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

Percentage Accurate: 99.8% → 99.9%

Time: 13.6s

Alternatives: 11

Speedup: 3.6×

• 28.5% 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

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

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 (pow x 2.0)))))))))

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 * pow(x, 2.0)))))));
}

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 * Math.pow(x, 2.0)))))));
}

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 * math.pow(x, 2.0)))))))

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 * (x ^ 2.0))))))))
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 ^ 2.0)))))));
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[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

   1. *-un-lft-identity 99.4%

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

   2. add-sqr-sqrt 31.3%

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

   3. fabs-sqr 31.3%

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

   4. add-sqr-sqrt 99.4%

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

6. Applied egg-rr 99.4%

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

   1. *-lft-identity 99.4%

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

8. Simplified 99.4%

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

   1. div-inv 99.9%

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

   2. add-sqr-sqrt 31.3%

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

   3. fabs-sqr 31.3%

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

   4. add-sqr-sqrt 99.9%

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

   5. metadata-eval 99.9%

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

   6. sqrt-div 99.9%

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

   7. *-commutative 99.9%

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

   8. inv-pow 99.9%

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

   9. sqrt-pow1 99.9%

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

   10. metadata-eval 99.9%

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

   11. add-sqr-sqrt 31.3%

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

   12. fabs-sqr 31.3%

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

   13. add-sqr-sqrt 99.9%

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

10. Applied egg-rr 99.9%

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

    1. fma-undefine 99.9%

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

12. Applied egg-rr 99.9%

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

13. Final simplification 99.9%

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

Alternative 3: 99.4% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

   1. *-un-lft-identity 99.4%

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

   2. add-sqr-sqrt 31.3%

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

   3. fabs-sqr 31.3%

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

   4. add-sqr-sqrt 99.4%

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

6. Applied egg-rr 99.4%

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

   1. *-lft-identity 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in x around 0 99.4%

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

    1. *-commutative 99.4%

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

11. Simplified 99.4%

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

12. Final simplification 99.4%

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

Alternative 4: 99.1% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

   1. *-un-lft-identity 99.4%

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

   2. add-sqr-sqrt 31.3%

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

   3. fabs-sqr 31.3%

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

   4. add-sqr-sqrt 99.4%

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

6. Applied egg-rr 99.4%

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

   1. *-lft-identity 99.4%

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

8. Simplified 99.4%

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

   1. div-inv 99.9%

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

   2. add-sqr-sqrt 31.3%

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

   3. fabs-sqr 31.3%

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

   4. add-sqr-sqrt 99.9%

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

   5. metadata-eval 99.9%

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

   6. sqrt-div 99.9%

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

   7. *-commutative 99.9%

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

   8. inv-pow 99.9%

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

   9. sqrt-pow1 99.9%

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

   10. metadata-eval 99.9%

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

   11. add-sqr-sqrt 31.3%

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

   12. fabs-sqr 31.3%

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

   13. add-sqr-sqrt 99.9%

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

10. Applied egg-rr 99.9%

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

11. Taylor expanded in x around inf 99.1%

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

12. Final simplification 99.1%

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

Alternative 5: 98.6% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

   1. *-un-lft-identity 99.4%

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

   2. add-sqr-sqrt 31.3%

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

   3. fabs-sqr 31.3%

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

   4. add-sqr-sqrt 99.4%

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

6. Applied egg-rr 99.4%

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

   1. *-lft-identity 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in x around inf 98.7%

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

10. Final simplification 98.7%

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

Alternative 6: 93.7% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot 2\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{0.4444444444444444 \cdot \frac{{x}^{6}}{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.4)
   (* (fabs x) (fabs (* (sqrt (/ 1.0 PI)) 2.0)))
   (fabs (sqrt (* 0.4444444444444444 (/ (pow x 6.0) PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.4) {
		tmp = fabs(x) * fabs((sqrt((1.0 / ((double) M_PI))) * 2.0));
	} else {
		tmp = fabs(sqrt((0.4444444444444444 * (pow(x, 6.0) / ((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.4) {
		tmp = Math.abs(x) * Math.abs((Math.sqrt((1.0 / Math.PI)) * 2.0));
	} else {
		tmp = Math.abs(Math.sqrt((0.4444444444444444 * (Math.pow(x, 6.0) / Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.4:
		tmp = math.fabs(x) * math.fabs((math.sqrt((1.0 / math.pi)) * 2.0))
	else:
		tmp = math.fabs(math.sqrt((0.4444444444444444 * (math.pow(x, 6.0) / math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.4)
		tmp = Float64(abs(x) * abs(Float64(sqrt(Float64(1.0 / pi)) * 2.0)));
	else
		tmp = abs(sqrt(Float64(0.4444444444444444 * Float64((x ^ 6.0) / pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.4)
		tmp = abs(x) * abs((sqrt((1.0 / pi)) * 2.0));
	else
		tmp = abs(sqrt((0.4444444444444444 * ((x ^ 6.0) / pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.4], N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[Sqrt[N[(0.4444444444444444 * N[(N[Power[x, 6.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.40000000000000002

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.3%

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

   6. Taylor expanded in x around 0 98.2%

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

   7. Taylor expanded in x around 0 98.2%

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

      1. *-commutative 98.2%

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

   9. Simplified 98.2%

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

   if 0.40000000000000002 < (fabs.f64 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 99.9%

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

   5. Taylor expanded in x around inf 75.9%

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

      1. *-commutative 75.9%

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

      2. unpow2 75.9%

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

      3. sqr-abs 75.9%

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

      4. unpow3 75.9%

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

      5. unpow3 75.9%

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

      6. fabs-mul 75.9%

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

      7. unpow2 75.9%

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

      8. fabs-mul 75.9%

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

      9. unpow2 75.9%

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

      10. unpow3 75.9%

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

      11. metadata-eval 75.9%

          \[\leadsto \left|0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|{x}^{\color{blue}{\left(2 \cdot 1.5\right)}}\right|\right)\right| \]

      12. pow-sqr 0.0%

          \[\leadsto \left|0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|\color{blue}{{x}^{1.5} \cdot {x}^{1.5}}\right|\right)\right| \]

      13. fabs-sqr 0.0%

          \[\leadsto \left|0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left({x}^{1.5} \cdot {x}^{1.5}\right)}\right)\right| \]

      14. pow-sqr 75.9%

          \[\leadsto \left|0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \color{blue}{{x}^{\left(2 \cdot 1.5\right)}}\right)\right| \]

      15. metadata-eval 75.9%

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

   7. Simplified 75.9%

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

      1. pow1 75.9%

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

      2. inv-pow 75.9%

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

      3. sqrt-pow1 75.9%

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

      4. metadata-eval 75.9%

         \[\leadsto \left|0.6666666666666666 \cdot {\left({\pi}^{\color{blue}{-0.5}} \cdot {x}^{3}\right)}^{1}\right| \]

   9. Applied egg-rr 75.9%

      \[\leadsto \left|0.6666666666666666 \cdot \color{blue}{{\left({\pi}^{-0.5} \cdot {x}^{3}\right)}^{1}}\right| \]
   10. Step-by-step derivation

       1. unpow1 75.9%

          \[\leadsto \left|0.6666666666666666 \cdot \color{blue}{\left({\pi}^{-0.5} \cdot {x}^{3}\right)}\right| \]

       2. *-commutative 75.9%

          \[\leadsto \left|0.6666666666666666 \cdot \color{blue}{\left({x}^{3} \cdot {\pi}^{-0.5}\right)}\right| \]

   11. Simplified 75.9%

       \[\leadsto \left|0.6666666666666666 \cdot \color{blue}{\left({x}^{3} \cdot {\pi}^{-0.5}\right)}\right| \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \left|\color{blue}{\sqrt{0.6666666666666666 \cdot \left({x}^{3} \cdot {\pi}^{-0.5}\right)} \cdot \sqrt{0.6666666666666666 \cdot \left({x}^{3} \cdot {\pi}^{-0.5}\right)}}\right| \]

       2. sqrt-unprod 85.2%

          \[\leadsto \left|\color{blue}{\sqrt{\left(0.6666666666666666 \cdot \left({x}^{3} \cdot {\pi}^{-0.5}\right)\right) \cdot \left(0.6666666666666666 \cdot \left({x}^{3} \cdot {\pi}^{-0.5}\right)\right)}}\right| \]

       3. *-commutative 85.2%

          \[\leadsto \left|\sqrt{\color{blue}{\left(\left({x}^{3} \cdot {\pi}^{-0.5}\right) \cdot 0.6666666666666666\right)} \cdot \left(0.6666666666666666 \cdot \left({x}^{3} \cdot {\pi}^{-0.5}\right)\right)}\right| \]

       4. *-commutative 85.2%

          \[\leadsto \left|\sqrt{\left(\left({x}^{3} \cdot {\pi}^{-0.5}\right) \cdot 0.6666666666666666\right) \cdot \color{blue}{\left(\left({x}^{3} \cdot {\pi}^{-0.5}\right) \cdot 0.6666666666666666\right)}}\right| \]

       5. swap-sqr 85.2%

          \[\leadsto \left|\sqrt{\color{blue}{\left(\left({x}^{3} \cdot {\pi}^{-0.5}\right) \cdot \left({x}^{3} \cdot {\pi}^{-0.5}\right)\right) \cdot \left(0.6666666666666666 \cdot 0.6666666666666666\right)}}\right| \]

       6. swap-sqr 85.2%

          \[\leadsto \left|\sqrt{\color{blue}{\left(\left({x}^{3} \cdot {x}^{3}\right) \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)\right)} \cdot \left(0.6666666666666666 \cdot 0.6666666666666666\right)}\right| \]

       7. pow-prod-up 85.2%

          \[\leadsto \left|\sqrt{\left(\color{blue}{{x}^{\left(3 + 3\right)}} \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)\right) \cdot \left(0.6666666666666666 \cdot 0.6666666666666666\right)}\right| \]

       8. metadata-eval 85.2%

          \[\leadsto \left|\sqrt{\left({x}^{\color{blue}{6}} \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)\right) \cdot \left(0.6666666666666666 \cdot 0.6666666666666666\right)}\right| \]

       9. pow-prod-up 85.2%

          \[\leadsto \left|\sqrt{\left({x}^{6} \cdot \color{blue}{{\pi}^{\left(-0.5 + -0.5\right)}}\right) \cdot \left(0.6666666666666666 \cdot 0.6666666666666666\right)}\right| \]

       10. metadata-eval 85.2%

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

       11. metadata-eval 85.2%

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

   13. Applied egg-rr 85.2%

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

       1. associate-*l* 85.2%

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

       2. unpow-1 85.2%

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

       3. associate-*l/ 85.2%

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

       4. metadata-eval 85.2%

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

   15. Simplified 85.2%

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

   16. Taylor expanded in x around 0 85.2%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot 2\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{0.4444444444444444 \cdot \frac{{x}^{6}}{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.3% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

   1. *-un-lft-identity 99.4%

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

   2. add-sqr-sqrt 31.3%

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

   3. fabs-sqr 31.3%

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

   4. add-sqr-sqrt 99.4%

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

6. Applied egg-rr 99.4%

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

   1. *-lft-identity 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in x around inf 98.7%

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

10. Taylor expanded in x around 0 98.2%

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

11. Final simplification 98.2%

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

Alternative 8: 34.0% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 93.8%

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

6. 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. expm1-log1p-u 93.1%

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

   2. expm1-undefine 34.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}\right|\right)} - 1} \]

   3. add-sqr-sqrt 2.4%

      \[\leadsto e^{\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left|\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}\right|\right)} - 1 \]

   4. fabs-sqr 2.4%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left|\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}\right|\right)} - 1 \]

   5. add-sqr-sqrt 4.3%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{x} \cdot \left|\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}\right|\right)} - 1 \]

   6. add-sqr-sqrt 4.3%

      \[\leadsto e^{\mathsf{log1p}\left(x \cdot \left|\color{blue}{\sqrt{\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}} \cdot \sqrt{\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}}}\right|\right)} - 1 \]

   7. fabs-sqr 4.3%

      \[\leadsto e^{\mathsf{log1p}\left(x \cdot \color{blue}{\left(\sqrt{\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}} \cdot \sqrt{\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}}\right)}\right)} - 1 \]

   8. add-sqr-sqrt 4.3%

      \[\leadsto e^{\mathsf{log1p}\left(x \cdot \color{blue}{\frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}}}\right)} - 1 \]

   9. fma-define 4.3%

      \[\leadsto e^{\mathsf{log1p}\left(x \cdot \frac{\color{blue}{\mathsf{fma}\left(0.2, {x}^{4}, 2\right)}}{\sqrt{\pi}}\right)} - 1 \]

8. Applied egg-rr 4.3%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 2\right)}{\sqrt{\pi}}\right)} - 1} \]
9. Step-by-step derivation

   1. log1p-undefine 4.3%

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

   2. rem-exp-log 4.3%

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

   3. +-commutative 4.3%

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

   4. associate--l+ 32.3%

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

   5. metadata-eval 32.3%

      \[\leadsto x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 2\right)}{\sqrt{\pi}} + \color{blue}{0} \]

   6. metadata-eval 32.3%

      \[\leadsto x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 2\right)}{\sqrt{\pi}} + \color{blue}{\left(-0\right)} \]

   7. sub-neg 32.3%

      \[\leadsto \color{blue}{x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 2\right)}{\sqrt{\pi}} - 0} \]

   8. --rgt-identity 32.3%

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

   9. associate-*r/ 32.1%

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

   10. associate-*l/ 32.1%

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

10. Simplified 32.1%

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

11. Final simplification 32.1%

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

Alternative 9: 34.0% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 93.8%

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

6. 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. *-commutative 93.1%

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

   2. add-sqr-sqrt 91.9%

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

   3. fabs-sqr 91.9%

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

   4. add-sqr-sqrt 93.1%

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

   5. add-sqr-sqrt 30.7%

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

   6. fabs-sqr 30.7%

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

   7. add-sqr-sqrt 32.3%

      \[\leadsto \frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}} \cdot \color{blue}{x} \]

   8. associate-*l/ 32.1%

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

   9. fma-define 32.1%

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

8. Applied egg-rr 32.1%

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

9. Final simplification 32.1%

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

Alternative 10: 32.2% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \left|\sqrt{0.4444444444444444 \cdot \frac{{x}^{6}}{\pi}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs (sqrt (* 0.4444444444444444 (/ (pow x 6.0) PI)))))

C

double code(double x) {
	return fabs(sqrt((0.4444444444444444 * (pow(x, 6.0) / ((double) M_PI)))));
}

Java

public static double code(double x) {
	return Math.abs(Math.sqrt((0.4444444444444444 * (Math.pow(x, 6.0) / Math.PI))));
}

Python

def code(x):
	return math.fabs(math.sqrt((0.4444444444444444 * (math.pow(x, 6.0) / math.pi))))

Julia

function code(x)
	return abs(sqrt(Float64(0.4444444444444444 * Float64((x ^ 6.0) / pi))))
end

MATLAB

function tmp = code(x)
	tmp = abs(sqrt((0.4444444444444444 * ((x ^ 6.0) / pi))));
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around inf 30.5%

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

   1. *-commutative 30.5%

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

   2. unpow2 30.5%

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

   3. sqr-abs 30.5%

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

   4. unpow3 30.5%

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

   5. unpow3 30.5%

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

   6. fabs-mul 30.5%

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

   7. unpow2 30.5%

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

   8. fabs-mul 30.5%

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

   9. unpow2 30.5%

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

   10. unpow3 30.5%

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

   11. metadata-eval 30.5%

       \[\leadsto \left|0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|{x}^{\color{blue}{\left(2 \cdot 1.5\right)}}\right|\right)\right| \]

   12. pow-sqr 1.9%

       \[\leadsto \left|0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|\color{blue}{{x}^{1.5} \cdot {x}^{1.5}}\right|\right)\right| \]

   13. fabs-sqr 1.9%

       \[\leadsto \left|0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left({x}^{1.5} \cdot {x}^{1.5}\right)}\right)\right| \]

   14. pow-sqr 30.5%

       \[\leadsto \left|0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \color{blue}{{x}^{\left(2 \cdot 1.5\right)}}\right)\right| \]

   15. metadata-eval 30.5%

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

7. Simplified 30.5%

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

   1. pow1 30.5%

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

   2. inv-pow 30.5%

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

   3. sqrt-pow1 30.5%

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

   4. metadata-eval 30.5%

      \[\leadsto \left|0.6666666666666666 \cdot {\left({\pi}^{\color{blue}{-0.5}} \cdot {x}^{3}\right)}^{1}\right| \]

9. Applied egg-rr 30.5%

   \[\leadsto \left|0.6666666666666666 \cdot \color{blue}{{\left({\pi}^{-0.5} \cdot {x}^{3}\right)}^{1}}\right| \]
10. Step-by-step derivation

    1. unpow1 30.5%

       \[\leadsto \left|0.6666666666666666 \cdot \color{blue}{\left({\pi}^{-0.5} \cdot {x}^{3}\right)}\right| \]

    2. *-commutative 30.5%

       \[\leadsto \left|0.6666666666666666 \cdot \color{blue}{\left({x}^{3} \cdot {\pi}^{-0.5}\right)}\right| \]

11. Simplified 30.5%

    \[\leadsto \left|0.6666666666666666 \cdot \color{blue}{\left({x}^{3} \cdot {\pi}^{-0.5}\right)}\right| \]
12. Step-by-step derivation

    1. add-sqr-sqrt 3.4%

       \[\leadsto \left|\color{blue}{\sqrt{0.6666666666666666 \cdot \left({x}^{3} \cdot {\pi}^{-0.5}\right)} \cdot \sqrt{0.6666666666666666 \cdot \left({x}^{3} \cdot {\pi}^{-0.5}\right)}}\right| \]

    2. sqrt-unprod 33.6%

       \[\leadsto \left|\color{blue}{\sqrt{\left(0.6666666666666666 \cdot \left({x}^{3} \cdot {\pi}^{-0.5}\right)\right) \cdot \left(0.6666666666666666 \cdot \left({x}^{3} \cdot {\pi}^{-0.5}\right)\right)}}\right| \]

    3. *-commutative 33.6%

       \[\leadsto \left|\sqrt{\color{blue}{\left(\left({x}^{3} \cdot {\pi}^{-0.5}\right) \cdot 0.6666666666666666\right)} \cdot \left(0.6666666666666666 \cdot \left({x}^{3} \cdot {\pi}^{-0.5}\right)\right)}\right| \]

    4. *-commutative 33.6%

       \[\leadsto \left|\sqrt{\left(\left({x}^{3} \cdot {\pi}^{-0.5}\right) \cdot 0.6666666666666666\right) \cdot \color{blue}{\left(\left({x}^{3} \cdot {\pi}^{-0.5}\right) \cdot 0.6666666666666666\right)}}\right| \]

    5. swap-sqr 33.6%

       \[\leadsto \left|\sqrt{\color{blue}{\left(\left({x}^{3} \cdot {\pi}^{-0.5}\right) \cdot \left({x}^{3} \cdot {\pi}^{-0.5}\right)\right) \cdot \left(0.6666666666666666 \cdot 0.6666666666666666\right)}}\right| \]

    6. swap-sqr 33.6%

       \[\leadsto \left|\sqrt{\color{blue}{\left(\left({x}^{3} \cdot {x}^{3}\right) \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)\right)} \cdot \left(0.6666666666666666 \cdot 0.6666666666666666\right)}\right| \]

    7. pow-prod-up 33.6%

       \[\leadsto \left|\sqrt{\left(\color{blue}{{x}^{\left(3 + 3\right)}} \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)\right) \cdot \left(0.6666666666666666 \cdot 0.6666666666666666\right)}\right| \]

    8. metadata-eval 33.6%

       \[\leadsto \left|\sqrt{\left({x}^{\color{blue}{6}} \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)\right) \cdot \left(0.6666666666666666 \cdot 0.6666666666666666\right)}\right| \]

    9. pow-prod-up 33.6%

       \[\leadsto \left|\sqrt{\left({x}^{6} \cdot \color{blue}{{\pi}^{\left(-0.5 + -0.5\right)}}\right) \cdot \left(0.6666666666666666 \cdot 0.6666666666666666\right)}\right| \]

    10. metadata-eval 33.6%

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

    11. metadata-eval 33.6%

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

13. Applied egg-rr 33.6%

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

    1. associate-*l* 33.6%

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

    2. unpow-1 33.6%

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

    3. associate-*l/ 33.6%

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

    4. metadata-eval 33.6%

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

15. Simplified 33.6%

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

16. Taylor expanded in x around 0 33.6%

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

17. Final simplification 33.6%

    \[\leadsto \left|\sqrt{0.4444444444444444 \cdot \frac{{x}^{6}}{\pi}}\right| \]
18. Add Preprocessing

Alternative 11: 32.2% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \left|0.6666666666666666 \cdot \sqrt{\frac{{x}^{6}}{\pi}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs (* 0.6666666666666666 (sqrt (/ (pow x 6.0) PI)))))

C

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

Java

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

Python

def code(x):
	return math.fabs((0.6666666666666666 * math.sqrt((math.pow(x, 6.0) / math.pi))))

Julia

function code(x)
	return abs(Float64(0.6666666666666666 * sqrt(Float64((x ^ 6.0) / pi))))
end

MATLAB

function tmp = code(x)
	tmp = abs((0.6666666666666666 * sqrt(((x ^ 6.0) / pi))));
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around inf 30.5%

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

   1. *-commutative 30.5%

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

   2. unpow2 30.5%

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

   3. sqr-abs 30.5%

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

   4. unpow3 30.5%

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

   5. unpow3 30.5%

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

   6. fabs-mul 30.5%

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

   7. unpow2 30.5%

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

   8. fabs-mul 30.5%

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

   9. unpow2 30.5%

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

   10. unpow3 30.5%

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

   11. metadata-eval 30.5%

       \[\leadsto \left|0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|{x}^{\color{blue}{\left(2 \cdot 1.5\right)}}\right|\right)\right| \]

   12. pow-sqr 1.9%

       \[\leadsto \left|0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|\color{blue}{{x}^{1.5} \cdot {x}^{1.5}}\right|\right)\right| \]

   13. fabs-sqr 1.9%

       \[\leadsto \left|0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left({x}^{1.5} \cdot {x}^{1.5}\right)}\right)\right| \]

   14. pow-sqr 30.5%

       \[\leadsto \left|0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \color{blue}{{x}^{\left(2 \cdot 1.5\right)}}\right)\right| \]

   15. metadata-eval 30.5%

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

7. Simplified 30.5%

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

   1. pow1 30.5%

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

   2. inv-pow 30.5%

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

   3. sqrt-pow1 30.5%

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

   4. metadata-eval 30.5%

      \[\leadsto \left|0.6666666666666666 \cdot {\left({\pi}^{\color{blue}{-0.5}} \cdot {x}^{3}\right)}^{1}\right| \]

9. Applied egg-rr 30.5%

   \[\leadsto \left|0.6666666666666666 \cdot \color{blue}{{\left({\pi}^{-0.5} \cdot {x}^{3}\right)}^{1}}\right| \]
10. Step-by-step derivation

    1. unpow1 30.5%

       \[\leadsto \left|0.6666666666666666 \cdot \color{blue}{\left({\pi}^{-0.5} \cdot {x}^{3}\right)}\right| \]

    2. *-commutative 30.5%

       \[\leadsto \left|0.6666666666666666 \cdot \color{blue}{\left({x}^{3} \cdot {\pi}^{-0.5}\right)}\right| \]

11. Simplified 30.5%

    \[\leadsto \left|0.6666666666666666 \cdot \color{blue}{\left({x}^{3} \cdot {\pi}^{-0.5}\right)}\right| \]
12. Step-by-step derivation

    1. add-sqr-sqrt 3.4%

       \[\leadsto \left|0.6666666666666666 \cdot \color{blue}{\left(\sqrt{{x}^{3} \cdot {\pi}^{-0.5}} \cdot \sqrt{{x}^{3} \cdot {\pi}^{-0.5}}\right)}\right| \]

    2. sqrt-unprod 33.6%

       \[\leadsto \left|0.6666666666666666 \cdot \color{blue}{\sqrt{\left({x}^{3} \cdot {\pi}^{-0.5}\right) \cdot \left({x}^{3} \cdot {\pi}^{-0.5}\right)}}\right| \]

    3. swap-sqr 33.6%

       \[\leadsto \left|0.6666666666666666 \cdot \sqrt{\color{blue}{\left({x}^{3} \cdot {x}^{3}\right) \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)}}\right| \]

    4. pow-prod-up 33.6%

       \[\leadsto \left|0.6666666666666666 \cdot \sqrt{\color{blue}{{x}^{\left(3 + 3\right)}} \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)}\right| \]

    5. metadata-eval 33.6%

       \[\leadsto \left|0.6666666666666666 \cdot \sqrt{{x}^{\color{blue}{6}} \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)}\right| \]

    6. pow-prod-up 33.6%

       \[\leadsto \left|0.6666666666666666 \cdot \sqrt{{x}^{6} \cdot \color{blue}{{\pi}^{\left(-0.5 + -0.5\right)}}}\right| \]

    7. metadata-eval 33.6%

       \[\leadsto \left|0.6666666666666666 \cdot \sqrt{{x}^{6} \cdot {\pi}^{\color{blue}{-1}}}\right| \]

13. Applied egg-rr 33.6%

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

    1. metadata-eval 33.6%

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

    2. pow-sqr 33.6%

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

    3. cube-prod 33.6%

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

    4. unpow2 33.6%

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

    5. unpow-1 33.6%

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

    6. associate-*r/ 33.6%

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

    7. *-rgt-identity 33.6%

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

    8. unpow2 33.6%

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

    9. cube-prod 33.6%

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

    10. pow-sqr 33.6%

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

    11. metadata-eval 33.6%

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

15. Simplified 33.6%

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

16. Final simplification 33.6%

    \[\leadsto \left|0.6666666666666666 \cdot \sqrt{\frac{{x}^{6}}{\pi}}\right| \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024074 
(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)))))))