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

Percentage Accurate: 99.8% → 99.9%

Time: 20.8s

Alternatives: 5

Speedup: 3.5×

• 28.9% 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, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
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. pow2 99.9%

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

7. Applied egg-rr 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 63.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{+25}:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left({x}^{2}, 0.6666666666666666, 2\right)\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|0.2 \cdot \sqrt{\frac{{x}^{10}}{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 1e+25)
   (*
    x
    (/
     (fma 0.2 (pow x 4.0) (fma (pow x 2.0) 0.6666666666666666 2.0))
     (sqrt PI)))
   (fabs (* 0.2 (sqrt (/ (pow x 10.0) PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 1e+25) {
		tmp = x * (fma(0.2, pow(x, 4.0), fma(pow(x, 2.0), 0.6666666666666666, 2.0)) / sqrt(((double) M_PI)));
	} else {
		tmp = fabs((0.2 * sqrt((pow(x, 10.0) / ((double) M_PI)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 1e+25)
		tmp = Float64(x * Float64(fma(0.2, (x ^ 4.0), fma((x ^ 2.0), 0.6666666666666666, 2.0)) / sqrt(pi)));
	else
		tmp = abs(Float64(0.2 * sqrt(Float64((x ^ 10.0) / pi))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 1e+25], N[(x * N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision] + N[(N[Power[x, 2.0], $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(0.2 * N[Sqrt[N[(N[Power[x, 10.0], $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000009e25

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 94.8%

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

      1. pow1 94.8%

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

      2. add-sqr-sqrt 48.0%

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

      3. fabs-sqr 48.0%

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

      4. add-sqr-sqrt 50.0%

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

      5. add-sqr-sqrt 49.2%

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

      6. fabs-sqr 49.2%

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

      7. add-sqr-sqrt 50.0%

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

      8. fma-define 50.0%

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

      9. pow2 50.0%

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

   7. Applied egg-rr 50.0%

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

      1. unpow1 50.0%

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

      2. fma-define 50.0%

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

      3. *-commutative 50.0%

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

      4. fma-define 50.0%

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

   9. Simplified 50.0%

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

   if 1.00000000000000009e25 < (fabs.f64 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{3} \cdot \left({\left(\left|x\right|\right)}^{3} \cdot \left|x\right|\right), \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{3} \cdot \left(x \cdot x\right), \mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)\right)\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 92.9%

      \[\leadsto \left|\color{blue}{0.2 \cdot \left(\left({x}^{2} \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   6. Step-by-step derivation

      1. associate-*r* 92.9%

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

      2. *-commutative 92.9%

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

      3. *-commutative 92.9%

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

      4. unpow2 92.9%

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

      5. associate-*r* 92.9%

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

      6. unpow3 92.9%

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

      7. fabs-mul 92.9%

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

      8. unpow2 92.9%

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

      9. fabs-mul 92.9%

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

      10. unpow2 92.9%

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

      11. unpow3 92.9%

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

      12. unpow1 92.9%

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

      13. sqr-pow 72.2%

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

      14. fabs-sqr 72.2%

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

      15. sqr-pow 92.9%

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

      16. unpow1 92.9%

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

      17. pow-plus 92.9%

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

      18. metadata-eval 92.9%

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

      19. pow-plus 92.9%

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

      20. metadata-eval 92.9%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{\color{blue}{5}}\right)\right| \]

   7. Simplified 92.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5}\right)}\right| \]
   8. Step-by-step derivation

      1. pow1 92.9%

         \[\leadsto \left|\color{blue}{{\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5}\right)\right)}^{1}}\right| \]

      2. inv-pow 92.9%

         \[\leadsto \left|{\left(\sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left(0.2 \cdot {x}^{5}\right)\right)}^{1}\right| \]

      3. sqrt-pow1 92.9%

         \[\leadsto \left|{\left(\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left(0.2 \cdot {x}^{5}\right)\right)}^{1}\right| \]

      4. metadata-eval 92.9%

         \[\leadsto \left|{\left({\pi}^{\color{blue}{-0.5}} \cdot \left(0.2 \cdot {x}^{5}\right)\right)}^{1}\right| \]

   9. Applied egg-rr 92.9%

      \[\leadsto \left|\color{blue}{{\left({\pi}^{-0.5} \cdot \left(0.2 \cdot {x}^{5}\right)\right)}^{1}}\right| \]
   10. Step-by-step derivation

       1. unpow1 92.9%

          \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(0.2 \cdot {x}^{5}\right)}\right| \]

       2. *-commutative 92.9%

          \[\leadsto \left|\color{blue}{\left(0.2 \cdot {x}^{5}\right) \cdot {\pi}^{-0.5}}\right| \]

       3. associate-*l* 92.9%

          \[\leadsto \left|\color{blue}{0.2 \cdot \left({x}^{5} \cdot {\pi}^{-0.5}\right)}\right| \]

   11. Simplified 92.9%

       \[\leadsto \left|\color{blue}{0.2 \cdot \left({x}^{5} \cdot {\pi}^{-0.5}\right)}\right| \]
   12. Step-by-step derivation

       1. associate-*r* 92.9%

          \[\leadsto \left|\color{blue}{\left(0.2 \cdot {x}^{5}\right) \cdot {\pi}^{-0.5}}\right| \]

       2. expm1-log1p-u 0.0%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.2 \cdot {x}^{5}\right)\right)} \cdot {\pi}^{-0.5}\right| \]

       3. metadata-eval 0.0%

          \[\leadsto \left|\mathsf{expm1}\left(\mathsf{log1p}\left(0.2 \cdot {x}^{5}\right)\right) \cdot {\pi}^{\color{blue}{\left(-0.5\right)}}\right| \]

       4. pow-flip 0.0%

          \[\leadsto \left|\mathsf{expm1}\left(\mathsf{log1p}\left(0.2 \cdot {x}^{5}\right)\right) \cdot \color{blue}{\frac{1}{{\pi}^{0.5}}}\right| \]

       5. pow1/2 0.0%

          \[\leadsto \left|\mathsf{expm1}\left(\mathsf{log1p}\left(0.2 \cdot {x}^{5}\right)\right) \cdot \frac{1}{\color{blue}{\sqrt{\pi}}}\right| \]

       6. un-div-inv 0.0%

          \[\leadsto \left|\color{blue}{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(0.2 \cdot {x}^{5}\right)\right)}{\sqrt{\pi}}}\right| \]

       7. expm1-log1p-u 92.9%

          \[\leadsto \left|\frac{\color{blue}{0.2 \cdot {x}^{5}}}{\sqrt{\pi}}\right| \]

   13. Applied egg-rr 92.9%

       \[\leadsto \left|\color{blue}{\frac{0.2 \cdot {x}^{5}}{\sqrt{\pi}}}\right| \]
   14. Step-by-step derivation

       1. associate-/l* 92.9%

          \[\leadsto \left|\color{blue}{0.2 \cdot \frac{{x}^{5}}{\sqrt{\pi}}}\right| \]

   15. Simplified 92.9%

       \[\leadsto \left|\color{blue}{0.2 \cdot \frac{{x}^{5}}{\sqrt{\pi}}}\right| \]
   16. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \left|0.2 \cdot \color{blue}{\left(\sqrt{\frac{{x}^{5}}{\sqrt{\pi}}} \cdot \sqrt{\frac{{x}^{5}}{\sqrt{\pi}}}\right)}\right| \]

       2. sqrt-unprod 97.7%

          \[\leadsto \left|0.2 \cdot \color{blue}{\sqrt{\frac{{x}^{5}}{\sqrt{\pi}} \cdot \frac{{x}^{5}}{\sqrt{\pi}}}}\right| \]

       3. frac-times 97.7%

          \[\leadsto \left|0.2 \cdot \sqrt{\color{blue}{\frac{{x}^{5} \cdot {x}^{5}}{\sqrt{\pi} \cdot \sqrt{\pi}}}}\right| \]

       4. pow-prod-up 97.7%

          \[\leadsto \left|0.2 \cdot \sqrt{\frac{\color{blue}{{x}^{\left(5 + 5\right)}}}{\sqrt{\pi} \cdot \sqrt{\pi}}}\right| \]

       5. metadata-eval 97.7%

          \[\leadsto \left|0.2 \cdot \sqrt{\frac{{x}^{\color{blue}{10}}}{\sqrt{\pi} \cdot \sqrt{\pi}}}\right| \]

       6. add-sqr-sqrt 97.7%

          \[\leadsto \left|0.2 \cdot \sqrt{\frac{{x}^{10}}{\color{blue}{\pi}}}\right| \]

   17. Applied egg-rr 97.7%

       \[\leadsto \left|0.2 \cdot \color{blue}{\sqrt{\frac{{x}^{10}}{\pi}}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{+25}:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left({x}^{2}, 0.6666666666666666, 2\right)\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|0.2 \cdot \sqrt{\frac{{x}^{10}}{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around inf 98.6%

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

6. Final simplification 98.6%

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

Alternative 4: 34.4% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + x \cdot 2\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (sqrt (/ 1.0 PI)) (+ (* 0.2 (pow x 5.0)) (* x 2.0))))

C

double code(double x) {
	return sqrt((1.0 / ((double) M_PI))) * ((0.2 * pow(x, 5.0)) + (x * 2.0));
}

Java

public static double code(double x) {
	return Math.sqrt((1.0 / Math.PI)) * ((0.2 * Math.pow(x, 5.0)) + (x * 2.0));
}

Python

def code(x):
	return math.sqrt((1.0 / math.pi)) * ((0.2 * math.pow(x, 5.0)) + (x * 2.0))

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(0.2 * (x ^ 5.0)) + Float64(x * 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 / pi)) * ((0.2 * (x ^ 5.0)) + (x * 2.0));
end

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 93.9%

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

6. Taylor expanded in x around 0 93.1%

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

   1. log1p-expm1-u 94.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left|x\right| \cdot \left|\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}\right|\right)\right)} \]

   2. log1p-undefine 35.9%

      \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(\left|x\right| \cdot \left|\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}\right|\right)\right)} \]

   3. add-sqr-sqrt 2.3%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left|\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}\right|\right)\right) \]

   4. fabs-sqr 2.3%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left|\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}\right|\right)\right) \]

   5. add-sqr-sqrt 3.9%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(\color{blue}{x} \cdot \left|\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}\right|\right)\right) \]

   6. add-sqr-sqrt 3.9%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(x \cdot \left|\color{blue}{\sqrt{\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}} \cdot \sqrt{\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}}}\right|\right)\right) \]

   7. fabs-sqr 3.9%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(x \cdot \color{blue}{\left(\sqrt{\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}} \cdot \sqrt{\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}}\right)}\right)\right) \]

   8. add-sqr-sqrt 3.9%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(x \cdot \color{blue}{\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}}\right)\right) \]

   9. fma-define 3.9%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(x \cdot \frac{\color{blue}{\mathsf{fma}\left(0.2, {x}^{4}, 2\right)}}{\sqrt{\pi}}\right)\right) \]

8. Applied egg-rr 3.9%

   \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 2\right)}{\sqrt{\pi}}\right)\right)} \]

9. Taylor expanded in x around 0 34.4%

   \[\leadsto \color{blue}{0.2 \cdot \left({x}^{5} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)} \]
10. Step-by-step derivation

    1. associate-*r* 34.4%

       \[\leadsto \color{blue}{\left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}}} + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right) \]

    2. associate-*r* 34.4%

       \[\leadsto \left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} \]

    3. distribute-rgt-out 34.4%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 2 \cdot x\right)} \]

    4. *-commutative 34.4%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + \color{blue}{x \cdot 2}\right) \]

11. Simplified 34.4%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + x \cdot 2\right)} \]

12. Final simplification 34.4%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + x \cdot 2\right) \]
13. Add Preprocessing

Alternative 5: 34.5% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 2.0 (* x (sqrt (/ 1.0 PI)))))

C

double code(double x) {
	return 2.0 * (x * sqrt((1.0 / ((double) M_PI))));
}

Java

public static double code(double x) {
	return 2.0 * (x * Math.sqrt((1.0 / Math.PI)));
}

Python

def code(x):
	return 2.0 * (x * math.sqrt((1.0 / math.pi)))

Julia

function code(x)
	return Float64(2.0 * Float64(x * sqrt(Float64(1.0 / pi))))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 * (x * sqrt((1.0 / pi)));
end

Wolfram

code[x_] := N[(2.0 * N[(x * N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 93.9%

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

6. Taylor expanded in x around 0 93.1%

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

   1. log1p-expm1-u 94.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left|x\right| \cdot \left|\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}\right|\right)\right)} \]

   2. log1p-undefine 35.9%

      \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(\left|x\right| \cdot \left|\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}\right|\right)\right)} \]

   3. add-sqr-sqrt 2.3%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left|\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}\right|\right)\right) \]

   4. fabs-sqr 2.3%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left|\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}\right|\right)\right) \]

   5. add-sqr-sqrt 3.9%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(\color{blue}{x} \cdot \left|\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}\right|\right)\right) \]

   6. add-sqr-sqrt 3.9%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(x \cdot \left|\color{blue}{\sqrt{\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}} \cdot \sqrt{\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}}}\right|\right)\right) \]

   7. fabs-sqr 3.9%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(x \cdot \color{blue}{\left(\sqrt{\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}} \cdot \sqrt{\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}}\right)}\right)\right) \]

   8. add-sqr-sqrt 3.9%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(x \cdot \color{blue}{\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}}\right)\right) \]

   9. fma-define 3.9%

      \[\leadsto \log \left(1 + \mathsf{expm1}\left(x \cdot \frac{\color{blue}{\mathsf{fma}\left(0.2, {x}^{4}, 2\right)}}{\sqrt{\pi}}\right)\right) \]

8. Applied egg-rr 3.9%

   \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 2\right)}{\sqrt{\pi}}\right)\right)} \]

9. Taylor expanded in x around 0 34.5%

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

    1. *-commutative 34.5%

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

11. Simplified 34.5%

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

12. Final simplification 34.5%

    \[\leadsto 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024044 
(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)))))))