.pow(x, 2.0) / (Math.PI * 0.25)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2e-34:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.sqrt((math.pow(x, 2.0) / (math.pi * 0.25)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e-34)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = sqrt(Float64((x ^ 2.0) / Float64(pi * 0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2e-34)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = sqrt(((x ^ 2.0) / (pi * 0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2e-34], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[x, 2.0], $MachinePrecision] / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999999999986e-34

   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 99.4%

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

   5. Taylor expanded in x around 0 63.9%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{0.5 \cdot \sqrt{\pi}}\right|} \]
   6. Step-by-step derivation

      1. div-inv 64.4%

         \[\leadsto \color{blue}{\left|x\right| \cdot \frac{1}{\left|0.5 \cdot \sqrt{\pi}\right|}} \]

      2. add-sqr-sqrt 23.1%

         \[\leadsto \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \frac{1}{\left|0.5 \cdot \sqrt{\pi}\right|} \]

      3. fabs-sqr 23.1%

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

      4. add-sqr-sqrt 25.0%

         \[\leadsto \color{blue}{x} \cdot \frac{1}{\left|0.5 \cdot \sqrt{\pi}\right|} \]

      5. add-sqr-sqrt 25.0%

         \[\leadsto x \cdot \frac{1}{\left|\color{blue}{\sqrt{0.5 \cdot \sqrt{\pi}} \cdot \sqrt{0.5 \cdot \sqrt{\pi}}}\right|} \]

      6. fabs-sqr 25.0%

         \[\leadsto x \cdot \frac{1}{\color{blue}{\sqrt{0.5 \cdot \sqrt{\pi}} \cdot \sqrt{0.5 \cdot \sqrt{\pi}}}} \]

      7. add-sqr-sqrt 25.0%

         \[\leadsto x \cdot \frac{1}{\color{blue}{0.5 \cdot \sqrt{\pi}}} \]

   7. Applied egg-rr 25.0%

      \[\leadsto \color{blue}{x \cdot \frac{1}{0.5 \cdot \sqrt{\pi}}} \]
   8. Step-by-step derivation

      1. associate-/r* 25.0%

         \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{0.5}}{\sqrt{\pi}}} \]

      2. metadata-eval 25.0%

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

   9. Simplified 25.0%

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

   if 1.99999999999999986e-34 < x

   1. Initial program 99.6%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 84.1%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{0.5 \cdot \sqrt{\pi}}\right|} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 83.8%

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

      2. fabs-sqr 83.8%

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

      3. add-sqr-sqrt 83.6%

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

      4. fabs-sqr 83.6%

         \[\leadsto \frac{\sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{0.5 \cdot \sqrt{\pi}} \cdot \sqrt{0.5 \cdot \sqrt{\pi}}}} \]

      5. add-sqr-sqrt 83.8%

         \[\leadsto \frac{\sqrt{x} \cdot \sqrt{x}}{\color{blue}{0.5 \cdot \sqrt{\pi}}} \]

      6. add-cube-cbrt 82.5%

         \[\leadsto \frac{\sqrt{x} \cdot \sqrt{x}}{\color{blue}{\left(\sqrt[3]{0.5 \cdot \sqrt{\pi}} \cdot \sqrt[3]{0.5 \cdot \sqrt{\pi}}\right) \cdot \sqrt[3]{0.5 \cdot \sqrt{\pi}}}} \]

      7. add-sqr-sqrt 82.5%

         \[\leadsto \frac{\color{blue}{x}}{\left(\sqrt[3]{0.5 \cdot \sqrt{\pi}} \cdot \sqrt[3]{0.5 \cdot \sqrt{\pi}}\right) \cdot \sqrt[3]{0.5 \cdot \sqrt{\pi}}} \]

      8. *-un-lft-identity 82.5%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{\left(\sqrt[3]{0.5 \cdot \sqrt{\pi}} \cdot \sqrt[3]{0.5 \cdot \sqrt{\pi}}\right) \cdot \sqrt[3]{0.5 \cdot \sqrt{\pi}}} \]

      9. times-frac 82.7%

         \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{0.5 \cdot \sqrt{\pi}} \cdot \sqrt[3]{0.5 \cdot \sqrt{\pi}}} \cdot \frac{x}{\sqrt[3]{0.5 \cdot \sqrt{\pi}}}} \]

      10. cbrt-unprod 83.2%

          \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{\left(0.5 \cdot \sqrt{\pi}\right) \cdot \left(0.5 \cdot \sqrt{\pi}\right)}}} \cdot \frac{x}{\sqrt[3]{0.5 \cdot \sqrt{\pi}}} \]

      11. pow2 83.2%

          \[\leadsto \frac{1}{\sqrt[3]{\color{blue}{{\left(0.5 \cdot \sqrt{\pi}\right)}^{2}}}} \cdot \frac{x}{\sqrt[3]{0.5 \cdot \sqrt{\pi}}} \]

   7. Applied egg-rr 83.2%

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{{\left(0.5 \cdot \sqrt{\pi}\right)}^{2}}} \cdot \frac{x}{\sqrt[3]{0.5 \cdot \sqrt{\pi}}}} \]
   8. Step-by-step derivation

      1. associate-*l/ 83.4%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{\sqrt[3]{0.5 \cdot \sqrt{\pi}}}}{\sqrt[3]{{\left(0.5 \cdot \sqrt{\pi}\right)}^{2}}}} \]

      2. *-lft-identity 83.4%

         \[\leadsto \frac{\color{blue}{\frac{x}{\sqrt[3]{0.5 \cdot \sqrt{\pi}}}}}{\sqrt[3]{{\left(0.5 \cdot \sqrt{\pi}\right)}^{2}}} \]

      3. *-commutative 83.4%

         \[\leadsto \frac{\frac{x}{\sqrt[3]{\color{blue}{\sqrt{\pi} \cdot 0.5}}}}{\sqrt[3]{{\left(0.5 \cdot \sqrt{\pi}\right)}^{2}}} \]

      4. unpow2 83.4%

         \[\leadsto \frac{\frac{x}{\sqrt[3]{\sqrt{\pi} \cdot 0.5}}}{\sqrt[3]{\color{blue}{\left(0.5 \cdot \sqrt{\pi}\right) \cdot \left(0.5 \cdot \sqrt{\pi}\right)}}} \]

      5. *-commutative 83.4%

         \[\leadsto \frac{\frac{x}{\sqrt[3]{\sqrt{\pi} \cdot 0.5}}}{\sqrt[3]{\color{blue}{\left(\sqrt{\pi} \cdot 0.5\right)} \cdot \left(0.5 \cdot \sqrt{\pi}\right)}} \]

      6. *-commutative 83.4%

         \[\leadsto \frac{\frac{x}{\sqrt[3]{\sqrt{\pi} \cdot 0.5}}}{\sqrt[3]{\left(\sqrt{\pi} \cdot 0.5\right) \cdot \color{blue}{\left(\sqrt{\pi} \cdot 0.5\right)}}} \]

      7. swap-sqr 83.4%

         \[\leadsto \frac{\frac{x}{\sqrt[3]{\sqrt{\pi} \cdot 0.5}}}{\sqrt[3]{\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \left(0.5 \cdot 0.5\right)}}} \]

      8. rem-square-sqrt 83.2%

         \[\leadsto \frac{\frac{x}{\sqrt[3]{\sqrt{\pi} \cdot 0.5}}}{\sqrt[3]{\color{blue}{\pi} \cdot \left(0.5 \cdot 0.5\right)}} \]

      9. metadata-eval 83.2%

         \[\leadsto \frac{\frac{x}{\sqrt[3]{\sqrt{\pi} \cdot 0.5}}}{\sqrt[3]{\pi \cdot \color{blue}{0.25}}} \]

   9. Simplified 83.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt[3]{\sqrt{\pi} \cdot 0.5}}}{\sqrt[3]{\pi \cdot 0.25}}} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 83.2%

          \[\leadsto \color{blue}{\sqrt{\frac{\frac{x}{\sqrt[3]{\sqrt{\pi} \cdot 0.5}}}{\sqrt[3]{\pi \cdot 0.25}}} \cdot \sqrt{\frac{\frac{x}{\sqrt[3]{\sqrt{\pi} \cdot 0.5}}}{\sqrt[3]{\pi \cdot 0.25}}}} \]

       2. sqrt-unprod 83.2%

          \[\leadsto \color{blue}{\sqrt{\frac{\frac{x}{\sqrt[3]{\sqrt{\pi} \cdot 0.5}}}{\sqrt[3]{\pi \cdot 0.25}} \cdot \frac{\frac{x}{\sqrt[3]{\sqrt{\pi} \cdot 0.5}}}{\sqrt[3]{\pi \cdot 0.25}}}} \]

       3. associate-/l/ 83.1%

          \[\leadsto \sqrt{\color{blue}{\frac{x}{\sqrt[3]{\pi \cdot 0.25} \cdot \sqrt[3]{\sqrt{\pi} \cdot 0.5}}} \cdot \frac{\frac{x}{\sqrt[3]{\sqrt{\pi} \cdot 0.5}}}{\sqrt[3]{\pi \cdot 0.25}}} \]

       4. associate-/l/ 83.0%

          \[\leadsto \sqrt{\frac{x}{\sqrt[3]{\pi \cdot 0.25} \cdot \sqrt[3]{\sqrt{\pi} \cdot 0.5}} \cdot \color{blue}{\frac{x}{\sqrt[3]{\pi \cdot 0.25} \cdot \sqrt[3]{\sqrt{\pi} \cdot 0.5}}}} \]

       5. frac-times 83.0%

          \[\leadsto \sqrt{\color{blue}{\frac{x \cdot x}{\left(\sqrt[3]{\pi \cdot 0.25} \cdot \sqrt[3]{\sqrt{\pi} \cdot 0.5}\right) \cdot \left(\sqrt[3]{\pi \cdot 0.25} \cdot \sqrt[3]{\sqrt{\pi} \cdot 0.5}\right)}}} \]

       6. pow2 83.0%

          \[\leadsto \sqrt{\frac{\color{blue}{{x}^{2}}}{\left(\sqrt[3]{\pi \cdot 0.25} \cdot \sqrt[3]{\sqrt{\pi} \cdot 0.5}\right) \cdot \left(\sqrt[3]{\pi \cdot 0.25} \cdot \sqrt[3]{\sqrt{\pi} \cdot 0.5}\right)}} \]

   11. Applied egg-rr 84.2%

       \[\leadsto \color{blue}{\sqrt{\frac{{x}^{2}}{\pi \cdot 0.25}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-34}:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{x}^{2}}{\pi \cdot 0.25}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 34.2% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{0.047619047619047616 \cdot {x}^{6} + 2}{\sqrt{\pi}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(x * N[(N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $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 99.4%

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

5. Taylor expanded in x around 0 98.0%

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

   1. div-inv 98.5%

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

   2. add-sqr-sqrt 26.6%

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

   3. fabs-sqr 26.6%

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

   4. add-sqr-sqrt 28.3%

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

   5. add-sqr-sqrt 28.3%

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

   6. fabs-sqr 28.3%

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

   7. add-sqr-sqrt 28.3%

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

   8. clear-num 28.3%

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

7. Applied egg-rr 28.3%

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

8. Taylor expanded in x around inf 28.3%

   \[\leadsto x \cdot \frac{\color{blue}{0.047619047619047616 \cdot {x}^{6}} + 2}{\sqrt{\pi}} \]

9. Final simplification 28.3%

   \[\leadsto x \cdot \frac{0.047619047619047616 \cdot {x}^{6} + 2}{\sqrt{\pi}} \]
10. Add Preprocessing

Alternative 7: 34.0% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(2.0 * N[(x / N[Sqrt[Pi], $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 99.4%

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

5. Taylor expanded in x around 0 65.1%

   \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{0.5 \cdot \sqrt{\pi}}\right|} \]
6. Step-by-step derivation

   1. add-sqr-sqrt 26.6%

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

   2. fabs-sqr 26.6%

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

   3. add-sqr-sqrt 26.6%

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

   4. fabs-sqr 26.6%

      \[\leadsto \frac{\sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{0.5 \cdot \sqrt{\pi}} \cdot \sqrt{0.5 \cdot \sqrt{\pi}}}} \]

   5. add-sqr-sqrt 28.4%

      \[\leadsto \frac{\color{blue}{x}}{\sqrt{0.5 \cdot \sqrt{\pi}} \cdot \sqrt{0.5 \cdot \sqrt{\pi}}} \]

   6. *-un-lft-identity 28.4%

      \[\leadsto \frac{\color{blue}{1 \cdot x}}{\sqrt{0.5 \cdot \sqrt{\pi}} \cdot \sqrt{0.5 \cdot \sqrt{\pi}}} \]

   7. add-sqr-sqrt 28.2%

      \[\leadsto \frac{1 \cdot x}{\color{blue}{0.5 \cdot \sqrt{\pi}}} \]

   8. times-frac 28.2%

      \[\leadsto \color{blue}{\frac{1}{0.5} \cdot \frac{x}{\sqrt{\pi}}} \]

   9. metadata-eval 28.2%

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

7. Applied egg-rr 28.2%

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

8. Final simplification 28.2%

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

Alternative 8: 34.2% 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 99.9%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 65.1%

   \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{0.5 \cdot \sqrt{\pi}}\right|} \]
6. Step-by-step derivation

   1. div-inv 65.6%

      \[\leadsto \color{blue}{\left|x\right| \cdot \frac{1}{\left|0.5 \cdot \sqrt{\pi}\right|}} \]

   2. add-sqr-sqrt 26.6%

      \[\leadsto \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \frac{1}{\left|0.5 \cdot \sqrt{\pi}\right|} \]

   3. fabs-sqr 26.6%

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

   4. add-sqr-sqrt 28.4%

      \[\leadsto \color{blue}{x} \cdot \frac{1}{\left|0.5 \cdot \sqrt{\pi}\right|} \]

   5. add-sqr-sqrt 28.4%

      \[\leadsto x \cdot \frac{1}{\left|\color{blue}{\sqrt{0.5 \cdot \sqrt{\pi}} \cdot \sqrt{0.5 \cdot \sqrt{\pi}}}\right|} \]

   6. fabs-sqr 28.4%

      \[\leadsto x \cdot \frac{1}{\color{blue}{\sqrt{0.5 \cdot \sqrt{\pi}} \cdot \sqrt{0.5 \cdot \sqrt{\pi}}}} \]

   7. add-sqr-sqrt 28.4%

      \[\leadsto x \cdot \frac{1}{\color{blue}{0.5 \cdot \sqrt{\pi}}} \]

7. Applied egg-rr 28.4%

   \[\leadsto \color{blue}{x \cdot \frac{1}{0.5 \cdot \sqrt{\pi}}} \]
8. Step-by-step derivation

   1. associate-/r* 28.4%

      \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{0.5}}{\sqrt{\pi}}} \]

   2. metadata-eval 28.4%

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

9. Simplified 28.4%

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

10. Final simplification 28.4%

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

Reproduce

herbie shell --seed 2024039 
(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)))))))