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

Percentage Accurate: 99.8% → 99.9%

Time: 7.6s

Alternatives: 10

Speedup: 3.7×

• 28.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[Abs[N[(x * N[(1.0 / N[(N[Sqrt[Pi], $MachinePrecision] / N[(N[(N[(x * 0.6666666666666666), $MachinePrecision] * x + 2.0), $MachinePrecision] + N[(0.2 * N[Power[x, 4.0], $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Simplified 99.4%

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

   1. associate-/l* 99.4%

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

   2. add-sqr-sqrt 34.0%

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

   3. fabs-sqr 34.0%

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

   4. add-sqr-sqrt 99.4%

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

   5. div-inv 99.9%

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

   6. add-sqr-sqrt 34.3%

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

   7. fabs-sqr 34.3%

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

   8. add-sqr-sqrt 99.9%

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

   9. add-sqr-sqrt 99.8%

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

   10. pow2 99.8%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return Math.abs(((x * ((2.0 + (0.6666666666666666 * (x * x))) + ((0.047619047619047616 * Math.pow(x, 6.0)) + (0.2 * Math.pow(x, 4.0))))) * Math.pow(Math.PI, -0.5)));
}

Python

def code(x):
	return math.fabs(((x * ((2.0 + (0.6666666666666666 * (x * x))) + ((0.047619047619047616 * math.pow(x, 6.0)) + (0.2 * math.pow(x, 4.0))))) * math.pow(math.pi, -0.5)))

Julia

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

MATLAB

function tmp = code(x)
	tmp = abs(((x * ((2.0 + (0.6666666666666666 * (x * x))) + ((0.047619047619047616 * (x ^ 6.0)) + (0.2 * (x ^ 4.0))))) * (pi ^ -0.5)));
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.4%

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

   1. div-inv 99.8%

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

5. Applied egg-rr 99.8%

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

6. Taylor expanded in x around 0 99.8%

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

   1. fma-udef 92.3%

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

   2. *-commutative 92.3%

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

   3. associate-*r* 92.3%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 3: 99.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.4%

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

   1. div-inv 99.8%

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

5. Applied egg-rr 99.8%

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

6. Taylor expanded in x around inf 98.9%

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

7. Final simplification 98.9%

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

Alternative 4: 94.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x \cdot \left(\left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {x}^{4}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.7)
   (fabs
    (*
     (pow PI -0.5)
     (* x (+ (+ 2.0 (* 0.6666666666666666 (* x x))) (* 0.2 (pow x 4.0))))))
   (fabs (* (pow x 7.0) (/ 0.047619047619047616 (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.7) {
		tmp = fabs((pow(((double) M_PI), -0.5) * (x * ((2.0 + (0.6666666666666666 * (x * x))) + (0.2 * pow(x, 4.0))))));
	} else {
		tmp = fabs((pow(x, 7.0) * (0.047619047619047616 / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.7) {
		tmp = Math.abs((Math.pow(Math.PI, -0.5) * (x * ((2.0 + (0.6666666666666666 * (x * x))) + (0.2 * Math.pow(x, 4.0))))));
	} else {
		tmp = Math.abs((Math.pow(x, 7.0) * (0.047619047619047616 / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.7:
		tmp = math.fabs((math.pow(math.pi, -0.5) * (x * ((2.0 + (0.6666666666666666 * (x * x))) + (0.2 * math.pow(x, 4.0))))))
	else:
		tmp = math.fabs((math.pow(x, 7.0) * (0.047619047619047616 / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.7)
		tmp = abs(Float64((pi ^ -0.5) * Float64(x * Float64(Float64(2.0 + Float64(0.6666666666666666 * Float64(x * x))) + Float64(0.2 * (x ^ 4.0))))));
	else
		tmp = abs(Float64((x ^ 7.0) * Float64(0.047619047619047616 / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.7)
		tmp = abs(((pi ^ -0.5) * (x * ((2.0 + (0.6666666666666666 * (x * x))) + (0.2 * (x ^ 4.0))))));
	else
		tmp = abs(((x ^ 7.0) * (0.047619047619047616 / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.7], N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * N[(N[(2.0 + N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Power[x, 7.0], $MachinePrecision] * N[(0.047619047619047616 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7000000000000002

   1. Initial program 99.5%

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

   3. Simplified 99.4%

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

      1. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 92.3%

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

      1. fma-udef 92.3%

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

      2. *-commutative 92.3%

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

      3. associate-*r* 92.3%

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

   8. Applied egg-rr 92.3%

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

   if 2.7000000000000002 < x

   1. Initial program 99.5%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 98.3%

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

      1. *-commutative 98.3%

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

      2. *-commutative 98.3%

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

      3. associate-+r+ 98.3%

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

      4. distribute-lft-in 98.3%

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

      5. fma-def 98.3%

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

      6. rem-square-sqrt 34.0%

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

      7. fabs-sqr 34.0%

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

      8. rem-square-sqrt 76.0%

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

      9. +-commutative 76.0%

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

      10. fma-def 76.0%

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

      11. rem-square-sqrt 34.2%

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

      12. fabs-sqr 34.2%

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

      13. rem-square-sqrt 76.0%

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

      14. *-commutative 76.0%

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

   6. Simplified 98.0%

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

   7. Taylor expanded in x around inf 34.8%

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

      1. expm1-log1p-u 3.9%

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

      2. expm1-udef 3.7%

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

      3. sqrt-div 3.7%

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

      4. metadata-eval 3.7%

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

   9. Applied egg-rr 3.7%

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

       1. expm1-def 3.9%

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

       2. expm1-log1p 34.8%

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

       3. associate-*l/ 34.8%

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

       4. *-lft-identity 34.8%

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

       5. *-commutative 34.8%

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

       6. associate-*r/ 34.8%

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

   11. Simplified 34.8%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x \cdot \left(\left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {x}^{4}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 5: 89.3% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (fabs
    (* (sqrt (/ 1.0 PI)) (+ (* x 2.0) (* 0.6666666666666666 (pow x 3.0)))))
   (fabs (* (pow x 7.0) (/ 0.047619047619047616 (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * ((x * 2.0) + (0.6666666666666666 * pow(x, 3.0)))));
	} else {
		tmp = fabs((pow(x, 7.0) * (0.047619047619047616 / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * ((x * 2.0) + (0.6666666666666666 * Math.pow(x, 3.0)))));
	} else {
		tmp = Math.abs((Math.pow(x, 7.0) * (0.047619047619047616 / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.2:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * ((x * 2.0) + (0.6666666666666666 * math.pow(x, 3.0)))))
	else:
		tmp = math.fabs((math.pow(x, 7.0) * (0.047619047619047616 / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(x * 2.0) + Float64(0.6666666666666666 * (x ^ 3.0)))));
	else
		tmp = abs(Float64((x ^ 7.0) * Float64(0.047619047619047616 / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.2)
		tmp = abs((sqrt((1.0 / pi)) * ((x * 2.0) + (0.6666666666666666 * (x ^ 3.0)))));
	else
		tmp = abs(((x ^ 7.0) * (0.047619047619047616 / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(x * 2.0), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Power[x, 7.0], $MachinePrecision] * N[(0.047619047619047616 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.5%

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

   3. Simplified 99.4%

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

      1. associate-/l* 99.4%

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

      2. add-sqr-sqrt 34.0%

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

      3. fabs-sqr 34.0%

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

      4. add-sqr-sqrt 99.4%

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

      5. div-inv 99.9%

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

      6. add-sqr-sqrt 34.3%

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

      7. fabs-sqr 34.3%

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

      8. add-sqr-sqrt 99.9%

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

      9. add-sqr-sqrt 99.8%

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

      10. pow2 99.8%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 88.3%

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

      1. +-commutative 88.3%

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

      2. associate-*r* 88.3%

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

      3. associate-*r* 88.3%

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

      4. distribute-rgt-out 88.3%

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

   8. Simplified 88.3%

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

   if 2.2000000000000002 < x

   1. Initial program 99.5%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 98.3%

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

      1. *-commutative 98.3%

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

      2. *-commutative 98.3%

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

      3. associate-+r+ 98.3%

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

      4. distribute-lft-in 98.3%

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

      5. fma-def 98.3%

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

      6. rem-square-sqrt 34.0%

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

      7. fabs-sqr 34.0%

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

      8. rem-square-sqrt 76.0%

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

      9. +-commutative 76.0%

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

      10. fma-def 76.0%

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

      11. rem-square-sqrt 34.2%

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

      12. fabs-sqr 34.2%

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

      13. rem-square-sqrt 76.0%

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

      14. *-commutative 76.0%

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

   6. Simplified 98.0%

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

   7. Taylor expanded in x around inf 34.8%

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

      1. expm1-log1p-u 3.9%

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

      2. expm1-udef 3.7%

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

      3. sqrt-div 3.7%

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

      4. metadata-eval 3.7%

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

   9. Applied egg-rr 3.7%

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

       1. expm1-def 3.9%

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

       2. expm1-log1p 34.8%

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

       3. associate-*l/ 34.8%

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

       4. *-lft-identity 34.8%

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

       5. *-commutative 34.8%

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

       6. associate-*r/ 34.8%

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

   11. Simplified 34.8%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 6: 89.3% accurate, 4.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (fabs (* x (* (pow PI -0.5) (fma x (* x 0.6666666666666666) 2.0))))
   (fabs (* (pow x 7.0) (/ 0.047619047619047616 (sqrt PI))))))

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[Abs[N[(x * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * N[(x * 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Power[x, 7.0], $MachinePrecision] * N[(0.047619047619047616 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.5%

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

   3. Simplified 99.4%

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

      1. associate-/l* 99.4%

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

      2. add-sqr-sqrt 34.0%

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

      3. fabs-sqr 34.0%

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

      4. add-sqr-sqrt 99.4%

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

      5. div-inv 99.9%

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

      6. add-sqr-sqrt 34.3%

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

      7. fabs-sqr 34.3%

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

      8. add-sqr-sqrt 99.9%

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

      9. add-sqr-sqrt 99.8%

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

      10. pow2 99.8%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 88.3%

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

      1. +-commutative 88.3%

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

      2. associate-*r* 88.3%

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

      3. associate-*r* 88.3%

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

      4. distribute-rgt-out 88.3%

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

      5. unpow3 88.3%

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

      6. unpow2 88.3%

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

      7. associate-*r* 88.3%

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

      8. unpow2 88.3%

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

      9. associate-*l* 88.3%

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

      10. *-commutative 88.3%

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

      11. distribute-rgt-out 88.3%

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

      12. +-commutative 88.3%

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

      13. *-commutative 88.3%

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

      14. fma-def 88.3%

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

   8. Simplified 88.3%

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

      1. expm1-log1p-u 67.5%

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

      2. expm1-udef 5.9%

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

      3. pow1/2 5.9%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1}{\pi}\right)}^{0.5}} \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)\right)\right)} - 1\right| \]

      4. inv-pow 5.9%

         \[\leadsto \left|e^{\mathsf{log1p}\left({\color{blue}{\left({\pi}^{-1}\right)}}^{0.5} \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)\right)\right)} - 1\right| \]

      5. pow-pow 5.9%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{{\pi}^{\left(-1 \cdot 0.5\right)}} \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)\right)\right)} - 1\right| \]

      6. metadata-eval 5.9%

         \[\leadsto \left|e^{\mathsf{log1p}\left({\pi}^{\color{blue}{-0.5}} \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)\right)\right)} - 1\right| \]

   10. Applied egg-rr 5.9%

       \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)\right)\right)} - 1}\right| \]
   11. Step-by-step derivation

       1. expm1-def 67.5%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)\right)\right)\right)}\right| \]

       2. expm1-log1p 88.3%

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

       3. *-commutative 88.3%

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

       4. associate-*l* 88.3%

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

   12. Simplified 88.3%

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

   if 2.2000000000000002 < x

   1. Initial program 99.5%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 98.3%

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

      1. *-commutative 98.3%

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

      2. *-commutative 98.3%

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

      3. associate-+r+ 98.3%

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

      4. distribute-lft-in 98.3%

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

      5. fma-def 98.3%

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

      6. rem-square-sqrt 34.0%

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

      7. fabs-sqr 34.0%

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

      8. rem-square-sqrt 76.0%

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

      9. +-commutative 76.0%

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

      10. fma-def 76.0%

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

      11. rem-square-sqrt 34.2%

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

      12. fabs-sqr 34.2%

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

      13. rem-square-sqrt 76.0%

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

      14. *-commutative 76.0%

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

   6. Simplified 98.0%

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

   7. Taylor expanded in x around inf 34.8%

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

      1. expm1-log1p-u 3.9%

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

      2. expm1-udef 3.7%

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

      3. sqrt-div 3.7%

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

      4. metadata-eval 3.7%

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

   9. Applied egg-rr 3.7%

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

       1. expm1-def 3.9%

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

       2. expm1-log1p 34.8%

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

       3. associate-*l/ 34.8%

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

       4. *-lft-identity 34.8%

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

       5. *-commutative 34.8%

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

       6. associate-*r/ 34.8%

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

   11. Simplified 34.8%

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

4. Final simplification 88.3%

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

Alternative 7: 89.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(2 + x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (fabs (* (sqrt (/ 1.0 PI)) (* x (+ 2.0 (* x (* x 0.6666666666666666))))))
   (fabs (* (pow x 7.0) (/ 0.047619047619047616 (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * (x * (2.0 + (x * (x * 0.6666666666666666))))));
	} else {
		tmp = fabs((pow(x, 7.0) * (0.047619047619047616 / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * (x * (2.0 + (x * (x * 0.6666666666666666))))));
	} else {
		tmp = Math.abs((Math.pow(x, 7.0) * (0.047619047619047616 / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.2:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * (x * (2.0 + (x * (x * 0.6666666666666666))))))
	else:
		tmp = math.fabs((math.pow(x, 7.0) * (0.047619047619047616 / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(x * Float64(2.0 + Float64(x * Float64(x * 0.6666666666666666))))));
	else
		tmp = abs(Float64((x ^ 7.0) * Float64(0.047619047619047616 / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.2)
		tmp = abs((sqrt((1.0 / pi)) * (x * (2.0 + (x * (x * 0.6666666666666666))))));
	else
		tmp = abs(((x ^ 7.0) * (0.047619047619047616 / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x * N[(2.0 + N[(x * N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Power[x, 7.0], $MachinePrecision] * N[(0.047619047619047616 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.5%

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

   3. Simplified 99.4%

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

      1. associate-/l* 99.4%

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

      2. add-sqr-sqrt 34.0%

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

      3. fabs-sqr 34.0%

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

      4. add-sqr-sqrt 99.4%

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

      5. div-inv 99.9%

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

      6. add-sqr-sqrt 34.3%

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

      7. fabs-sqr 34.3%

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

      8. add-sqr-sqrt 99.9%

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

      9. add-sqr-sqrt 99.8%

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

      10. pow2 99.8%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 88.3%

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

      1. +-commutative 88.3%

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

      2. associate-*r* 88.3%

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

      3. associate-*r* 88.3%

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

      4. distribute-rgt-out 88.3%

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

      5. unpow3 88.3%

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

      6. unpow2 88.3%

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

      7. associate-*r* 88.3%

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

      8. unpow2 88.3%

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

      9. associate-*l* 88.3%

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

      10. *-commutative 88.3%

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

      11. distribute-rgt-out 88.3%

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

      12. +-commutative 88.3%

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

      13. *-commutative 88.3%

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

      14. fma-def 88.3%

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

   8. Simplified 88.3%

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

      1. fma-udef 88.3%

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

   10. Applied egg-rr 88.3%

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

   if 2.2000000000000002 < x

   1. Initial program 99.5%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 98.3%

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

      1. *-commutative 98.3%

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

      2. *-commutative 98.3%

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

      3. associate-+r+ 98.3%

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

      4. distribute-lft-in 98.3%

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

      5. fma-def 98.3%

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

      6. rem-square-sqrt 34.0%

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

      7. fabs-sqr 34.0%

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

      8. rem-square-sqrt 76.0%

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

      9. +-commutative 76.0%

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

      10. fma-def 76.0%

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

      11. rem-square-sqrt 34.2%

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

      12. fabs-sqr 34.2%

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

      13. rem-square-sqrt 76.0%

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

      14. *-commutative 76.0%

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

   6. Simplified 98.0%

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

   7. Taylor expanded in x around inf 34.8%

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

      1. expm1-log1p-u 3.9%

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

      2. expm1-udef 3.7%

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

      3. sqrt-div 3.7%

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

      4. metadata-eval 3.7%

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

   9. Applied egg-rr 3.7%

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

       1. expm1-def 3.9%

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

       2. expm1-log1p 34.8%

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

       3. associate-*l/ 34.8%

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

       4. *-lft-identity 34.8%

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

       5. *-commutative 34.8%

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

       6. associate-*r/ 34.8%

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

   11. Simplified 34.8%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(2 + x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 8: 67.8% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.75)
   (fabs (* x (/ 2.0 (sqrt PI))))
   (fabs (* (sqrt (/ 1.0 PI)) (* x (* x (* x 0.6666666666666666)))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.75) {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * (x * (x * (x * 0.6666666666666666)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.75) {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * (x * (x * (x * 0.6666666666666666)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.75:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * (x * (x * (x * 0.6666666666666666)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.75)
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(x * Float64(x * Float64(x * 0.6666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.75)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = abs((sqrt((1.0 / pi)) * (x * (x * (x * 0.6666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.75], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x * N[(x * N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.75

   1. Initial program 99.5%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 98.3%

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

      1. *-commutative 98.3%

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

      2. *-commutative 98.3%

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

      3. associate-+r+ 98.3%

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

      4. distribute-lft-in 98.3%

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

      5. fma-def 98.3%

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

      6. rem-square-sqrt 34.0%

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

      7. fabs-sqr 34.0%

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

      8. rem-square-sqrt 76.0%

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

      9. +-commutative 76.0%

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

      10. fma-def 76.0%

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

      11. rem-square-sqrt 34.2%

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

      12. fabs-sqr 34.2%

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

      13. rem-square-sqrt 76.0%

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

      14. *-commutative 76.0%

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

   6. Simplified 98.0%

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

   7. Taylor expanded in x around 0 68.8%

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

      1. expm1-log1p-u 67.0%

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

      2. expm1-udef 5.6%

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

      3. sqrt-div 5.6%

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

      4. metadata-eval 5.6%

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

      5. *-commutative 5.6%

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

   9. Applied egg-rr 5.6%

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

       1. expm1-def 67.0%

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

       2. expm1-log1p 68.8%

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

       3. associate-*l/ 68.4%

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

       4. *-lft-identity 68.4%

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

       5. *-commutative 68.4%

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

   11. Simplified 68.4%

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

       1. expm1-log1p-u 66.6%

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

       2. expm1-udef 5.6%

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

       3. associate-/l* 5.6%

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

   13. Applied egg-rr 5.6%

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

       1. expm1-def 66.6%

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

       2. expm1-log1p 68.3%

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

       3. associate-/r/ 68.8%

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

   15. Simplified 68.8%

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

   if 1.75 < x

   1. Initial program 99.5%

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

   3. Simplified 99.4%

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

      1. associate-/l* 99.4%

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

      2. add-sqr-sqrt 34.0%

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

      3. fabs-sqr 34.0%

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

      4. add-sqr-sqrt 99.4%

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

      5. div-inv 99.9%

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

      6. add-sqr-sqrt 34.3%

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

      7. fabs-sqr 34.3%

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

      8. add-sqr-sqrt 99.9%

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

      9. add-sqr-sqrt 99.8%

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

      10. pow2 99.8%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 88.3%

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

      1. +-commutative 88.3%

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

      2. associate-*r* 88.3%

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

      3. associate-*r* 88.3%

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

      4. distribute-rgt-out 88.3%

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

      5. unpow3 88.3%

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

      6. unpow2 88.3%

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

      7. associate-*r* 88.3%

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

      8. unpow2 88.3%

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

      9. associate-*l* 88.3%

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

      10. *-commutative 88.3%

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

      11. distribute-rgt-out 88.3%

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

      12. +-commutative 88.3%

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

      13. *-commutative 88.3%

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

      14. fma-def 88.3%

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

   8. Simplified 88.3%

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

   9. Taylor expanded in x around inf 25.0%

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

       1. unpow2 25.0%

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

       2. *-commutative 25.0%

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

       3. associate-*r* 25.0%

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

   11. Simplified 25.0%

       \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot 0.6666666666666666\right)\right)}\right)\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)\right|\\ \end{array} \]

Alternative 9: 89.3% accurate, 6.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (fabs (* (sqrt (/ 1.0 PI)) (* x (+ 2.0 (* x (* x 0.6666666666666666)))))))

C

double code(double x) {
	return fabs((sqrt((1.0 / ((double) M_PI))) * (x * (2.0 + (x * (x * 0.6666666666666666))))));
}

Java

public static double code(double x) {
	return Math.abs((Math.sqrt((1.0 / Math.PI)) * (x * (2.0 + (x * (x * 0.6666666666666666))))));
}

Python

def code(x):
	return math.fabs((math.sqrt((1.0 / math.pi)) * (x * (2.0 + (x * (x * 0.6666666666666666))))))

Julia

function code(x)
	return abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(x * Float64(2.0 + Float64(x * Float64(x * 0.6666666666666666))))))
end

MATLAB

function tmp = code(x)
	tmp = abs((sqrt((1.0 / pi)) * (x * (2.0 + (x * (x * 0.6666666666666666))))));
end

Wolfram

code[x_] := N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x * N[(2.0 + N[(x * N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Simplified 99.4%

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

   1. associate-/l* 99.4%

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

   2. add-sqr-sqrt 34.0%

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

   3. fabs-sqr 34.0%

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

   4. add-sqr-sqrt 99.4%

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

   5. div-inv 99.9%

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

   6. add-sqr-sqrt 34.3%

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

   7. fabs-sqr 34.3%

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

   8. add-sqr-sqrt 99.9%

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

   9. add-sqr-sqrt 99.8%

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

   10. pow2 99.8%

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

5. Applied egg-rr 99.9%

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

6. Taylor expanded in x around 0 88.3%

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

   1. +-commutative 88.3%

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

   2. associate-*r* 88.3%

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

   3. associate-*r* 88.3%

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

   4. distribute-rgt-out 88.3%

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

   5. unpow3 88.3%

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

   6. unpow2 88.3%

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

   7. associate-*r* 88.3%

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

   8. unpow2 88.3%

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

   9. associate-*l* 88.3%

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

   10. *-commutative 88.3%

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

   11. distribute-rgt-out 88.3%

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

   12. +-commutative 88.3%

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

   13. *-commutative 88.3%

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

   14. fma-def 88.3%

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

8. Simplified 88.3%

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

   1. fma-udef 88.3%

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

10. Applied egg-rr 88.3%

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

11. Final simplification 88.3%

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

Alternative 10: 67.8% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around 0 98.3%

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

   1. *-commutative 98.3%

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

   2. *-commutative 98.3%

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

   3. associate-+r+ 98.3%

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

   4. distribute-lft-in 98.3%

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

   5. fma-def 98.3%

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

   6. rem-square-sqrt 34.0%

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

   7. fabs-sqr 34.0%

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

   8. rem-square-sqrt 76.0%

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

   9. +-commutative 76.0%

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

   10. fma-def 76.0%

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

   11. rem-square-sqrt 34.2%

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

   12. fabs-sqr 34.2%

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

   13. rem-square-sqrt 76.0%

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

   14. *-commutative 76.0%

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

6. Simplified 98.0%

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

7. Taylor expanded in x around 0 68.8%

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

   1. expm1-log1p-u 67.0%

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

   2. expm1-udef 5.6%

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

   3. sqrt-div 5.6%

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

   4. metadata-eval 5.6%

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

   5. *-commutative 5.6%

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

9. Applied egg-rr 5.6%

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

    1. expm1-def 67.0%

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

    2. expm1-log1p 68.8%

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

    3. associate-*l/ 68.4%

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

    4. *-lft-identity 68.4%

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

    5. *-commutative 68.4%

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

11. Simplified 68.4%

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

    1. expm1-log1p-u 66.6%

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

    2. expm1-udef 5.6%

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

    3. associate-/l* 5.6%

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

13. Applied egg-rr 5.6%

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

    1. expm1-def 66.6%

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

    2. expm1-log1p 68.3%

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

    3. associate-/r/ 68.8%

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

15. Simplified 68.8%

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

16. Final simplification 68.8%

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

Reproduce

herbie shell --seed 2023263 
(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)))))))