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

Percentage Accurate: 99.8% → 99.9%

Time: 10.6s

Alternatives: 10

Speedup: 3.5×

• 28.7% 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, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. fma-undefine 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 99.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. pow1 99.8%

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

   2. mul-fabs 99.8%

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

   3. +-commutative 99.8%

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

   4. pow2 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow1 99.8%

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

   2. associate-*r/ 99.4%

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

   3. fabs-div 99.4%

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

   4. rem-square-sqrt 99.6%

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

   5. fabs-sqr 99.6%

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

   6. rem-square-sqrt 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in x around 0 99.8%

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

10. Final simplification 99.8%

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

Alternative 3: 99.4% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. pow1 99.8%

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

   2. mul-fabs 99.8%

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

   3. +-commutative 99.8%

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

   4. pow2 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow1 99.8%

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

   2. associate-*r/ 99.4%

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

   3. fabs-div 99.4%

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

   4. rem-square-sqrt 99.6%

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

   5. fabs-sqr 99.6%

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

   6. rem-square-sqrt 99.4%

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

8. Simplified 99.4%

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

   1. fma-undefine 99.8%

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

10. Applied egg-rr 99.4%

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

    1. fma-undefine 99.4%

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

12. Applied egg-rr 99.4%

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

13. Final simplification 99.4%

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

Alternative 4: 34.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. pow1 99.8%

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

   2. mul-fabs 99.8%

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

   3. +-commutative 99.8%

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

   4. pow2 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow1 99.8%

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

   2. associate-*r/ 99.4%

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

   3. fabs-div 99.4%

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

   4. rem-square-sqrt 99.6%

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

   5. fabs-sqr 99.6%

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

   6. rem-square-sqrt 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in x around 0 98.3%

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

    1. fma-undefine 99.8%

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

11. Applied egg-rr 98.3%

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

    1. *-un-lft-identity 98.3%

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

    2. add-sqr-sqrt 33.6%

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

    3. fabs-sqr 33.6%

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

    4. add-sqr-sqrt 34.7%

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

    5. fma-define 34.7%

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

13. Applied egg-rr 34.7%

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

    1. *-lft-identity 34.7%

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

    2. associate-*r/ 34.9%

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

    3. fma-undefine 34.9%

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

    4. +-commutative 34.9%

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

    5. associate-+r+ 34.9%

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

    6. +-commutative 34.9%

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

    7. associate-+l+ 34.9%

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

    8. fma-define 34.9%

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

    9. +-commutative 34.9%

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

    10. fma-define 34.9%

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

15. Simplified 34.9%

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

16. Final simplification 34.9%

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

Alternative 5: 34.1% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. pow1 99.8%

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

   2. mul-fabs 99.8%

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

   3. +-commutative 99.8%

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

   4. pow2 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow1 99.8%

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

   2. associate-*r/ 99.4%

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

   3. fabs-div 99.4%

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

   4. rem-square-sqrt 99.6%

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

   5. fabs-sqr 99.6%

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

   6. rem-square-sqrt 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in x around 0 98.3%

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

    1. fma-undefine 99.8%

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

11. Applied egg-rr 98.3%

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

    1. *-un-lft-identity 98.3%

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

    2. add-sqr-sqrt 33.6%

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

    3. fabs-sqr 33.6%

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

    4. add-sqr-sqrt 34.7%

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

    5. fma-define 34.7%

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

13. Applied egg-rr 34.7%

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

    1. *-lft-identity 34.7%

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

    2. associate-*r/ 34.9%

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

    3. fma-undefine 34.9%

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

    4. +-commutative 34.9%

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

    5. associate-+r+ 34.9%

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

    6. +-commutative 34.9%

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

    7. fma-undefine 34.9%

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

15. Simplified 34.9%

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

16. Final simplification 34.9%

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

Alternative 6: 67.7% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 63.3%

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

      1. associate-*r* 63.3%

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

      2. *-commutative 63.3%

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

      3. unpow1 63.3%

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

      4. sqr-pow 33.6%

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

      5. fabs-sqr 33.6%

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

      6. sqr-pow 63.3%

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

      7. unpow1 63.3%

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

   7. Simplified 63.3%

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

      1. pow1 63.3%

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

      2. associate-*r* 63.3%

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

      3. inv-pow 63.3%

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

      4. sqrt-pow1 63.3%

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

      5. metadata-eval 63.3%

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

   9. Applied egg-rr 63.3%

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

       1. unpow1 63.3%

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

       2. associate-*l* 63.3%

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

   11. Simplified 63.3%

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

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

      1. pow1 99.8%

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

      2. mul-fabs 99.8%

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

      3. +-commutative 99.8%

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

      4. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow1 99.8%

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

      2. associate-*r/ 99.4%

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

      3. fabs-div 99.4%

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

      4. rem-square-sqrt 99.6%

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

      5. fabs-sqr 99.6%

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

      6. rem-square-sqrt 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in x around 0 98.3%

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

   10. Taylor expanded in x around inf 98.0%

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

   11. Taylor expanded in x around inf 40.7%

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

4. Final simplification 63.3%

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

Alternative 7: 98.5% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. pow1 99.8%

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

   2. mul-fabs 99.8%

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

   3. +-commutative 99.8%

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

   4. pow2 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow1 99.8%

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

   2. associate-*r/ 99.4%

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

   3. fabs-div 99.4%

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

   4. rem-square-sqrt 99.6%

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

   5. fabs-sqr 99.6%

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

   6. rem-square-sqrt 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in x around 0 98.3%

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

10. Taylor expanded in x around inf 98.0%

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

11. Final simplification 98.0%

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

Alternative 8: 33.8% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. pow1 99.8%

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

   2. mul-fabs 99.8%

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

   3. +-commutative 99.8%

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

   4. pow2 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow1 99.8%

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

   2. associate-*r/ 99.4%

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

   3. fabs-div 99.4%

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

   4. rem-square-sqrt 99.6%

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

   5. fabs-sqr 99.6%

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

   6. rem-square-sqrt 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in x around 0 98.3%

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

10. Taylor expanded in x around inf 98.0%

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

    1. *-un-lft-identity 98.0%

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

    2. add-sqr-sqrt 33.6%

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

    3. fabs-sqr 33.6%

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

    4. add-sqr-sqrt 34.7%

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

    5. +-commutative 34.7%

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

    6. fma-define 34.7%

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

12. Applied egg-rr 34.7%

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

    1. *-lft-identity 34.7%

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

    2. *-commutative 34.7%

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

    3. associate-/l* 34.7%

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

14. Simplified 34.7%

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

15. Final simplification 34.7%

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

Alternative 9: 67.7% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 63.3%

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

   1. associate-*r* 63.3%

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

   2. *-commutative 63.3%

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

   3. unpow1 63.3%

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

   4. sqr-pow 33.6%

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

   5. fabs-sqr 33.6%

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

   6. sqr-pow 63.3%

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

   7. unpow1 63.3%

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

7. Simplified 63.3%

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

   1. pow1 63.3%

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

   2. associate-*r* 63.3%

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

   3. inv-pow 63.3%

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

   4. sqrt-pow1 63.3%

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

   5. metadata-eval 63.3%

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

9. Applied egg-rr 63.3%

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

    1. unpow1 63.3%

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

    2. associate-*l* 63.3%

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

11. Simplified 63.3%

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

12. Final simplification 63.3%

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

Alternative 10: 67.2% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. pow1 99.8%

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

   2. mul-fabs 99.8%

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

   3. +-commutative 99.8%

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

   4. pow2 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow1 99.8%

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

   2. associate-*r/ 99.4%

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

   3. fabs-div 99.4%

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

   4. rem-square-sqrt 99.6%

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

   5. fabs-sqr 99.6%

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

   6. rem-square-sqrt 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in x around 0 98.3%

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

10. Taylor expanded in x around inf 98.0%

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

11. Taylor expanded in x around 0 62.9%

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

    1. *-commutative 62.9%

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

13. Simplified 62.9%

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

14. Final simplification 62.9%

    \[\leadsto \frac{\left|x \cdot 2\right|}{\sqrt{\pi}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(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)))))))