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

Percentage Accurate: 99.8% → 99.8%

Time: 10.8s

Alternatives: 8

Speedup: 3.1×

• 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

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

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 5.0)))
    (* 0.047619047619047616 (* (fabs x) (* (* x x) (* (* x 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, 5.0))) + (0.047619047619047616 * (fabs(x) * ((x * x) * ((x * 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 * (x ^ 5.0))) + Float64(0.047619047619047616 * Float64(abs(x) * Float64(Float64(x * x) * Float64(Float64(x * 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[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 99.9%

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

   1. unpow2 99.9%

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

   2. associate-*r* 99.9%

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

   3. rem-square-sqrt 38.2%

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

   4. fabs-sqr 38.2%

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

   5. rem-square-sqrt 76.0%

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

   6. associate-*r* 76.0%

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

   7. cube-mult 76.0%

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

6. Simplified 76.0%

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

7. Taylor expanded in x around 0 76.0%

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

   1. *-commutative 76.0%

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

   2. unpow1 76.0%

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

   3. sqr-pow 38.2%

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

   4. fabs-sqr 38.2%

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

   5. sqr-pow 72.5%

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

   6. unpow1 72.5%

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

   7. pow-plus 72.5%

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

   8. metadata-eval 72.5%

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

9. Simplified 72.5%

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

10. Final simplification 72.5%

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

Alternative 2: 99.9% accurate, 3.1× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (fabs
  (*
   (* x (pow PI -0.5))
   (+
    (fma 0.6666666666666666 (* x x) 2.0)
    (+ (* 0.2 (pow x 4.0)) (* 0.047619047619047616 (pow x 6.0)))))))

C

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

Julia

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[Abs[N[(N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] + N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

   1. div-inv 99.9%

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

   2. add-sqr-sqrt 38.0%

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

   3. fabs-sqr 38.0%

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

   4. add-sqr-sqrt 99.9%

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

   5. pow1/2 99.9%

      \[\leadsto \left|\left(x \cdot \frac{1}{\color{blue}{{\pi}^{0.5}}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   6. pow-flip 99.9%

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

   7. metadata-eval 99.9%

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

5. Applied egg-rr 99.9%

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

6. Taylor expanded in x around 0 99.9%

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

7. Final simplification 99.9%

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

Alternative 3: 99.2% accurate, 3.8× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (fabs
  (*
   (* x (pow PI -0.5))
   (+
    (fma 0.6666666666666666 (* x x) 2.0)
    (* 0.047619047619047616 (pow x 6.0))))))

C

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

Julia

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[Abs[N[(N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

   1. div-inv 99.9%

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

   2. add-sqr-sqrt 38.0%

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

   3. fabs-sqr 38.0%

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

   4. add-sqr-sqrt 99.9%

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

   5. pow1/2 99.9%

      \[\leadsto \left|\left(x \cdot \frac{1}{\color{blue}{{\pi}^{0.5}}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   6. pow-flip 99.9%

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

   7. metadata-eval 99.9%

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

5. Applied egg-rr 99.9%

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

6. Taylor expanded in x around inf 98.5%

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

7. Final simplification 98.5%

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

Alternative 4: 98.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around inf 97.4%

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

5. Final simplification 97.4%

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

Alternative 5: 99.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (fabs (* (pow PI -0.5) (+ (* 0.6666666666666666 (pow x 3.0)) (* 2.0 x))))
   (fabs (* 0.047619047619047616 (/ (pow x 7.0) (sqrt PI))))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = fabs((pow(((double) M_PI), -0.5) * ((0.6666666666666666 * pow(x, 3.0)) + (2.0 * x))));
	} else {
		tmp = fabs((0.047619047619047616 * (pow(x, 7.0) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = Math.abs((Math.pow(Math.PI, -0.5) * ((0.6666666666666666 * Math.pow(x, 3.0)) + (2.0 * x))));
	} else {
		tmp = Math.abs((0.047619047619047616 * (Math.pow(x, 7.0) / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 2.2:
		tmp = math.fabs((math.pow(math.pi, -0.5) * ((0.6666666666666666 * math.pow(x, 3.0)) + (2.0 * x))))
	else:
		tmp = math.fabs((0.047619047619047616 * (math.pow(x, 7.0) / math.sqrt(math.pi))))
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = abs(Float64((pi ^ -0.5) * Float64(Float64(0.6666666666666666 * (x ^ 3.0)) + Float64(2.0 * x))));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64((x ^ 7.0) / sqrt(pi))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.2)
		tmp = abs(((pi ^ -0.5) * ((0.6666666666666666 * (x ^ 3.0)) + (2.0 * x))));
	else
		tmp = abs((0.047619047619047616 * ((x ^ 7.0) / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 2.2], N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[Power[x, 7.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 91.5%

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

      1. +-commutative 91.5%

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

      2. associate-*r* 91.5%

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

      3. associate-*r* 91.5%

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

      4. distribute-rgt-out 91.5%

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

      5. *-commutative 91.5%

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

   6. Simplified 91.5%

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

      1. distribute-lft-in 91.5%

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

      2. pow1/2 91.5%

         \[\leadsto \left|\color{blue}{{\left(\frac{1}{\pi}\right)}^{0.5}} \cdot \left(x \cdot 2\right) + \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right)\right| \]

      3. inv-pow 91.5%

         \[\leadsto \left|{\color{blue}{\left({\pi}^{-1}\right)}}^{0.5} \cdot \left(x \cdot 2\right) + \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right)\right| \]

      4. pow-pow 91.5%

         \[\leadsto \left|\color{blue}{{\pi}^{\left(-1 \cdot 0.5\right)}} \cdot \left(x \cdot 2\right) + \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right)\right| \]

      5. metadata-eval 91.5%

         \[\leadsto \left|{\pi}^{\color{blue}{-0.5}} \cdot \left(x \cdot 2\right) + \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right)\right| \]

      6. pow1/2 91.5%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \left(x \cdot 2\right) + \color{blue}{{\left(\frac{1}{\pi}\right)}^{0.5}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right)\right| \]

      7. inv-pow 91.5%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \left(x \cdot 2\right) + {\color{blue}{\left({\pi}^{-1}\right)}}^{0.5} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right)\right| \]

      8. pow-pow 91.5%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \left(x \cdot 2\right) + \color{blue}{{\pi}^{\left(-1 \cdot 0.5\right)}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right)\right| \]

      9. metadata-eval 91.5%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \left(x \cdot 2\right) + {\pi}^{\color{blue}{-0.5}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right)\right| \]

   8. Applied egg-rr 91.5%

      \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(x \cdot 2\right) + {\pi}^{-0.5} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right)}\right| \]
   9. Step-by-step derivation

      1. distribute-lft-out 91.5%

         \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)}\right| \]

   10. Simplified 91.5%

       \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)}\right| \]

   if 2.2000000000000002 < x

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around inf 34.5%

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

      1. associate-*r* 34.5%

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

   6. Simplified 34.5%

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

      1. expm1-log1p-u 4.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)}\right| \]

      2. expm1-udef 3.8%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}\right)} - 1}\right| \]

      3. associate-*l* 3.8%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)}\right)} - 1\right| \]

      4. sqrt-div 3.8%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)\right)} - 1\right| \]

      5. metadata-eval 3.8%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)\right)} - 1\right| \]

      6. un-div-inv 3.8%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \color{blue}{\frac{{x}^{7}}{\sqrt{\pi}}}\right)} - 1\right| \]

   8. Applied egg-rr 3.8%

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

      1. expm1-def 4.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\right)\right)}\right| \]

      2. expm1-log1p 34.5%

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

   10. Simplified 34.5%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 6: 98.7% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = abs((0.047619047619047616 * ((x ^ 7.0) / sqrt(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[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[Power[x, 7.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 69.1%

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

      1. associate-*r* 69.1%

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

   6. Simplified 69.1%

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

      1. expm1-log1p-u 67.4%

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

      2. expm1-udef 5.4%

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

      3. *-commutative 5.4%

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

      4. *-commutative 5.4%

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

      5. pow1/2 5.4%

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

      6. inv-pow 5.4%

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

      7. pow-pow 5.4%

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

      8. metadata-eval 5.4%

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

   8. Applied egg-rr 5.4%

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

      1. expm1-def 67.4%

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

      2. expm1-log1p 69.1%

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

   10. Simplified 69.1%

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

       1. expm1-log1p-u 67.4%

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

       2. expm1-udef 5.4%

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

       3. associate-*r* 5.4%

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

       4. metadata-eval 5.4%

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

       5. pow-flip 5.4%

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

       6. pow1/2 5.4%

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

       7. associate-/r/ 5.4%

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

       8. clear-num 5.4%

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

       9. *-commutative 5.4%

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

   12. Applied egg-rr 5.4%

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

       1. expm1-def 66.9%

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

       2. expm1-log1p 68.6%

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

       3. associate-*r/ 68.6%

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

       4. associate-/l* 68.7%

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

   14. Simplified 68.7%

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

       1. associate-/r/ 69.1%

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

   16. Applied egg-rr 69.1%

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

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around inf 34.5%

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

      1. associate-*r* 34.5%

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

   6. Simplified 34.5%

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

      1. expm1-log1p-u 4.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)}\right| \]

      2. expm1-udef 3.8%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}\right)} - 1}\right| \]

      3. associate-*l* 3.8%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)}\right)} - 1\right| \]

      4. sqrt-div 3.8%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)\right)} - 1\right| \]

      5. metadata-eval 3.8%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)\right)} - 1\right| \]

      6. un-div-inv 3.8%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \color{blue}{\frac{{x}^{7}}{\sqrt{\pi}}}\right)} - 1\right| \]

   8. Applied egg-rr 3.8%

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

      1. expm1-def 4.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\right)\right)}\right| \]

      2. expm1-log1p 34.5%

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

   10. Simplified 34.5%

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

4. Final simplification 69.1%

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

Alternative 7: 83.0% accurate, 6.3× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.05:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{4 \cdot \frac{x \cdot x}{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.05)
   (fabs (* x (/ 2.0 (sqrt PI))))
   (fabs (sqrt (* 4.0 (/ (* x x) PI))))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.05) {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs(sqrt((4.0 * ((x * x) / ((double) M_PI)))));
	}
	return tmp;
}

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.05) {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs(Math.sqrt((4.0 * ((x * x) / Math.PI))));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.05:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs(math.sqrt((4.0 * ((x * x) / math.pi))))
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.05)
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(sqrt(Float64(4.0 * Float64(Float64(x * x) / pi))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.05)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = abs(sqrt((4.0 * ((x * x) / pi))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.05], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[Sqrt[N[(4.0 * N[(N[(x * x), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.050000000000000003

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 69.3%

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

      1. associate-*r* 69.3%

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

   6. Simplified 69.3%

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

      1. expm1-log1p-u 67.5%

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

      2. expm1-udef 5.3%

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

      3. *-commutative 5.3%

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

      4. *-commutative 5.3%

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

      5. pow1/2 5.3%

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

      6. inv-pow 5.3%

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

      7. pow-pow 5.3%

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

      8. metadata-eval 5.3%

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

   8. Applied egg-rr 5.3%

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

      1. expm1-def 67.5%

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

      2. expm1-log1p 69.3%

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

   10. Simplified 69.3%

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

       1. expm1-log1p-u 67.5%

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

       2. expm1-udef 5.3%

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

       3. associate-*r* 5.3%

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

       4. metadata-eval 5.3%

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

       5. pow-flip 5.3%

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

       6. pow1/2 5.3%

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

       7. associate-/r/ 5.3%

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

       8. clear-num 5.3%

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

       9. *-commutative 5.3%

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

   12. Applied egg-rr 5.3%

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

       1. expm1-def 67.1%

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

       2. expm1-log1p 68.8%

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

       3. associate-*r/ 68.8%

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

       4. associate-/l* 68.8%

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

   14. Simplified 68.8%

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

       1. associate-/r/ 69.3%

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

   16. Applied egg-rr 69.3%

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

   if 0.050000000000000003 < x

   1. Initial program 98.4%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in x around 0 26.6%

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

      1. associate-*r* 26.6%

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

   6. Simplified 26.6%

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

      1. expm1-log1p-u 26.6%

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

      2. expm1-udef 26.6%

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

      3. *-commutative 26.6%

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

      4. *-commutative 26.6%

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

      5. pow1/2 26.6%

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

      6. inv-pow 26.6%

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

      7. pow-pow 26.6%

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

      8. metadata-eval 26.6%

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

   8. Applied egg-rr 26.6%

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

      1. expm1-def 26.6%

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

      2. expm1-log1p 26.6%

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

   10. Simplified 26.6%

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

       1. add-sqr-sqrt 26.6%

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

       2. pow1/2 26.6%

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

       3. pow1/2 26.6%

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

       4. pow-prod-down 26.6%

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

   12. Applied egg-rr 26.6%

       \[\leadsto \left|\color{blue}{{\left(\frac{x \cdot x}{\pi} \cdot 4\right)}^{0.5}}\right| \]
   13. Step-by-step derivation

       1. unpow1/2 26.6%

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

   14. Simplified 26.6%

       \[\leadsto \left|\color{blue}{\sqrt{\frac{x \cdot x}{\pi} \cdot 4}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.05:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{4 \cdot \frac{x \cdot x}{\pi}}\right|\\ \end{array} \]

Alternative 8: 67.4% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around 0 69.1%

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

   1. associate-*r* 69.1%

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

6. Simplified 69.1%

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

   1. expm1-log1p-u 67.4%

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

   2. expm1-udef 5.4%

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

   3. *-commutative 5.4%

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

   4. *-commutative 5.4%

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

   5. pow1/2 5.4%

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

   6. inv-pow 5.4%

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

   7. pow-pow 5.4%

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

   8. metadata-eval 5.4%

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

8. Applied egg-rr 5.4%

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

   1. expm1-def 67.4%

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

   2. expm1-log1p 69.1%

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

10. Simplified 69.1%

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

    1. expm1-log1p-u 67.4%

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

    2. expm1-udef 5.4%

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

    3. associate-*r* 5.4%

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

    4. metadata-eval 5.4%

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

    5. pow-flip 5.4%

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

    6. pow1/2 5.4%

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

    7. associate-/r/ 5.4%

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

    8. clear-num 5.4%

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

    9. *-commutative 5.4%

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

12. Applied egg-rr 5.4%

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

    1. expm1-def 66.9%

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

    2. expm1-log1p 68.6%

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

    3. associate-*r/ 68.6%

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

    4. associate-/l* 68.7%

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

14. Simplified 68.7%

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

    1. associate-/r/ 69.1%

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

16. Applied egg-rr 69.1%

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

17. Final simplification 69.1%

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

Reproduce

herbie shell --seed 2023213 
(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)))))))