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

Percentage Accurate: 99.8% → 99.9%

Time: 9.4s

Alternatives: 12

Speedup: 3.7×

• 29.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return Math.abs(((x * Math.sqrt((1.0 / Math.PI))) * (((x * (x * 0.6666666666666666)) + 2.0) + ((0.2 * Math.pow(x, 4.0)) + (0.047619047619047616 * Math.pow(x, 6.0))))));
}

Python

def code(x):
	return math.fabs(((x * math.sqrt((1.0 / math.pi))) * (((x * (x * 0.6666666666666666)) + 2.0) + ((0.2 * math.pow(x, 4.0)) + (0.047619047619047616 * math.pow(x, 6.0))))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = abs(((x * sqrt((1.0 / pi))) * (((x * (x * 0.6666666666666666)) + 2.0) + ((0.2 * (x ^ 4.0)) + (0.047619047619047616 * (x ^ 6.0))))));
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around 0 99.8%

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

   1. unpow1 99.8%

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

   2. sqr-pow 34.1%

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

   3. fabs-sqr 34.1%

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

   4. sqr-pow 99.8%

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

   5. unpow1 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in x around 0 99.8%

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

   1. fma-udef 99.8%

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

   2. associate-*r* 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 99.4% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around 0 99.8%

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

   1. unpow1 99.8%

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

   2. sqr-pow 34.1%

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

   3. fabs-sqr 34.1%

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

   4. sqr-pow 99.8%

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

   5. unpow1 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in x around 0 99.8%

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

   1. fma-udef 99.8%

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

   2. associate-*r* 99.8%

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

9. Applied egg-rr 99.8%

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

    1. expm1-log1p-u 68.6%

       \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \cdot \left(\left(\left(0.6666666666666666 \cdot x\right) \cdot x + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    2. expm1-udef 5.3%

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

    3. sqrt-div 5.3%

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

    4. metadata-eval 5.3%

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

    5. un-div-inv 5.3%

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

11. Applied egg-rr 5.3%

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

    1. expm1-def 68.2%

       \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}}\right)\right)} \cdot \left(\left(\left(0.6666666666666666 \cdot x\right) \cdot x + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

    2. expm1-log1p 99.4%

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

13. Simplified 99.4%

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

14. Final simplification 99.4%

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

Alternative 3: 98.4% accurate, 3.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (fabs (/ (fma 2.0 x (* 0.047619047619047616 (pow x 7.0))) (sqrt PI))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around inf 98.9%

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

5. Final simplification 98.9%

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

Alternative 4: 74.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ t_1 := x \cdot \left(x \cdot 0.6666666666666666\right)\\ \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|t_0 \cdot \left(x \cdot \frac{t_1 \cdot t_1 - 4}{t_1 - 2}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_0 \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))) (t_1 (* x (* x 0.6666666666666666))))
   (if (<= x 2.2)
     (fabs (* t_0 (* x (/ (- (* t_1 t_1) 4.0) (- t_1 2.0)))))
     (fabs (* t_0 (* 0.047619047619047616 (pow x 7.0)))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double t_1 = x * (x * 0.6666666666666666);
	double tmp;
	if (x <= 2.2) {
		tmp = fabs((t_0 * (x * (((t_1 * t_1) - 4.0) / (t_1 - 2.0)))));
	} else {
		tmp = fabs((t_0 * (0.047619047619047616 * pow(x, 7.0))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	double t_1 = x * (x * 0.6666666666666666);
	double tmp;
	if (x <= 2.2) {
		tmp = Math.abs((t_0 * (x * (((t_1 * t_1) - 4.0) / (t_1 - 2.0)))));
	} else {
		tmp = Math.abs((t_0 * (0.047619047619047616 * Math.pow(x, 7.0))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / math.pi))
	t_1 = x * (x * 0.6666666666666666)
	tmp = 0
	if x <= 2.2:
		tmp = math.fabs((t_0 * (x * (((t_1 * t_1) - 4.0) / (t_1 - 2.0)))))
	else:
		tmp = math.fabs((t_0 * (0.047619047619047616 * math.pow(x, 7.0))))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / pi))
	t_1 = Float64(x * Float64(x * 0.6666666666666666))
	tmp = 0.0
	if (x <= 2.2)
		tmp = abs(Float64(t_0 * Float64(x * Float64(Float64(Float64(t_1 * t_1) - 4.0) / Float64(t_1 - 2.0)))));
	else
		tmp = abs(Float64(t_0 * Float64(0.047619047619047616 * (x ^ 7.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 / pi));
	t_1 = x * (x * 0.6666666666666666);
	tmp = 0.0;
	if (x <= 2.2)
		tmp = abs((t_0 * (x * (((t_1 * t_1) - 4.0) / (t_1 - 2.0)))));
	else
		tmp = abs((t_0 * (0.047619047619047616 * (x ^ 7.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.2], N[Abs[N[(t$95$0 * N[(x * N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - 4.0), $MachinePrecision] / N[(t$95$1 - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$0 * N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 91.4%

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

      1. associate-*r* 91.4%

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

      2. unpow2 91.4%

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

      3. associate-*r* 91.4%

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

      4. distribute-rgt-out 91.4%

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

      5. +-commutative 91.4%

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

      6. associate-*r* 91.4%

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

      7. *-commutative 91.4%

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

      8. associate-*l* 91.4%

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

      9. *-commutative 91.4%

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

      10. distribute-lft-in 91.4%

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

      11. fma-udef 91.4%

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

   6. Simplified 91.4%

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

      1. fma-udef 91.4%

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

      2. flip-+ 77.9%

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

      3. associate-*r* 77.9%

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

      4. associate-*r* 77.9%

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

      5. metadata-eval 77.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \frac{\left(\left(0.6666666666666666 \cdot x\right) \cdot x\right) \cdot \left(\left(0.6666666666666666 \cdot x\right) \cdot x\right) - \color{blue}{4}}{0.6666666666666666 \cdot \left(x \cdot x\right) - 2}\right)\right| \]

      6. associate-*r* 77.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \frac{\left(\left(0.6666666666666666 \cdot x\right) \cdot x\right) \cdot \left(\left(0.6666666666666666 \cdot x\right) \cdot x\right) - 4}{\color{blue}{\left(0.6666666666666666 \cdot x\right) \cdot x} - 2}\right)\right| \]

   8. Applied egg-rr 77.9%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \color{blue}{\frac{\left(\left(0.6666666666666666 \cdot x\right) \cdot x\right) \cdot \left(\left(0.6666666666666666 \cdot x\right) \cdot x\right) - 4}{\left(0.6666666666666666 \cdot x\right) \cdot x - 2}}\right)\right| \]

   if 2.2000000000000002 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 34.8%

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

      1. *-commutative 34.8%

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

      2. associate-*l* 34.8%

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

      3. *-commutative 34.8%

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

      4. unpow1 34.8%

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

      5. sqr-pow 1.9%

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

      6. fabs-sqr 1.9%

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

      7. sqr-pow 34.8%

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

      8. unpow1 34.8%

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

      9. pow-plus 34.8%

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

      10. unpow1 34.8%

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

      11. sqr-pow 1.9%

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

      12. fabs-sqr 1.9%

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

      13. sqr-pow 34.8%

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

      14. unpow1 34.8%

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

      15. metadata-eval 34.8%

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

      16. associate-*l* 34.8%

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

      17. *-commutative 34.8%

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

      18. associate-*l* 34.8%

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

      19. *-commutative 34.8%

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

   6. Simplified 34.8%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \frac{\left(x \cdot \left(x \cdot 0.6666666666666666\right)\right) \cdot \left(x \cdot \left(x \cdot 0.6666666666666666\right)\right) - 4}{x \cdot \left(x \cdot 0.6666666666666666\right) - 2}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \end{array} \]

Alternative 5: 74.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot 0.6666666666666666\right)\\ \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \frac{t_0 \cdot t_0 - 4}{t_0 - 2}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x 0.6666666666666666))))
   (if (<= x 2.2)
     (fabs (* (sqrt (/ 1.0 PI)) (* x (/ (- (* t_0 t_0) 4.0) (- t_0 2.0)))))
     (fabs (/ (* 0.047619047619047616 (pow x 7.0)) (sqrt PI))))))

C

double code(double x) {
	double t_0 = x * (x * 0.6666666666666666);
	double tmp;
	if (x <= 2.2) {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * (x * (((t_0 * t_0) - 4.0) / (t_0 - 2.0)))));
	} else {
		tmp = fabs(((0.047619047619047616 * pow(x, 7.0)) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = x * (x * 0.6666666666666666);
	double tmp;
	if (x <= 2.2) {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * (x * (((t_0 * t_0) - 4.0) / (t_0 - 2.0)))));
	} else {
		tmp = Math.abs(((0.047619047619047616 * Math.pow(x, 7.0)) / Math.sqrt(Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * 0.6666666666666666)
	tmp = 0
	if x <= 2.2:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * (x * (((t_0 * t_0) - 4.0) / (t_0 - 2.0)))))
	else:
		tmp = math.fabs(((0.047619047619047616 * math.pow(x, 7.0)) / math.sqrt(math.pi)))
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * 0.6666666666666666))
	tmp = 0.0
	if (x <= 2.2)
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(x * Float64(Float64(Float64(t_0 * t_0) - 4.0) / Float64(t_0 - 2.0)))));
	else
		tmp = abs(Float64(Float64(0.047619047619047616 * (x ^ 7.0)) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * 0.6666666666666666);
	tmp = 0.0;
	if (x <= 2.2)
		tmp = abs((sqrt((1.0 / pi)) * (x * (((t_0 * t_0) - 4.0) / (t_0 - 2.0)))));
	else
		tmp = abs(((0.047619047619047616 * (x ^ 7.0)) / sqrt(pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.2], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x * N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - 4.0), $MachinePrecision] / N[(t$95$0 - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 91.4%

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

      1. associate-*r* 91.4%

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

      2. unpow2 91.4%

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

      3. associate-*r* 91.4%

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

      4. distribute-rgt-out 91.4%

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

      5. +-commutative 91.4%

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

      6. associate-*r* 91.4%

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

      7. *-commutative 91.4%

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

      8. associate-*l* 91.4%

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

      9. *-commutative 91.4%

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

      10. distribute-lft-in 91.4%

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

      11. fma-udef 91.4%

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

   6. Simplified 91.4%

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

      1. fma-udef 91.4%

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

      2. flip-+ 77.9%

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

      3. associate-*r* 77.9%

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

      4. associate-*r* 77.9%

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

      5. metadata-eval 77.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \frac{\left(\left(0.6666666666666666 \cdot x\right) \cdot x\right) \cdot \left(\left(0.6666666666666666 \cdot x\right) \cdot x\right) - \color{blue}{4}}{0.6666666666666666 \cdot \left(x \cdot x\right) - 2}\right)\right| \]

      6. associate-*r* 77.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \frac{\left(\left(0.6666666666666666 \cdot x\right) \cdot x\right) \cdot \left(\left(0.6666666666666666 \cdot x\right) \cdot x\right) - 4}{\color{blue}{\left(0.6666666666666666 \cdot x\right) \cdot x} - 2}\right)\right| \]

   8. Applied egg-rr 77.9%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \color{blue}{\frac{\left(\left(0.6666666666666666 \cdot x\right) \cdot x\right) \cdot \left(\left(0.6666666666666666 \cdot x\right) \cdot x\right) - 4}{\left(0.6666666666666666 \cdot x\right) \cdot x - 2}}\right)\right| \]

   if 2.2000000000000002 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 35.1%

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

      1. associate-*r* 35.1%

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

      2. associate-*r* 35.1%

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

      3. distribute-rgt-out 35.1%

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

      4. fma-def 35.1%

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

   6. Simplified 35.1%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)}\right| \]
   7. Step-by-step derivation

      1. expm1-log1p-u 4.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}\right| \]

      2. expm1-udef 3.7%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)} - 1}\right| \]

      3. *-commutative 3.7%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}}}\right)} - 1\right| \]

      4. sqrt-div 3.7%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)} - 1\right| \]

      5. metadata-eval 3.7%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right) \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)} - 1\right| \]

      6. un-div-inv 3.7%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)}{\sqrt{\pi}}}\right)} - 1\right| \]

   8. Applied egg-rr 3.7%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)}{\sqrt{\pi}}\right)} - 1}\right| \]
   9. Step-by-step derivation

      1. expm1-def 4.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)}{\sqrt{\pi}}\right)\right)}\right| \]

      2. expm1-log1p 35.1%

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

      3. fma-def 35.1%

         \[\leadsto \left|\frac{\color{blue}{0.047619047619047616 \cdot {x}^{7} + 0.2 \cdot {x}^{5}}}{\sqrt{\pi}}\right| \]

      4. +-commutative 35.1%

         \[\leadsto \left|\frac{\color{blue}{0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}}}{\sqrt{\pi}}\right| \]

      5. fma-def 35.1%

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

   10. Simplified 35.1%

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

   11. Taylor expanded in x around inf 34.8%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \frac{\left(x \cdot \left(x \cdot 0.6666666666666666\right)\right) \cdot \left(x \cdot \left(x \cdot 0.6666666666666666\right)\right) - 4}{x \cdot \left(x \cdot 0.6666666666666666\right) - 2}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 6: 74.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ t_1 := x \cdot \left(x \cdot 0.6666666666666666\right)\\ \mathbf{if}\;x \leq 10^{+102}:\\ \;\;\;\;\left|t_0 \cdot \left(x \cdot \frac{t_1 \cdot t_1 - 4}{t_1 - 2}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_0 \cdot \left(x \cdot t_1\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))) (t_1 (* x (* x 0.6666666666666666))))
   (if (<= x 1e+102)
     (fabs (* t_0 (* x (/ (- (* t_1 t_1) 4.0) (- t_1 2.0)))))
     (fabs (* t_0 (* x t_1))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double t_1 = x * (x * 0.6666666666666666);
	double tmp;
	if (x <= 1e+102) {
		tmp = fabs((t_0 * (x * (((t_1 * t_1) - 4.0) / (t_1 - 2.0)))));
	} else {
		tmp = fabs((t_0 * (x * t_1)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	double t_1 = x * (x * 0.6666666666666666);
	double tmp;
	if (x <= 1e+102) {
		tmp = Math.abs((t_0 * (x * (((t_1 * t_1) - 4.0) / (t_1 - 2.0)))));
	} else {
		tmp = Math.abs((t_0 * (x * t_1)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / math.pi))
	t_1 = x * (x * 0.6666666666666666)
	tmp = 0
	if x <= 1e+102:
		tmp = math.fabs((t_0 * (x * (((t_1 * t_1) - 4.0) / (t_1 - 2.0)))))
	else:
		tmp = math.fabs((t_0 * (x * t_1)))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / pi))
	t_1 = Float64(x * Float64(x * 0.6666666666666666))
	tmp = 0.0
	if (x <= 1e+102)
		tmp = abs(Float64(t_0 * Float64(x * Float64(Float64(Float64(t_1 * t_1) - 4.0) / Float64(t_1 - 2.0)))));
	else
		tmp = abs(Float64(t_0 * Float64(x * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 / pi));
	t_1 = x * (x * 0.6666666666666666);
	tmp = 0.0;
	if (x <= 1e+102)
		tmp = abs((t_0 * (x * (((t_1 * t_1) - 4.0) / (t_1 - 2.0)))));
	else
		tmp = abs((t_0 * (x * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1e+102], N[Abs[N[(t$95$0 * N[(x * N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - 4.0), $MachinePrecision] / N[(t$95$1 - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$0 * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999977e101

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 91.4%

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

      1. associate-*r* 91.4%

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

      2. unpow2 91.4%

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

      3. associate-*r* 91.4%

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

      4. distribute-rgt-out 91.4%

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

      5. +-commutative 91.4%

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

      6. associate-*r* 91.4%

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

      7. *-commutative 91.4%

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

      8. associate-*l* 91.4%

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

      9. *-commutative 91.4%

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

      10. distribute-lft-in 91.4%

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

      11. fma-udef 91.4%

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

   6. Simplified 91.4%

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

      1. fma-udef 91.4%

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

      2. flip-+ 77.9%

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

      3. associate-*r* 77.9%

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

      4. associate-*r* 77.9%

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

      5. metadata-eval 77.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \frac{\left(\left(0.6666666666666666 \cdot x\right) \cdot x\right) \cdot \left(\left(0.6666666666666666 \cdot x\right) \cdot x\right) - \color{blue}{4}}{0.6666666666666666 \cdot \left(x \cdot x\right) - 2}\right)\right| \]

      6. associate-*r* 77.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \frac{\left(\left(0.6666666666666666 \cdot x\right) \cdot x\right) \cdot \left(\left(0.6666666666666666 \cdot x\right) \cdot x\right) - 4}{\color{blue}{\left(0.6666666666666666 \cdot x\right) \cdot x} - 2}\right)\right| \]

   8. Applied egg-rr 77.9%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \color{blue}{\frac{\left(\left(0.6666666666666666 \cdot x\right) \cdot x\right) \cdot \left(\left(0.6666666666666666 \cdot x\right) \cdot x\right) - 4}{\left(0.6666666666666666 \cdot x\right) \cdot x - 2}}\right)\right| \]

   if 9.99999999999999977e101 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 91.4%

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

      1. associate-*r* 91.4%

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

      2. unpow2 91.4%

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

      3. associate-*r* 91.4%

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

      4. distribute-rgt-out 91.4%

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

      5. +-commutative 91.4%

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

      6. associate-*r* 91.4%

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

      7. *-commutative 91.4%

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

      8. associate-*l* 91.4%

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

      9. *-commutative 91.4%

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

      10. distribute-lft-in 91.4%

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

      11. fma-udef 91.4%

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

   6. Simplified 91.4%

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

   7. Taylor expanded in x around inf 27.2%

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

      1. *-commutative 27.2%

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

      2. unpow2 27.2%

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

      3. associate-*r* 27.2%

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

   9. Simplified 27.2%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+102}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \frac{\left(x \cdot \left(x \cdot 0.6666666666666666\right)\right) \cdot \left(x \cdot \left(x \cdot 0.6666666666666666\right)\right) - 4}{x \cdot \left(x \cdot 0.6666666666666666\right) - 2}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)\right|\\ \end{array} \]

Alternative 7: 89.4% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around 0 91.4%

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

   1. associate-*r* 91.4%

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

   2. unpow2 91.4%

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

   3. associate-*r* 91.4%

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

   4. distribute-rgt-out 91.4%

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

   5. +-commutative 91.4%

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

   6. associate-*r* 91.4%

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

   7. *-commutative 91.4%

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

   8. associate-*l* 91.4%

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

   9. *-commutative 91.4%

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

   10. distribute-lft-in 91.4%

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

   11. fma-udef 91.4%

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

6. Simplified 91.4%

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

   1. fma-udef 91.4%

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

   2. distribute-lft-in 91.4%

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

   3. associate-*r* 91.4%

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

8. Applied egg-rr 91.4%

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

   1. distribute-lft-in 91.4%

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

   2. inv-pow 91.4%

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

   3. sqrt-pow1 91.4%

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

   4. metadata-eval 91.4%

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

   5. *-commutative 91.4%

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

   6. *-commutative 91.4%

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

   7. inv-pow 91.4%

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

   8. sqrt-pow1 91.4%

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

   9. metadata-eval 91.4%

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

10. Applied egg-rr 91.4%

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

    1. distribute-lft-out 91.4%

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

    2. associate-*r* 91.4%

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

    3. *-commutative 91.4%

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

12. Simplified 91.4%

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

13. Final simplification 91.4%

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

Alternative 8: 67.4% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.75)
   (fabs (* (pow PI -0.5) (* x 2.0)))
   (fabs (* (sqrt (/ 1.0 PI)) (* x (* x (* x 0.6666666666666666)))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.75) {
		tmp = fabs((pow(((double) M_PI), -0.5) * (x * 2.0)));
	} 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((Math.pow(Math.PI, -0.5) * (x * 2.0)));
	} 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((math.pow(math.pi, -0.5) * (x * 2.0)))
	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((pi ^ -0.5) * Float64(x * 2.0)));
	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(((pi ^ -0.5) * (x * 2.0)));
	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[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * 2.0), $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 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 70.1%

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

      1. associate-*r* 70.1%

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

      2. *-commutative 70.1%

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

      3. *-commutative 70.1%

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

      4. unpow1 70.1%

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

      5. sqr-pow 33.8%

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

      6. fabs-sqr 33.8%

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

      7. sqr-pow 70.1%

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

      8. unpow1 70.1%

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

   6. Simplified 70.1%

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

      1. *-commutative 70.1%

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

      2. sqrt-div 70.1%

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

      3. metadata-eval 70.1%

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

      4. un-div-inv 69.7%

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

   8. Applied egg-rr 69.7%

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

      1. div-inv 70.1%

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

      2. pow1/2 70.1%

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

      3. pow-flip 70.1%

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

      4. metadata-eval 70.1%

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

   10. Applied egg-rr 70.1%

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

   if 1.75 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 91.4%

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

      1. associate-*r* 91.4%

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

      2. unpow2 91.4%

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

      3. associate-*r* 91.4%

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

      4. distribute-rgt-out 91.4%

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

      5. +-commutative 91.4%

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

      6. associate-*r* 91.4%

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

      7. *-commutative 91.4%

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

      8. associate-*l* 91.4%

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

      9. *-commutative 91.4%

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

      10. distribute-lft-in 91.4%

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

      11. fma-udef 91.4%

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

   6. Simplified 91.4%

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

   7. Taylor expanded in x around inf 27.2%

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

      1. *-commutative 27.2%

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

      2. unpow2 27.2%

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

      3. associate-*r* 27.2%

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

   9. Simplified 27.2%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.6666666666666666\right)\right)\right)\right|\\ \end{array} \]

Alternative 9: 89.4% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around 0 91.4%

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

   1. associate-*r* 91.4%

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

   2. unpow2 91.4%

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

   3. associate-*r* 91.4%

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

   4. distribute-rgt-out 91.4%

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

   5. +-commutative 91.4%

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

   6. associate-*r* 91.4%

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

   7. *-commutative 91.4%

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

   8. associate-*l* 91.4%

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

   9. *-commutative 91.4%

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

   10. distribute-lft-in 91.4%

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

   11. fma-udef 91.4%

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

6. Simplified 91.4%

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

   1. fma-udef 99.8%

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

   2. associate-*r* 99.8%

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

8. Applied egg-rr 91.4%

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

9. Final simplification 91.4%

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

Alternative 10: 67.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-13}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{4 \cdot \left(x \cdot x\right)}{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4e-13)
   (fabs (* (pow PI -0.5) (* x 2.0)))
   (fabs (sqrt (/ (* 4.0 (* x x)) PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 4e-13) {
		tmp = fabs((pow(((double) M_PI), -0.5) * (x * 2.0)));
	} else {
		tmp = fabs(sqrt(((4.0 * (x * x)) / ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 4e-13) {
		tmp = Math.abs((Math.pow(Math.PI, -0.5) * (x * 2.0)));
	} else {
		tmp = Math.abs(Math.sqrt(((4.0 * (x * x)) / Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4e-13:
		tmp = math.fabs((math.pow(math.pi, -0.5) * (x * 2.0)))
	else:
		tmp = math.fabs(math.sqrt(((4.0 * (x * x)) / math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4e-13)
		tmp = abs(Float64((pi ^ -0.5) * Float64(x * 2.0)));
	else
		tmp = abs(sqrt(Float64(Float64(4.0 * Float64(x * x)) / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4e-13)
		tmp = abs(((pi ^ -0.5) * (x * 2.0)));
	else
		tmp = abs(sqrt(((4.0 * (x * x)) / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4e-13], N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[Sqrt[N[(N[(4.0 * N[(x * x), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.0000000000000001e-13

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 70.2%

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

      1. associate-*r* 70.2%

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

      2. *-commutative 70.2%

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

      3. *-commutative 70.2%

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

      4. unpow1 70.2%

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

      5. sqr-pow 33.8%

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

      6. fabs-sqr 33.8%

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

      7. sqr-pow 70.2%

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

      8. unpow1 70.2%

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

   6. Simplified 70.2%

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

      1. *-commutative 70.2%

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

      2. sqrt-div 70.2%

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

      3. metadata-eval 70.2%

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

      4. un-div-inv 69.8%

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

   8. Applied egg-rr 69.8%

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

      1. div-inv 70.2%

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

      2. pow1/2 70.2%

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

      3. pow-flip 70.2%

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

      4. metadata-eval 70.2%

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

   10. Applied egg-rr 70.2%

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

   if 4.0000000000000001e-13 < x

   1. Initial program 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in x around 0 43.2%

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

      1. associate-*r* 43.2%

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

      2. *-commutative 43.2%

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

      3. *-commutative 43.2%

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

      4. unpow1 43.2%

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

      5. sqr-pow 43.2%

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

      6. fabs-sqr 43.2%

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

      7. sqr-pow 43.2%

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

      8. unpow1 43.2%

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

   6. Simplified 43.2%

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

      1. *-commutative 43.2%

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

      2. sqrt-div 43.2%

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

      3. metadata-eval 43.2%

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

      4. un-div-inv 43.2%

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

   8. Applied egg-rr 43.2%

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

      1. add-sqr-sqrt 43.2%

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

      2. sqrt-unprod 43.2%

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

      3. frac-times 43.2%

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

      4. *-commutative 43.2%

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

      5. *-commutative 43.2%

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

      6. swap-sqr 43.2%

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

      7. metadata-eval 43.2%

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

      8. add-sqr-sqrt 43.2%

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

   10. Applied egg-rr 43.2%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-13}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{4 \cdot \left(x \cdot x\right)}{\pi}}\right|\\ \end{array} \]

Alternative 11: 67.4% accurate, 6.4× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (fabs (* (pow PI -0.5) (* x 2.0))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around 0 70.1%

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

   1. associate-*r* 70.1%

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

   2. *-commutative 70.1%

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

   3. *-commutative 70.1%

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

   4. unpow1 70.1%

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

   5. sqr-pow 33.8%

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

   6. fabs-sqr 33.8%

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

   7. sqr-pow 70.1%

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

   8. unpow1 70.1%

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

6. Simplified 70.1%

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

   1. *-commutative 70.1%

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

   2. sqrt-div 70.1%

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

   3. metadata-eval 70.1%

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

   4. un-div-inv 69.7%

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

8. Applied egg-rr 69.7%

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

   1. div-inv 70.1%

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

   2. pow1/2 70.1%

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

   3. pow-flip 70.1%

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

   4. metadata-eval 70.1%

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

10. Applied egg-rr 70.1%

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

11. Final simplification 70.1%

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

Alternative 12: 66.9% accurate, 6.4× speedup

Math

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

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(Float64(x * 2.0) / sqrt(pi)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around 0 70.1%

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

   1. associate-*r* 70.1%

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

   2. *-commutative 70.1%

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

   3. *-commutative 70.1%

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

   4. unpow1 70.1%

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

   5. sqr-pow 33.8%

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

   6. fabs-sqr 33.8%

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

   7. sqr-pow 70.1%

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

   8. unpow1 70.1%

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

6. Simplified 70.1%

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

   1. *-commutative 70.1%

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

   2. sqrt-div 70.1%

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

   3. metadata-eval 70.1%

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

   4. un-div-inv 69.7%

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

8. Applied egg-rr 69.7%

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

9. Final simplification 69.7%

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

Reproduce

herbie shell --seed 2023261 
(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)))))))