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

Percentage Accurate: 99.8% → 99.9%

Time: 11.5s

Alternatives: 8

Speedup: 3.5×

• 29.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

Alternative 2: 99.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot {x}^{5}\right) + 0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot {x}^{3}\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt PI))
   (+
    (+ (+ (* x 2.0) (* 0.6666666666666666 (pow x 3.0))) (* 0.2 (pow x 5.0)))
    (* 0.047619047619047616 (* (* x x) (* (* x x) (pow x 3.0))))))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((x * 2.0) + (0.6666666666666666 * pow(x, 3.0))) + (0.2 * pow(x, 5.0))) + (0.047619047619047616 * ((x * x) * ((x * x) * pow(x, 3.0)))))));
}

Java

public static double code(double x) {
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((x * 2.0) + (0.6666666666666666 * Math.pow(x, 3.0))) + (0.2 * Math.pow(x, 5.0))) + (0.047619047619047616 * ((x * x) * ((x * x) * Math.pow(x, 3.0)))))));
}

Python

def code(x):
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((x * 2.0) + (0.6666666666666666 * math.pow(x, 3.0))) + (0.2 * math.pow(x, 5.0))) + (0.047619047619047616 * ((x * x) * ((x * x) * math.pow(x, 3.0)))))))

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(x * 2.0) + Float64(0.6666666666666666 * (x ^ 3.0))) + Float64(0.2 * (x ^ 5.0))) + Float64(0.047619047619047616 * Float64(Float64(x * x) * Float64(Float64(x * x) * (x ^ 3.0)))))))
end

MATLAB

function tmp = code(x)
	tmp = abs(((1.0 / sqrt(pi)) * ((((x * 2.0) + (0.6666666666666666 * (x ^ 3.0))) + (0.2 * (x ^ 5.0))) + (0.047619047619047616 * ((x * x) * ((x * x) * (x ^ 3.0)))))));
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(x * 2.0), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[Power[x, 3.0], $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%

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

   1. fma-undefine 99.8%

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

   2. add-sqr-sqrt 32.5%

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

   3. fabs-sqr 32.5%

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

   4. add-sqr-sqrt 99.5%

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

   5. add-sqr-sqrt 32.8%

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

   6. fabs-sqr 32.8%

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

   7. add-sqr-sqrt 75.6%

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

   8. cube-mult 75.6%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot \color{blue}{{x}^{3}}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

6. Applied egg-rr 75.6%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\color{blue}{\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)} + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

7. Taylor expanded in x around 0 75.6%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\color{blue}{\left({x}^{2} \cdot \left|x\right|\right)} \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]
8. Step-by-step derivation

   1. rem-square-sqrt 32.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left({x}^{2} \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   2. fabs-sqr 32.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left({x}^{2} \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   3. rem-square-sqrt 71.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left({x}^{2} \cdot \color{blue}{x}\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   4. pow-plus 71.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\color{blue}{{x}^{\left(2 + 1\right)}} \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   5. metadata-eval 71.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left({x}^{\color{blue}{3}} \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

9. Simplified 71.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\color{blue}{{x}^{3}} \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

10. Taylor expanded in x around 0 71.8%

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + \color{blue}{0.2 \cdot {x}^{5}}\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

11. Taylor expanded in x around 0 71.8%

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot {x}^{5}\right) + 0.047619047619047616 \cdot \left(\left(\color{blue}{\left({x}^{2} \cdot \left|x\right|\right)} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]
12. Step-by-step derivation

    1. rem-square-sqrt 32.8%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left({x}^{2} \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

    2. fabs-sqr 32.8%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left({x}^{2} \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

    3. rem-square-sqrt 71.8%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left({x}^{2} \cdot \color{blue}{x}\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

    4. pow-plus 71.8%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\color{blue}{{x}^{\left(2 + 1\right)}} \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

    5. metadata-eval 71.8%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left({x}^{\color{blue}{3}} \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

13. Simplified 99.8%

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot {x}^{5}\right) + 0.047619047619047616 \cdot \left(\left(\color{blue}{{x}^{3}} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

14. Final simplification 99.8%

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot {x}^{5}\right) + 0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot {x}^{3}\right)\right)\right)\right| \]
15. Add Preprocessing

Alternative 3: 99.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Taylor expanded in x around 0 98.9%

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

   1. rem-square-sqrt 32.4%

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

   2. fabs-sqr 32.4%

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

   3. rem-square-sqrt 98.9%

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

   4. +-commutative 98.9%

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

   5. fma-define 98.9%

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

   6. rem-square-sqrt 32.5%

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

   7. fabs-sqr 32.5%

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

   8. rem-square-sqrt 98.9%

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

   9. rem-square-sqrt 32.5%

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

   10. fabs-sqr 32.5%

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

   11. rem-square-sqrt 98.9%

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

7. Simplified 98.9%

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

   1. div-inv 99.3%

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

   2. metadata-eval 99.3%

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

   3. sqrt-div 99.3%

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

   4. *-commutative 99.3%

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

   5. *-commutative 99.3%

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

   6. associate-*l* 99.4%

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

   7. *-commutative 99.4%

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

   8. sqrt-div 99.4%

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

   9. metadata-eval 99.4%

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

   10. div-inv 99.4%

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

9. Applied egg-rr 99.4%

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

10. Final simplification 99.4%

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

Alternative 4: 99.0% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(x \cdot 2 + 0.2 \cdot t\_0\right) + 0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot t\_0\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) (* (fabs x) (* x x)))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+ (+ (* x 2.0) (* 0.2 t_0)) (* 0.047619047619047616 (* (* x x) t_0)))))))

C

double code(double x) {
	double t_0 = (x * x) * (fabs(x) * (x * x));
	return fabs(((1.0 / sqrt(((double) M_PI))) * (((x * 2.0) + (0.2 * t_0)) + (0.047619047619047616 * ((x * x) * t_0)))));
}

Java

public static double code(double x) {
	double t_0 = (x * x) * (Math.abs(x) * (x * x));
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * (((x * 2.0) + (0.2 * t_0)) + (0.047619047619047616 * ((x * x) * t_0)))));
}

Python

def code(x):
	t_0 = (x * x) * (math.fabs(x) * (x * x))
	return math.fabs(((1.0 / math.sqrt(math.pi)) * (((x * 2.0) + (0.2 * t_0)) + (0.047619047619047616 * ((x * x) * t_0)))))

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * Float64(abs(x) * Float64(x * x)))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(x * 2.0) + Float64(0.2 * t_0)) + Float64(0.047619047619047616 * Float64(Float64(x * x) * t_0)))))
end

MATLAB

function tmp = code(x)
	t_0 = (x * x) * (abs(x) * (x * x));
	tmp = abs(((1.0 / sqrt(pi)) * (((x * 2.0) + (0.2 * t_0)) + (0.047619047619047616 * ((x * x) * t_0)))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * 2.0), $MachinePrecision] + N[(0.2 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[(N[(x * x), $MachinePrecision] * t$95$0), $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%

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

5. Taylor expanded in x around 0 99.3%

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

   1. rem-square-sqrt 32.5%

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

   2. fabs-sqr 32.5%

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

   3. rem-square-sqrt 99.3%

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

7. Simplified 99.3%

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

8. Final simplification 99.3%

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

Alternative 5: 34.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.01:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.01)
   (* x (/ 2.0 (sqrt PI)))
   (* 0.047619047619047616 (* (pow PI -0.5) (pow x 7.0)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.01) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = 0.047619047619047616 * (pow(((double) M_PI), -0.5) * pow(x, 7.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.01) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = 0.047619047619047616 * (Math.pow(Math.PI, -0.5) * Math.pow(x, 7.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.01:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = 0.047619047619047616 * (math.pow(math.pi, -0.5) * math.pow(x, 7.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.01)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(0.047619047619047616 * Float64((pi ^ -0.5) * (x ^ 7.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.01)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = 0.047619047619047616 * ((pi ^ -0.5) * (x ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.01], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0100000000000000002

   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 99.2%

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

   5. Taylor expanded in x around 0 98.7%

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

      1. rem-square-sqrt 51.2%

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

      2. fabs-sqr 51.2%

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

      3. rem-square-sqrt 98.7%

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

      4. +-commutative 98.7%

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

      5. fma-define 98.7%

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

      6. rem-square-sqrt 51.3%

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

      7. fabs-sqr 51.3%

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

      8. rem-square-sqrt 98.7%

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

      9. rem-square-sqrt 51.3%

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

      10. fabs-sqr 51.3%

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

      11. rem-square-sqrt 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in x around 0 99.4%

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

      1. associate-*r* 99.4%

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

      2. *-commutative 99.4%

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

      3. unpow-1 99.4%

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

      4. metadata-eval 99.4%

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

      5. pow-sqr 99.4%

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

      6. rem-sqrt-square 99.4%

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

      7. rem-square-sqrt 99.4%

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

      8. fabs-sqr 99.4%

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

      9. rem-square-sqrt 99.4%

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

   10. Simplified 99.4%

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

       1. add-sqr-sqrt 51.3%

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

       2. fabs-sqr 51.3%

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

       3. add-sqr-sqrt 53.6%

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

       4. *-commutative 53.6%

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

       5. associate-*l* 53.6%

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

       6. metadata-eval 53.6%

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

       7. pow-flip 53.6%

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

       8. pow1/2 53.6%

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

       9. div-inv 53.2%

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

   12. Applied egg-rr 53.2%

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

       1. *-commutative 53.2%

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

       2. associate-*l/ 53.2%

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

       3. associate-/l* 53.6%

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

   14. Simplified 53.6%

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

   if 0.0100000000000000002 < (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 0 99.2%

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

      1. rem-square-sqrt 0.0%

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

      2. fabs-sqr 0.0%

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

      3. rem-square-sqrt 99.2%

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

      4. +-commutative 99.2%

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

      5. fma-define 99.2%

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

      6. rem-square-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. rem-square-sqrt 99.2%

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

      9. rem-square-sqrt 0.0%

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

      10. fabs-sqr 0.0%

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

      11. rem-square-sqrt 99.2%

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

   7. Simplified 99.2%

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

   8. Taylor expanded in x around inf 98.1%

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

      1. unpow-1 98.1%

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

      2. metadata-eval 98.1%

         \[\leadsto \left|0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{{\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right)\right| \]

      3. pow-sqr 98.1%

         \[\leadsto \left|0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}}\right)\right| \]

      4. rem-sqrt-square 98.1%

         \[\leadsto \left|0.047619047619047616 \cdot \left({x}^{7} \cdot \color{blue}{\left|{\pi}^{-0.5}\right|}\right)\right| \]

      5. rem-square-sqrt 98.1%

         \[\leadsto \left|0.047619047619047616 \cdot \left({x}^{7} \cdot \left|\color{blue}{\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}}\right|\right)\right| \]

      6. fabs-sqr 98.1%

         \[\leadsto \left|0.047619047619047616 \cdot \left({x}^{7} \cdot \color{blue}{\left(\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}\right)}\right)\right| \]

      7. rem-square-sqrt 98.1%

         \[\leadsto \left|0.047619047619047616 \cdot \left({x}^{7} \cdot \color{blue}{{\pi}^{-0.5}}\right)\right| \]

   10. Simplified 98.1%

       \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)}\right| \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \left|\color{blue}{\sqrt{0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)} \cdot \sqrt{0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)}}\right| \]

       2. fabs-sqr 0.0%

          \[\leadsto \color{blue}{\sqrt{0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)} \cdot \sqrt{0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)}} \]

       3. add-sqr-sqrt 0.1%

          \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)} \]

       4. *-commutative 0.1%

          \[\leadsto \color{blue}{\left({x}^{7} \cdot {\pi}^{-0.5}\right) \cdot 0.047619047619047616} \]

       5. *-commutative 0.1%

          \[\leadsto \color{blue}{\left({\pi}^{-0.5} \cdot {x}^{7}\right)} \cdot 0.047619047619047616 \]

   12. Applied egg-rr 0.1%

       \[\leadsto \color{blue}{\left({\pi}^{-0.5} \cdot {x}^{7}\right) \cdot 0.047619047619047616} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.01:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.01:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.0022675736961451248 \cdot \frac{{x}^{14}}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.01)
   (* x (/ 2.0 (sqrt PI)))
   (sqrt (* 0.0022675736961451248 (/ (pow x 14.0) PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.01) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = sqrt((0.0022675736961451248 * (pow(x, 14.0) / ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.01) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.sqrt((0.0022675736961451248 * (Math.pow(x, 14.0) / Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.01:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.sqrt((0.0022675736961451248 * (math.pow(x, 14.0) / math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.01)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = sqrt(Float64(0.0022675736961451248 * Float64((x ^ 14.0) / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.01)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = sqrt((0.0022675736961451248 * ((x ^ 14.0) / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.01], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.0022675736961451248 * N[(N[Power[x, 14.0], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0100000000000000002

   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 99.2%

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

   5. Taylor expanded in x around 0 98.7%

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

      1. rem-square-sqrt 51.2%

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

      2. fabs-sqr 51.2%

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

      3. rem-square-sqrt 98.7%

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

      4. +-commutative 98.7%

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

      5. fma-define 98.7%

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

      6. rem-square-sqrt 51.3%

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

      7. fabs-sqr 51.3%

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

      8. rem-square-sqrt 98.7%

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

      9. rem-square-sqrt 51.3%

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

      10. fabs-sqr 51.3%

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

      11. rem-square-sqrt 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in x around 0 99.4%

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

      1. associate-*r* 99.4%

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

      2. *-commutative 99.4%

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

      3. unpow-1 99.4%

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

      4. metadata-eval 99.4%

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

      5. pow-sqr 99.4%

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

      6. rem-sqrt-square 99.4%

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

      7. rem-square-sqrt 99.4%

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

      8. fabs-sqr 99.4%

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

      9. rem-square-sqrt 99.4%

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

   10. Simplified 99.4%

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

       1. add-sqr-sqrt 51.3%

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

       2. fabs-sqr 51.3%

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

       3. add-sqr-sqrt 53.6%

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

       4. *-commutative 53.6%

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

       5. associate-*l* 53.6%

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

       6. metadata-eval 53.6%

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

       7. pow-flip 53.6%

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

       8. pow1/2 53.6%

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

       9. div-inv 53.2%

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

   12. Applied egg-rr 53.2%

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

       1. *-commutative 53.2%

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

       2. associate-*l/ 53.2%

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

       3. associate-/l* 53.6%

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

   14. Simplified 53.6%

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

   if 0.0100000000000000002 < (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 0 99.2%

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

      1. rem-square-sqrt 0.0%

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

      2. fabs-sqr 0.0%

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

      3. rem-square-sqrt 99.2%

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

      4. +-commutative 99.2%

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

      5. fma-define 99.2%

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

      6. rem-square-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. rem-square-sqrt 99.2%

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

      9. rem-square-sqrt 0.0%

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

      10. fabs-sqr 0.0%

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

      11. rem-square-sqrt 99.2%

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

   7. Simplified 99.2%

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

   8. Taylor expanded in x around inf 98.1%

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

      1. unpow-1 98.1%

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

      2. metadata-eval 98.1%

         \[\leadsto \left|0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{{\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}}\right)\right| \]

      3. pow-sqr 98.1%

         \[\leadsto \left|0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}}\right)\right| \]

      4. rem-sqrt-square 98.1%

         \[\leadsto \left|0.047619047619047616 \cdot \left({x}^{7} \cdot \color{blue}{\left|{\pi}^{-0.5}\right|}\right)\right| \]

      5. rem-square-sqrt 98.1%

         \[\leadsto \left|0.047619047619047616 \cdot \left({x}^{7} \cdot \left|\color{blue}{\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}}\right|\right)\right| \]

      6. fabs-sqr 98.1%

         \[\leadsto \left|0.047619047619047616 \cdot \left({x}^{7} \cdot \color{blue}{\left(\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}\right)}\right)\right| \]

      7. rem-square-sqrt 98.1%

         \[\leadsto \left|0.047619047619047616 \cdot \left({x}^{7} \cdot \color{blue}{{\pi}^{-0.5}}\right)\right| \]

   10. Simplified 98.1%

       \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)}\right| \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \left|\color{blue}{\sqrt{0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)} \cdot \sqrt{0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)}}\right| \]

       2. fabs-sqr 0.0%

          \[\leadsto \color{blue}{\sqrt{0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)} \cdot \sqrt{0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)}} \]

       3. add-sqr-sqrt 0.1%

          \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)} \]

       4. associate-*r* 0.1%

          \[\leadsto \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot {\pi}^{-0.5}} \]

       5. *-commutative 0.1%

          \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)} \]

   12. Applied egg-rr 0.1%

       \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)} \]
   13. Step-by-step derivation

       1. pow1 0.1%

          \[\leadsto \color{blue}{{\left({\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right)}^{1}} \]

       2. *-commutative 0.1%

          \[\leadsto {\color{blue}{\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot {\pi}^{-0.5}\right)}}^{1} \]

       3. associate-*l* 0.1%

          \[\leadsto {\color{blue}{\left(0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)\right)}}^{1} \]

       4. *-commutative 0.1%

          \[\leadsto {\left(0.047619047619047616 \cdot \color{blue}{\left({\pi}^{-0.5} \cdot {x}^{7}\right)}\right)}^{1} \]

   14. Applied egg-rr 0.1%

       \[\leadsto \color{blue}{{\left(0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)\right)}^{1}} \]
   15. Step-by-step derivation

       1. unpow1 0.1%

          \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)} \]

       2. rem-square-sqrt 0.0%

          \[\leadsto \color{blue}{\sqrt{0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)} \cdot \sqrt{0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)}} \]

       3. fabs-sqr 0.0%

          \[\leadsto \color{blue}{\left|\sqrt{0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)} \cdot \sqrt{0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)}\right|} \]

       4. rem-square-sqrt 98.1%

          \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)}\right| \]

       5. rem-sqrt-square 89.3%

          \[\leadsto \color{blue}{\sqrt{\left(0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)\right) \cdot \left(0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)\right)}} \]

       6. swap-sqr 89.3%

          \[\leadsto \sqrt{\color{blue}{\left(0.047619047619047616 \cdot 0.047619047619047616\right) \cdot \left(\left({\pi}^{-0.5} \cdot {x}^{7}\right) \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)\right)}} \]

       7. metadata-eval 89.3%

          \[\leadsto \sqrt{\color{blue}{0.0022675736961451248} \cdot \left(\left({\pi}^{-0.5} \cdot {x}^{7}\right) \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)\right)} \]

       8. swap-sqr 89.3%

          \[\leadsto \sqrt{0.0022675736961451248 \cdot \color{blue}{\left(\left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right) \cdot \left({x}^{7} \cdot {x}^{7}\right)\right)}} \]

       9. pow-sqr 89.3%

          \[\leadsto \sqrt{0.0022675736961451248 \cdot \left(\color{blue}{{\pi}^{\left(2 \cdot -0.5\right)}} \cdot \left({x}^{7} \cdot {x}^{7}\right)\right)} \]

       10. metadata-eval 89.3%

           \[\leadsto \sqrt{0.0022675736961451248 \cdot \left({\pi}^{\color{blue}{-1}} \cdot \left({x}^{7} \cdot {x}^{7}\right)\right)} \]

       11. unpow-1 89.3%

           \[\leadsto \sqrt{0.0022675736961451248 \cdot \left(\color{blue}{\frac{1}{\pi}} \cdot \left({x}^{7} \cdot {x}^{7}\right)\right)} \]

       12. associate-*l/ 89.3%

           \[\leadsto \sqrt{0.0022675736961451248 \cdot \color{blue}{\frac{1 \cdot \left({x}^{7} \cdot {x}^{7}\right)}{\pi}}} \]

       13. *-lft-identity 89.3%

           \[\leadsto \sqrt{0.0022675736961451248 \cdot \frac{\color{blue}{{x}^{7} \cdot {x}^{7}}}{\pi}} \]

       14. pow-sqr 89.3%

           \[\leadsto \sqrt{0.0022675736961451248 \cdot \frac{\color{blue}{{x}^{\left(2 \cdot 7\right)}}}{\pi}} \]

       15. metadata-eval 89.3%

           \[\leadsto \sqrt{0.0022675736961451248 \cdot \frac{{x}^{\color{blue}{14}}}{\pi}} \]

   16. Simplified 89.3%

       \[\leadsto \color{blue}{\sqrt{0.0022675736961451248 \cdot \frac{{x}^{14}}{\pi}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.5% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = abs(((x / sqrt(pi)) * ((0.047619047619047616 * (x ^ 6.0)) + 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] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 98.9%

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

   1. rem-square-sqrt 32.4%

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

   2. fabs-sqr 32.4%

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

   3. rem-square-sqrt 98.9%

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

   4. +-commutative 98.9%

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

   5. fma-define 98.9%

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

   6. rem-square-sqrt 32.5%

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

   7. fabs-sqr 32.5%

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

   8. rem-square-sqrt 98.9%

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

   9. rem-square-sqrt 32.5%

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

   10. fabs-sqr 32.5%

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

   11. rem-square-sqrt 98.9%

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

7. Simplified 98.9%

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

   1. associate-/l* 99.4%

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

   2. div-inv 99.4%

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

   3. metadata-eval 99.4%

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

   4. sqrt-div 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-*r* 99.3%

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

   7. sqrt-div 99.3%

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

   8. metadata-eval 99.3%

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

   9. un-div-inv 98.9%

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

9. Applied egg-rr 98.9%

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

10. Taylor expanded in x around inf 98.5%

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

11. Final simplification 98.5%

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

Alternative 8: 34.1% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 98.9%

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

   1. rem-square-sqrt 32.4%

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

   2. fabs-sqr 32.4%

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

   3. rem-square-sqrt 98.9%

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

   4. +-commutative 98.9%

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

   5. fma-define 98.9%

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

   6. rem-square-sqrt 32.5%

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

   7. fabs-sqr 32.5%

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

   8. rem-square-sqrt 98.9%

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

   9. rem-square-sqrt 32.5%

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

   10. fabs-sqr 32.5%

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

   11. rem-square-sqrt 98.9%

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

7. Simplified 98.9%

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

8. Taylor expanded in x around 0 65.2%

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

   1. associate-*r* 65.2%

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

   2. *-commutative 65.2%

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

   3. unpow-1 65.2%

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

   4. metadata-eval 65.2%

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

   5. pow-sqr 65.2%

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

   6. rem-sqrt-square 65.2%

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

   7. rem-square-sqrt 65.2%

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

   8. fabs-sqr 65.2%

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

   9. rem-square-sqrt 65.2%

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

10. Simplified 65.2%

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

    1. add-sqr-sqrt 32.5%

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

    2. fabs-sqr 32.5%

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

    3. add-sqr-sqrt 34.1%

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

    4. *-commutative 34.1%

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

    5. associate-*l* 34.1%

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

    6. metadata-eval 34.1%

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

    7. pow-flip 34.1%

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

    8. pow1/2 34.1%

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

    9. div-inv 33.8%

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

12. Applied egg-rr 33.8%

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

    1. *-commutative 33.8%

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

    2. associate-*l/ 33.8%

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

    3. associate-/l* 34.1%

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

14. Simplified 34.1%

    \[\leadsto \color{blue}{x \cdot \frac{2}{\sqrt{\pi}}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(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)))))))