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

Percentage Accurate: 99.8% → 99.8%

Time: 13.5s

Alternatives: 10

Speedup: N/A×

• 29.1% 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: 99.8% accurate, 2.3× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt PI))
   (+
    (+
     (fma 2.0 (fabs x) (* 0.6666666666666666 (pow x 3.0)))
     (* 0.2 (* (pow x 3.0) (* x x))))
    (* 0.047619047619047616 (* (* x x) (* (* x x) (* (fabs x) (* x x)))))))))

C

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

Julia

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[(N[Power[x, 3.0], $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around 0 99.8%

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

   1. *-commutative 99.8%

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

   2. unpow1 99.8%

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

   3. sqr-pow 32.7%

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

   4. fabs-sqr 32.7%

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

   5. sqr-pow 77.1%

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

   6. unpow1 77.1%

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

   7. unpow2 77.1%

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

   8. cube-mult 77.1%

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

6. Simplified 77.1%

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

7. Taylor expanded in x around 0 77.1%

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

   1. unpow1 77.1%

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

   2. sqr-pow 32.7%

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

   3. fabs-sqr 32.7%

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

   4. sqr-pow 70.4%

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

   5. unpow1 70.4%

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

   6. pow-plus 70.4%

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

   7. metadata-eval 70.4%

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

9. Simplified 70.4%

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

10. Final simplification 70.4%

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

Alternative 2: 99.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.6666666666666666 \cdot {x}^{3} + \left(0.2 \cdot {x}^{5} + 2 \cdot x\right)\right)}{\sqrt{\pi}}\right| \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (fabs
  (/
   (fma
    0.047619047619047616
    (pow x 7.0)
    (+ (* 0.6666666666666666 (pow x 3.0)) (+ (* 0.2 (pow x 5.0)) (* 2.0 x))))
   (sqrt PI))))

C

x = abs(x);
double code(double x) {
	return fabs((fma(0.047619047619047616, pow(x, 7.0), ((0.6666666666666666 * pow(x, 3.0)) + ((0.2 * pow(x, 5.0)) + (2.0 * x)))) / sqrt(((double) M_PI))));
}

Julia

x = abs(x)
function code(x)
	return abs(Float64(fma(0.047619047619047616, (x ^ 7.0), Float64(Float64(0.6666666666666666 * (x ^ 3.0)) + Float64(Float64(0.2 * (x ^ 5.0)) + Float64(2.0 * x)))) / sqrt(pi)))
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[Abs[N[(N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision] + N[(N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around 0 99.8%

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

   1. *-commutative 99.8%

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

   2. unpow1 99.8%

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

   3. sqr-pow 32.7%

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

   4. fabs-sqr 32.7%

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

   5. sqr-pow 77.1%

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

   6. unpow1 77.1%

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

   7. unpow2 77.1%

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

   8. cube-mult 77.1%

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

6. Simplified 77.1%

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

   1. expm1-log1p-u 76.8%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right| \]

   2. expm1-udef 16.3%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)} - 1}\right| \]

8. Applied egg-rr 4.2%

   \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{3} \cdot {x}^{4}, \mathsf{fma}\left(0.2, x \cdot {x}^{4}, \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right)\right)\right)}{\sqrt{\pi}}\right)} - 1}\right| \]
9. Step-by-step derivation

   1. expm1-def 64.3%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{3} \cdot {x}^{4}, \mathsf{fma}\left(0.2, x \cdot {x}^{4}, \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right)\right)\right)}{\sqrt{\pi}}\right)\right)}\right| \]

   2. expm1-log1p 99.4%

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

   3. metadata-eval 99.4%

      \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{3} \cdot {x}^{\color{blue}{\left(3 + 1\right)}}, \mathsf{fma}\left(0.2, x \cdot {x}^{4}, \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right)\right)\right)}{\sqrt{\pi}}\right| \]

   4. pow-plus 99.4%

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

   5. associate-*l* 99.4%

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

   6. pow-sqr 99.4%

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

   7. metadata-eval 99.4%

      \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{\color{blue}{6}} \cdot x, \mathsf{fma}\left(0.2, x \cdot {x}^{4}, \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right)\right)\right)}{\sqrt{\pi}}\right| \]

   8. pow-plus 99.4%

      \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, \color{blue}{{x}^{\left(6 + 1\right)}}, \mathsf{fma}\left(0.2, x \cdot {x}^{4}, \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right)\right)\right)}{\sqrt{\pi}}\right| \]

   9. metadata-eval 99.4%

      \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{\color{blue}{7}}, \mathsf{fma}\left(0.2, x \cdot {x}^{4}, \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right)\right)\right)}{\sqrt{\pi}}\right| \]

   10. *-commutative 99.4%

       \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.2, \color{blue}{{x}^{4} \cdot x}, \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right)\right)\right)}{\sqrt{\pi}}\right| \]

   11. pow-plus 99.4%

       \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.2, \color{blue}{{x}^{\left(4 + 1\right)}}, \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right)\right)\right)}{\sqrt{\pi}}\right| \]

   12. metadata-eval 99.4%

       \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.2, {x}^{\color{blue}{5}}, \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right)\right)\right)}{\sqrt{\pi}}\right| \]

10. Simplified 99.4%

    \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.2, {x}^{5}, \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right)\right)\right)}{\sqrt{\pi}}}\right| \]
11. Step-by-step derivation

    1. fma-udef 99.4%

       \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{0.2 \cdot {x}^{5} + \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right)}\right)}{\sqrt{\pi}}\right| \]

    2. fma-udef 99.4%

       \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5} + \color{blue}{\left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)}\right)}{\sqrt{\pi}}\right| \]

    3. associate-+r+ 99.4%

       \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{\left(0.2 \cdot {x}^{5} + 2 \cdot x\right) + 0.6666666666666666 \cdot {x}^{3}}\right)}{\sqrt{\pi}}\right| \]

    4. *-commutative 99.4%

       \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \left(0.2 \cdot {x}^{5} + \color{blue}{x \cdot 2}\right) + 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}\right| \]

12. Applied egg-rr 99.4%

    \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{\left(0.2 \cdot {x}^{5} + x \cdot 2\right) + 0.6666666666666666 \cdot {x}^{3}}\right)}{\sqrt{\pi}}\right| \]

13. Final simplification 99.4%

    \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.6666666666666666 \cdot {x}^{3} + \left(0.2 \cdot {x}^{5} + 2 \cdot x\right)\right)}{\sqrt{\pi}}\right| \]

Alternative 3: 99.4% accurate, 2.7× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ x (sqrt PI))
   (+
    (fma 0.2 (pow x 4.0) (* 0.047619047619047616 (pow x 6.0)))
    (fma 0.6666666666666666 (* x x) 2.0)))))

C

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

Julia

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[Abs[N[(N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * 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]), $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%

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

   1. expm1-log1p-u 98.6%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left|x\right|}{\sqrt{\pi}}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2\right)\right| \]

   2. expm1-udef 38.6%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\left|x\right|}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2\right)\right| \]

5. Applied egg-rr 39.1%

   \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\left|x\right|}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]
6. Step-by-step derivation

   1. expm1-def 98.6%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left|x\right|}{\sqrt{\pi}}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2\right)\right| \]

   2. expm1-log1p 98.8%

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

   3. unpow1 98.8%

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

   4. sqr-pow 32.3%

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

   5. fabs-sqr 32.3%

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

   6. sqr-pow 98.8%

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

   7. unpow1 98.8%

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

7. Simplified 99.4%

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

8. Final simplification 99.4%

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

Alternative 4: 98.9% accurate, 3.2× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (fabs
  (*
   x
   (*
    (+ 2.0 (fma 0.2 (pow x 4.0) (* 0.047619047619047616 (pow x 6.0))))
    (sqrt (/ 1.0 PI))))))

C

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

Julia

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[Abs[N[(x * N[(N[(2.0 + N[(0.2 * N[Power[x, 4.0], $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / Pi), $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 99.4%

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

4. Taylor expanded in x around 0 99.2%

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

   1. *-commutative 99.2%

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

   2. associate-*l* 99.2%

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

   3. unpow1 99.2%

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

   4. sqr-pow 32.3%

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

   5. fabs-sqr 32.3%

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

   6. sqr-pow 99.2%

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

   7. unpow1 99.2%

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

6. Simplified 99.2%

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

7. Final simplification 99.2%

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

Alternative 5: 98.5% accurate, 3.8× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ x (sqrt PI))
   (+ 2.0 (+ (* 0.047619047619047616 (pow x 6.0)) (* 0.2 (pow x 4.0)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[Abs[N[(N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[Power[x, 4.0], $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 99.4%

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

4. Taylor expanded in x around 0 98.8%

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

   1. expm1-log1p-u 98.6%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left|x\right|}{\sqrt{\pi}}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2\right)\right| \]

   2. expm1-udef 38.6%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\left|x\right|}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2\right)\right| \]

6. Applied egg-rr 38.6%

   \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\left|x\right|}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2\right)\right| \]
7. Step-by-step derivation

   1. expm1-def 98.6%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left|x\right|}{\sqrt{\pi}}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2\right)\right| \]

   2. expm1-log1p 98.8%

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

   3. unpow1 98.8%

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

   4. sqr-pow 32.3%

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

   5. fabs-sqr 32.3%

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

   6. sqr-pow 98.8%

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

   7. unpow1 98.8%

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

8. Simplified 98.8%

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

   1. fma-udef 98.8%

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

10. Applied egg-rr 98.8%

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

11. Final simplification 98.8%

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

Alternative 6: 98.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 2 \cdot x\right)}{\sqrt{\pi}}\right| \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (fabs (/ (fma 0.047619047619047616 (pow x 7.0) (* 2.0 x)) (sqrt PI))))

C

x = abs(x);
double code(double x) {
	return fabs((fma(0.047619047619047616, pow(x, 7.0), (2.0 * x)) / sqrt(((double) M_PI))));
}

Julia

x = abs(x)
function code(x)
	return abs(Float64(fma(0.047619047619047616, (x ^ 7.0), Float64(2.0 * x)) / sqrt(pi)))
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[Abs[N[(N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around 0 99.8%

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

   1. *-commutative 99.8%

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

   2. unpow1 99.8%

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

   3. sqr-pow 32.7%

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

   4. fabs-sqr 32.7%

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

   5. sqr-pow 77.1%

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

   6. unpow1 77.1%

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

   7. unpow2 77.1%

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

   8. cube-mult 77.1%

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

6. Simplified 77.1%

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

   1. expm1-log1p-u 76.8%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right| \]

   2. expm1-udef 16.3%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)} - 1}\right| \]

8. Applied egg-rr 4.2%

   \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{3} \cdot {x}^{4}, \mathsf{fma}\left(0.2, x \cdot {x}^{4}, \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right)\right)\right)}{\sqrt{\pi}}\right)} - 1}\right| \]
9. Step-by-step derivation

   1. expm1-def 64.3%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{3} \cdot {x}^{4}, \mathsf{fma}\left(0.2, x \cdot {x}^{4}, \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right)\right)\right)}{\sqrt{\pi}}\right)\right)}\right| \]

   2. expm1-log1p 99.4%

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

   3. metadata-eval 99.4%

      \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{3} \cdot {x}^{\color{blue}{\left(3 + 1\right)}}, \mathsf{fma}\left(0.2, x \cdot {x}^{4}, \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right)\right)\right)}{\sqrt{\pi}}\right| \]

   4. pow-plus 99.4%

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

   5. associate-*l* 99.4%

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

   6. pow-sqr 99.4%

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

   7. metadata-eval 99.4%

      \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{\color{blue}{6}} \cdot x, \mathsf{fma}\left(0.2, x \cdot {x}^{4}, \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right)\right)\right)}{\sqrt{\pi}}\right| \]

   8. pow-plus 99.4%

      \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, \color{blue}{{x}^{\left(6 + 1\right)}}, \mathsf{fma}\left(0.2, x \cdot {x}^{4}, \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right)\right)\right)}{\sqrt{\pi}}\right| \]

   9. metadata-eval 99.4%

      \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{\color{blue}{7}}, \mathsf{fma}\left(0.2, x \cdot {x}^{4}, \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right)\right)\right)}{\sqrt{\pi}}\right| \]

   10. *-commutative 99.4%

       \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.2, \color{blue}{{x}^{4} \cdot x}, \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right)\right)\right)}{\sqrt{\pi}}\right| \]

   11. pow-plus 99.4%

       \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.2, \color{blue}{{x}^{\left(4 + 1\right)}}, \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right)\right)\right)}{\sqrt{\pi}}\right| \]

   12. metadata-eval 99.4%

       \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.2, {x}^{\color{blue}{5}}, \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right)\right)\right)}{\sqrt{\pi}}\right| \]

10. Simplified 99.4%

    \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.2, {x}^{5}, \mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3}\right)\right)\right)}{\sqrt{\pi}}}\right| \]

11. Taylor expanded in x around 0 98.7%

    \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{2 \cdot x}\right)}{\sqrt{\pi}}\right| \]

12. Final simplification 98.7%

    \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 2 \cdot x\right)}{\sqrt{\pi}}\right| \]

Alternative 7: 98.3% accurate, 4.8× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (fabs (* (/ x (sqrt PI)) (+ 2.0 (* 0.047619047619047616 (pow x 6.0))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
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 99.8%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around 0 98.8%

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

   1. expm1-log1p-u 98.6%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left|x\right|}{\sqrt{\pi}}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2\right)\right| \]

   2. expm1-udef 38.6%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\left|x\right|}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2\right)\right| \]

6. Applied egg-rr 38.6%

   \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\left|x\right|}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2\right)\right| \]
7. Step-by-step derivation

   1. expm1-def 98.6%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left|x\right|}{\sqrt{\pi}}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2\right)\right| \]

   2. expm1-log1p 98.8%

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

   3. unpow1 98.8%

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

   4. sqr-pow 32.3%

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

   5. fabs-sqr 32.3%

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

   6. sqr-pow 98.8%

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

   7. unpow1 98.8%

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

8. Simplified 98.8%

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

9. Taylor expanded in x around inf 98.7%

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

10. Final simplification 98.7%

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

Alternative 8: 98.7% accurate, 4.8× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (fabs (* (* 2.0 x) (pow PI -0.5)))
   (fabs (* 0.047619047619047616 (* (pow x 7.0) (pow PI -0.5))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.85], N[Abs[N[(N[(2.0 * x), $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[Power[x, 7.0], $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 66.5%

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

      1. *-commutative 66.5%

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

      2. *-commutative 66.5%

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

      3. *-commutative 66.5%

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

      4. associate-*l* 66.5%

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

      5. *-commutative 66.5%

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

      6. unpow1 66.5%

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

      7. sqr-pow 32.3%

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

      8. fabs-sqr 32.3%

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

      9. sqr-pow 66.5%

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

      10. unpow1 66.5%

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

   6. Simplified 66.5%

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

      1. expm1-log1p-u 64.5%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x\right)\right)\right)}\right| \]

      2. expm1-udef 4.1%

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

      3. inv-pow 4.1%

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

      4. sqrt-pow1 4.1%

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

      5. metadata-eval 4.1%

         \[\leadsto \left|e^{\mathsf{log1p}\left({\pi}^{\color{blue}{-0.5}} \cdot \left(2 \cdot x\right)\right)} - 1\right| \]

      6. *-commutative 4.1%

         \[\leadsto \left|e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \color{blue}{\left(x \cdot 2\right)}\right)} - 1\right| \]

   8. Applied egg-rr 4.1%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right)} - 1}\right| \]
   9. Step-by-step derivation

      1. expm1-def 64.5%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right)\right)}\right| \]

      2. expm1-log1p 66.5%

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

      3. *-commutative 66.5%

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

   10. Simplified 66.5%

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

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 38.4%

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

      1. expm1-log1p-u 38.0%

         \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)}\right| \]

      2. expm1-udef 37.9%

         \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)} - 1\right)}\right| \]

      3. *-commutative 37.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left({x}^{6} \cdot \left|x\right|\right)}\right)} - 1\right)\right| \]

      4. inv-pow 37.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)} - 1\right)\right| \]

      5. sqrt-pow1 37.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)} - 1\right)\right| \]

      6. metadata-eval 37.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left({\pi}^{\color{blue}{-0.5}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)} - 1\right)\right| \]

      7. *-commutative 37.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \color{blue}{\left(\left|x\right| \cdot {x}^{6}\right)}\right)} - 1\right)\right| \]

      8. add-sqr-sqrt 1.8%

         \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot {x}^{6}\right)\right)} - 1\right)\right| \]

      9. fabs-sqr 1.8%

         \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {x}^{6}\right)\right)} - 1\right)\right| \]

      10. add-sqr-sqrt 3.6%

          \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(\color{blue}{x} \cdot {x}^{6}\right)\right)} - 1\right)\right| \]

   6. Applied egg-rr 3.6%

      \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot {x}^{6}\right)\right)} - 1\right)}\right| \]
   7. Step-by-step derivation

      1. expm1-def 3.8%

         \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot {x}^{6}\right)\right)\right)}\right| \]

      2. expm1-log1p 38.4%

         \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\left({\pi}^{-0.5} \cdot \left(x \cdot {x}^{6}\right)\right)}\right| \]

      3. *-commutative 38.4%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot \color{blue}{\left({x}^{6} \cdot x\right)}\right)\right| \]

      4. pow-plus 38.4%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot \color{blue}{{x}^{\left(6 + 1\right)}}\right)\right| \]

      5. metadata-eval 38.4%

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

      6. *-commutative 38.4%

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

   8. Simplified 38.4%

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

4. Final simplification 66.5%

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

Alternative 9: 96.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|\left(2 \cdot x\right) \cdot {\pi}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \sqrt{\frac{{x}^{14}}{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (fabs (* (* 2.0 x) (pow PI -0.5)))
   (fabs (* 0.047619047619047616 (sqrt (/ (pow x 14.0) PI))))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = fabs(((2.0 * x) * pow(((double) M_PI), -0.5)));
	} else {
		tmp = fabs((0.047619047619047616 * sqrt((pow(x, 14.0) / ((double) M_PI)))));
	}
	return tmp;
}

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = Math.abs(((2.0 * x) * Math.pow(Math.PI, -0.5)));
	} else {
		tmp = Math.abs((0.047619047619047616 * Math.sqrt((Math.pow(x, 14.0) / Math.PI))));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = math.fabs(((2.0 * x) * math.pow(math.pi, -0.5)))
	else:
		tmp = math.fabs((0.047619047619047616 * math.sqrt((math.pow(x, 14.0) / math.pi))))
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = abs(Float64(Float64(2.0 * x) * (pi ^ -0.5)));
	else
		tmp = abs(Float64(0.047619047619047616 * sqrt(Float64((x ^ 14.0) / pi))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = abs(((2.0 * x) * (pi ^ -0.5)));
	else
		tmp = abs((0.047619047619047616 * sqrt(((x ^ 14.0) / pi))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.85], N[Abs[N[(N[(2.0 * x), $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[Sqrt[N[(N[Power[x, 14.0], $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 66.5%

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

      1. *-commutative 66.5%

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

      2. *-commutative 66.5%

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

      3. *-commutative 66.5%

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

      4. associate-*l* 66.5%

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

      5. *-commutative 66.5%

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

      6. unpow1 66.5%

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

      7. sqr-pow 32.3%

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

      8. fabs-sqr 32.3%

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

      9. sqr-pow 66.5%

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

      10. unpow1 66.5%

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

   6. Simplified 66.5%

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

      1. expm1-log1p-u 64.5%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x\right)\right)\right)}\right| \]

      2. expm1-udef 4.1%

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

      3. inv-pow 4.1%

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

      4. sqrt-pow1 4.1%

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

      5. metadata-eval 4.1%

         \[\leadsto \left|e^{\mathsf{log1p}\left({\pi}^{\color{blue}{-0.5}} \cdot \left(2 \cdot x\right)\right)} - 1\right| \]

      6. *-commutative 4.1%

         \[\leadsto \left|e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \color{blue}{\left(x \cdot 2\right)}\right)} - 1\right| \]

   8. Applied egg-rr 4.1%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right)} - 1}\right| \]
   9. Step-by-step derivation

      1. expm1-def 64.5%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right)\right)}\right| \]

      2. expm1-log1p 66.5%

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

      3. *-commutative 66.5%

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

   10. Simplified 66.5%

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

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 38.4%

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

      1. expm1-log1p-u 38.0%

         \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)}\right| \]

      2. expm1-udef 37.9%

         \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)} - 1\right)}\right| \]

      3. *-commutative 37.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left({x}^{6} \cdot \left|x\right|\right)}\right)} - 1\right)\right| \]

      4. inv-pow 37.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)} - 1\right)\right| \]

      5. sqrt-pow1 37.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)} - 1\right)\right| \]

      6. metadata-eval 37.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left({\pi}^{\color{blue}{-0.5}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)} - 1\right)\right| \]

      7. *-commutative 37.9%

         \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \color{blue}{\left(\left|x\right| \cdot {x}^{6}\right)}\right)} - 1\right)\right| \]

      8. add-sqr-sqrt 1.8%

         \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot {x}^{6}\right)\right)} - 1\right)\right| \]

      9. fabs-sqr 1.8%

         \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {x}^{6}\right)\right)} - 1\right)\right| \]

      10. add-sqr-sqrt 3.6%

          \[\leadsto \left|0.047619047619047616 \cdot \left(e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(\color{blue}{x} \cdot {x}^{6}\right)\right)} - 1\right)\right| \]

   6. Applied egg-rr 3.6%

      \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot {x}^{6}\right)\right)} - 1\right)}\right| \]
   7. Step-by-step derivation

      1. expm1-def 3.8%

         \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot {x}^{6}\right)\right)\right)}\right| \]

      2. expm1-log1p 38.4%

         \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\left({\pi}^{-0.5} \cdot \left(x \cdot {x}^{6}\right)\right)}\right| \]

   8. Simplified 38.4%

      \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\left({\pi}^{-0.5} \cdot \left(x \cdot {x}^{6}\right)\right)}\right| \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 3.6%

         \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\left(\sqrt{{\pi}^{-0.5} \cdot \left(x \cdot {x}^{6}\right)} \cdot \sqrt{{\pi}^{-0.5} \cdot \left(x \cdot {x}^{6}\right)}\right)}\right| \]

      2. sqrt-unprod 35.8%

         \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\sqrt{\left({\pi}^{-0.5} \cdot \left(x \cdot {x}^{6}\right)\right) \cdot \left({\pi}^{-0.5} \cdot \left(x \cdot {x}^{6}\right)\right)}}\right| \]

      3. metadata-eval 35.8%

         \[\leadsto \left|0.047619047619047616 \cdot \sqrt{\left({\pi}^{\color{blue}{\left(\frac{-1}{2}\right)}} \cdot \left(x \cdot {x}^{6}\right)\right) \cdot \left({\pi}^{-0.5} \cdot \left(x \cdot {x}^{6}\right)\right)}\right| \]

      4. sqrt-pow1 35.8%

         \[\leadsto \left|0.047619047619047616 \cdot \sqrt{\left(\color{blue}{\sqrt{{\pi}^{-1}}} \cdot \left(x \cdot {x}^{6}\right)\right) \cdot \left({\pi}^{-0.5} \cdot \left(x \cdot {x}^{6}\right)\right)}\right| \]

      5. inv-pow 35.8%

         \[\leadsto \left|0.047619047619047616 \cdot \sqrt{\left(\sqrt{\color{blue}{\frac{1}{\pi}}} \cdot \left(x \cdot {x}^{6}\right)\right) \cdot \left({\pi}^{-0.5} \cdot \left(x \cdot {x}^{6}\right)\right)}\right| \]

      6. metadata-eval 35.8%

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

      7. sqrt-pow1 35.8%

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

      8. inv-pow 35.8%

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

      9. swap-sqr 35.8%

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

      10. add-sqr-sqrt 35.8%

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

      11. pow1 35.8%

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

      12. pow-prod-up 35.8%

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

      13. metadata-eval 35.8%

          \[\leadsto \left|0.047619047619047616 \cdot \sqrt{\frac{1}{\pi} \cdot \left({x}^{\color{blue}{7}} \cdot \left(x \cdot {x}^{6}\right)\right)}\right| \]

      14. pow1 35.8%

          \[\leadsto \left|0.047619047619047616 \cdot \sqrt{\frac{1}{\pi} \cdot \left({x}^{7} \cdot \left(\color{blue}{{x}^{1}} \cdot {x}^{6}\right)\right)}\right| \]

      15. pow-prod-up 35.8%

          \[\leadsto \left|0.047619047619047616 \cdot \sqrt{\frac{1}{\pi} \cdot \left({x}^{7} \cdot \color{blue}{{x}^{\left(1 + 6\right)}}\right)}\right| \]

      16. metadata-eval 35.8%

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

   10. Applied egg-rr 35.8%

       \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\sqrt{\frac{1}{\pi} \cdot {x}^{14}}}\right| \]
   11. Step-by-step derivation

       1. associate-*l/ 35.8%

          \[\leadsto \left|0.047619047619047616 \cdot \sqrt{\color{blue}{\frac{1 \cdot {x}^{14}}{\pi}}}\right| \]

       2. *-lft-identity 35.8%

          \[\leadsto \left|0.047619047619047616 \cdot \sqrt{\frac{\color{blue}{{x}^{14}}}{\pi}}\right| \]

   12. Simplified 35.8%

       \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\sqrt{\frac{{x}^{14}}{\pi}}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|\left(2 \cdot x\right) \cdot {\pi}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \sqrt{\frac{{x}^{14}}{\pi}}\right|\\ \end{array} \]

Alternative 10: 66.9% accurate, 6.4× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (fabs (* (* 2.0 x) (pow PI -0.5))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[Abs[N[(N[(2.0 * x), $MachinePrecision] * N[Power[Pi, -0.5], $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. Taylor expanded in x around 0 66.5%

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

   1. *-commutative 66.5%

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

   2. *-commutative 66.5%

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

   3. *-commutative 66.5%

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

   4. associate-*l* 66.5%

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

   5. *-commutative 66.5%

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

   6. unpow1 66.5%

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

   7. sqr-pow 32.3%

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

   8. fabs-sqr 32.3%

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

   9. sqr-pow 66.5%

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

   10. unpow1 66.5%

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

6. Simplified 66.5%

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

   1. expm1-log1p-u 64.5%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x\right)\right)\right)}\right| \]

   2. expm1-udef 4.1%

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

   3. inv-pow 4.1%

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

   4. sqrt-pow1 4.1%

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

   5. metadata-eval 4.1%

      \[\leadsto \left|e^{\mathsf{log1p}\left({\pi}^{\color{blue}{-0.5}} \cdot \left(2 \cdot x\right)\right)} - 1\right| \]

   6. *-commutative 4.1%

      \[\leadsto \left|e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \color{blue}{\left(x \cdot 2\right)}\right)} - 1\right| \]

8. Applied egg-rr 4.1%

   \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right)} - 1}\right| \]
9. Step-by-step derivation

   1. expm1-def 64.5%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right)\right)}\right| \]

   2. expm1-log1p 66.5%

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

   3. *-commutative 66.5%

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

10. Simplified 66.5%

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

11. Final simplification 66.5%

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

Reproduce

herbie shell --seed 2023333 
(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)))))))