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

Percentage Accurate: 99.8% → 99.9%

Time: 33.2s

Alternatives: 12

Speedup: 0.6×

• 71.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq \frac{1}{2}\]

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
       (t_1 (* (* t_0 (fabs x)) (fabs x))))
  (fabs
   (*
    (/ 1 (sqrt PI))
    (+
     (+ (+ (* 2 (fabs x)) (* (/ 2 3) t_0)) (* (/ 1 5) t_1))
     (* (/ 1 21) (* (* 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 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2 / 3), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1 / 5), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1 / 21), $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

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

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
       (t_1 (* (* t_0 (fabs x)) (fabs x))))
  (fabs
   (*
    (/ 1 (sqrt PI))
    (+
     (+ (+ (* 2 (fabs x)) (* (/ 2 3) t_0)) (* (/ 1 5) t_1))
     (* (/ 1 21) (* (* 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 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2 / 3), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1 / 5), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1 / 21), $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, 0.6× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (pow (fabs x) 5)))
  (fabs
   (*
    (/ 1 (sqrt PI))
    (+
     (+
      (+
       (* 2 (fabs x))
       (* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x))))
      (* (/ 1 5) t_0))
     (* (/ 1 21) (* (* t_0 (fabs x)) (fabs x))))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Abs[x], $MachinePrecision], 5], $MachinePrecision]}, N[Abs[N[(N[(1 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2 / 3), $MachinePrecision] * N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1 / 5), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1 / 21), $MachinePrecision] * N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. pow3 N/A

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

   7. pow2 N/A

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

   8. pow-prod-up N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 99.8%

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

3. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \color{blue}{{\left(\left|x\right|\right)}^{5}}\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. pow3 N/A

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

   7. pow2 N/A

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

   8. pow-prod-up N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 99.9%

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

5. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5}\right) + \frac{1}{21} \cdot \left(\left(\color{blue}{{\left(\left|x\right|\right)}^{5}} \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
6. Add Preprocessing

Alternative 2: 99.8% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (* x x) x)))
  (*
   (/ 1 (sqrt PI))
   (fabs
    (+
     (* (fabs x) (+ (* (* 1/5 (* x x)) (* x x)) (* (* t_0 t_0) 1/21)))
     (* (fabs x) (+ (* (* x x) 2/3) 2)))))))

C

double code(double x) {
	double t_0 = (x * x) * x;
	return (1.0 / sqrt(((double) M_PI))) * fabs(((fabs(x) * (((0.2 * (x * x)) * (x * x)) + ((t_0 * t_0) * 0.047619047619047616))) + (fabs(x) * (((x * x) * 0.6666666666666666) + 2.0))));
}

Java

public static double code(double x) {
	double t_0 = (x * x) * x;
	return (1.0 / Math.sqrt(Math.PI)) * Math.abs(((Math.abs(x) * (((0.2 * (x * x)) * (x * x)) + ((t_0 * t_0) * 0.047619047619047616))) + (Math.abs(x) * (((x * x) * 0.6666666666666666) + 2.0))));
}

Python

def code(x):
	t_0 = (x * x) * x
	return (1.0 / math.sqrt(math.pi)) * math.fabs(((math.fabs(x) * (((0.2 * (x * x)) * (x * x)) + ((t_0 * t_0) * 0.047619047619047616))) + (math.fabs(x) * (((x * x) * 0.6666666666666666) + 2.0))))

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	return Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(Float64(abs(x) * Float64(Float64(Float64(0.2 * Float64(x * x)) * Float64(x * x)) + Float64(Float64(t_0 * t_0) * 0.047619047619047616))) + Float64(abs(x) * Float64(Float64(Float64(x * x) * 0.6666666666666666) + 2.0)))))
end

MATLAB

function tmp = code(x)
	t_0 = (x * x) * x;
	tmp = (1.0 / sqrt(pi)) * abs(((abs(x) * (((0.2 * (x * x)) * (x * x)) + ((t_0 * t_0) * 0.047619047619047616))) + (abs(x) * (((x * x) * 0.6666666666666666) + 2.0))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, N[(N[(1 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(N[(N[Abs[x], $MachinePrecision] * N[(N[(N[(1/5 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 1/21), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 2/3), $MachinePrecision] + 2), $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. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{\pi}} \cdot \left|\left|x\right| \cdot \left(\left(\frac{1}{5} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \frac{1}{21}\right) + \left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{3} + 2\right)\right|} \]
4. Add Preprocessing

Alternative 3: 99.6% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (fabs
 (*
  (-
   (* x (+ (* (* (* (* (* x x) x) x) x) 1/21) (* (* 1/5 (* x x)) x)))
   (- -2 (* 2/3 (* x x))))
  (* (/ (fabs x) PI) (sqrt PI)))))

C

double code(double x) {
	return fabs((((x * ((((((x * x) * x) * x) * x) * 0.047619047619047616) + ((0.2 * (x * x)) * x))) - (-2.0 - (0.6666666666666666 * (x * x)))) * ((fabs(x) / ((double) M_PI)) * sqrt(((double) M_PI)))));
}

Java

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

Python

def code(x):
	return math.fabs((((x * ((((((x * x) * x) * x) * x) * 0.047619047619047616) + ((0.2 * (x * x)) * x))) - (-2.0 - (0.6666666666666666 * (x * x)))) * ((math.fabs(x) / math.pi) * math.sqrt(math.pi))))

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(N[(N[(x * N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 1/21), $MachinePrecision] + N[(N[(1/5 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-2 - N[(2/3 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[x], $MachinePrecision] / Pi), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. fp-cancel-sign-sub-inv N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) - \left(\mathsf{neg}\left(\frac{1}{5}\right)\right) \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)} + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   4. lower--.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) - \left(\mathsf{neg}\left(\frac{1}{5}\right)\right) \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)} + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{\left(\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{3} + 2\right) - \left(\frac{-1}{5} \cdot \left|x\right|\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right)} + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. mul-fabs N/A

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

   6. lift-*.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. mul-fabs N/A

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

   9. lift-*.f64 N/A

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

   10. rem-sqrt-square-rev N/A

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

   11. sqr-abs-rev N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. mul-fabs N/A

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

   15. lift-*.f64 N/A

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

   16. mul-fabs N/A

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

   17. lift-fabs.f64 N/A

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

   18. lift-fabs.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. lift-fabs.f64 N/A

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

   21. lift-*.f64 N/A

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

   22. pow2 N/A

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.6%

   \[\leadsto \left|\color{blue}{\left(x \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \frac{1}{21} + \left(\frac{1}{5} \cdot \left(x \cdot x\right)\right) \cdot x\right) - \left(-2 - \frac{2}{3} \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\frac{\left|x\right|}{\pi} \cdot \sqrt{\pi}\right)}\right| \]
8. Add Preprocessing

Alternative 4: 99.4% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (/
 (fabs
  (*
   (-
    (* x (+ (* (* (* (* (* x x) x) x) x) 1/21) (* (* 1/5 (* x x)) x)))
    (- -2 (* 2/3 (* x x))))
   (fabs x)))
 (sqrt PI)))

C

double code(double x) {
	return fabs((((x * ((((((x * x) * x) * x) * x) * 0.047619047619047616) + ((0.2 * (x * x)) * x))) - (-2.0 - (0.6666666666666666 * (x * x)))) * fabs(x))) / sqrt(((double) M_PI));
}

Java

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

Python

def code(x):
	return math.fabs((((x * ((((((x * x) * x) * x) * x) * 0.047619047619047616) + ((0.2 * (x * x)) * x))) - (-2.0 - (0.6666666666666666 * (x * x)))) * math.fabs(x))) / math.sqrt(math.pi)

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Abs[N[(N[(N[(x * N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 1/21), $MachinePrecision] + N[(N[(1/5 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-2 - N[(2/3 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. fp-cancel-sign-sub-inv N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) - \left(\mathsf{neg}\left(\frac{1}{5}\right)\right) \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)} + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   4. lower--.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) - \left(\mathsf{neg}\left(\frac{1}{5}\right)\right) \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)} + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{\left(\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{3} + 2\right) - \left(\frac{-1}{5} \cdot \left|x\right|\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right)} + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. mul-fabs N/A

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

   6. lift-*.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. mul-fabs N/A

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

   9. lift-*.f64 N/A

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

   10. rem-sqrt-square-rev N/A

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

   11. sqr-abs-rev N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. mul-fabs N/A

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

   15. lift-*.f64 N/A

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

   16. mul-fabs N/A

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

   17. lift-fabs.f64 N/A

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

   18. lift-fabs.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. lift-fabs.f64 N/A

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

   21. lift-*.f64 N/A

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

   22. pow2 N/A

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\left|\left(x \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \frac{1}{21} + \left(\frac{1}{5} \cdot \left(x \cdot x\right)\right) \cdot x\right) - \left(-2 - \frac{2}{3} \cdot \left(x \cdot x\right)\right)\right) \cdot \left|x\right|\right|}{\sqrt{\pi}}} \]
8. Add Preprocessing

Alternative 5: 99.4% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (fabs
 (*
  (/ (fabs x) (sqrt PI))
  (-
   (* (- (* (* 1/5 (* x x)) x) (* -1/21 (* (* (* (* x x) x) x) x))) x)
   (- -2 (* 2/3 (* x x)))))))

C

double code(double x) {
	return fabs(((fabs(x) / sqrt(((double) M_PI))) * (((((0.2 * (x * x)) * x) - (-0.047619047619047616 * ((((x * x) * x) * x) * x))) * x) - (-2.0 - (0.6666666666666666 * (x * x))))));
}

Java

public static double code(double x) {
	return Math.abs(((Math.abs(x) / Math.sqrt(Math.PI)) * (((((0.2 * (x * x)) * x) - (-0.047619047619047616 * ((((x * x) * x) * x) * x))) * x) - (-2.0 - (0.6666666666666666 * (x * x))))));
}

Python

def code(x):
	return math.fabs(((math.fabs(x) / math.sqrt(math.pi)) * (((((0.2 * (x * x)) * x) - (-0.047619047619047616 * ((((x * x) * x) * x) * x))) * x) - (-2.0 - (0.6666666666666666 * (x * x))))))

Julia

function code(x)
	return abs(Float64(Float64(abs(x) / sqrt(pi)) * Float64(Float64(Float64(Float64(Float64(0.2 * Float64(x * x)) * x) - Float64(-0.047619047619047616 * Float64(Float64(Float64(Float64(x * x) * x) * x) * x))) * x) - Float64(-2.0 - Float64(0.6666666666666666 * Float64(x * x))))))
end

MATLAB

function tmp = code(x)
	tmp = abs(((abs(x) / sqrt(pi)) * (((((0.2 * (x * x)) * x) - (-0.047619047619047616 * ((((x * x) * x) * x) * x))) * x) - (-2.0 - (0.6666666666666666 * (x * x))))));
end

Wolfram

code[x_] := N[Abs[N[(N[(N[Abs[x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(1/5 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] - N[(-1/21 * N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] - N[(-2 - N[(2/3 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. fp-cancel-sign-sub-inv N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) - \left(\mathsf{neg}\left(\frac{1}{5}\right)\right) \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)} + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   4. lower--.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) - \left(\mathsf{neg}\left(\frac{1}{5}\right)\right) \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)} + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{\left(\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{3} + 2\right) - \left(\frac{-1}{5} \cdot \left|x\right|\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right)} + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. mul-fabs N/A

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

   6. lift-*.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. mul-fabs N/A

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

   9. lift-*.f64 N/A

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

   10. rem-sqrt-square-rev N/A

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

   11. sqr-abs-rev N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. mul-fabs N/A

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

   15. lift-*.f64 N/A

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

   16. mul-fabs N/A

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

   17. lift-fabs.f64 N/A

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

   18. lift-fabs.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. lift-fabs.f64 N/A

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

   21. lift-*.f64 N/A

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

   22. pow2 N/A

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.6%

   \[\leadsto \left|\color{blue}{\left(x \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \frac{1}{21} + \left(\frac{1}{5} \cdot \left(x \cdot x\right)\right) \cdot x\right) - \left(-2 - \frac{2}{3} \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\frac{\left|x\right|}{\pi} \cdot \sqrt{\pi}\right)}\right| \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.6%

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

9. Applied rewrites 99.4%

   \[\leadsto \left|\color{blue}{\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(\left(\left(\frac{1}{5} \cdot \left(x \cdot x\right)\right) \cdot x - \frac{-1}{21} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot x - \left(-2 - \frac{2}{3} \cdot \left(x \cdot x\right)\right)\right)}\right| \]
10. Add Preprocessing

Alternative 6: 99.1% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (*
 (/ (sqrt PI) PI)
 (fabs
  (*
   x
   (+
    (* x (+ (* (* (* x x) x) (* x (* 1/21 x))) (* (* 1/5 (* x x)) x)))
    2)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Sqrt[Pi], $MachinePrecision] / Pi), $MachinePrecision] * N[Abs[N[(x * N[(N[(x * N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * N[(1/21 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1/5 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2), $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. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

   \[\leadsto \frac{\left|\left|x\right| \cdot \left(\left(\frac{1}{5} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \frac{1}{21}\right) + \left|x\right| \cdot \color{blue}{2}\right|}{\sqrt{\pi}} \]
5. Step-by-step derivation

6. Applied rewrites 98.6%

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

8. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\frac{\sqrt{\pi}}{\pi} \cdot \left|x \cdot \left(x \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \frac{1}{21} + \left(\frac{1}{5} \cdot \left(x \cdot x\right)\right) \cdot x\right) + 2\right)\right|} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.1%

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

10. Applied rewrites 99.1%

    \[\leadsto \frac{\sqrt{\pi}}{\pi} \cdot \left|x \cdot \left(x \cdot \left(\color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot \left(\frac{1}{21} \cdot x\right)\right)} + \left(\frac{1}{5} \cdot \left(x \cdot x\right)\right) \cdot x\right) + 2\right)\right| \]
11. Add Preprocessing

Alternative 7: 99.1% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (*
 (/ (sqrt PI) PI)
 (fabs
  (*
   x
   (+
    (* x (+ (* (* 1/21 (* (* (* x x) x) x)) x) (* (* 1/5 (* x x)) x)))
    2)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Sqrt[Pi], $MachinePrecision] / Pi), $MachinePrecision] * N[Abs[N[(x * N[(N[(x * N[(N[(N[(1/21 * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] + N[(N[(1/5 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2), $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. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

   \[\leadsto \frac{\left|\left|x\right| \cdot \left(\left(\frac{1}{5} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \frac{1}{21}\right) + \left|x\right| \cdot \color{blue}{2}\right|}{\sqrt{\pi}} \]
5. Step-by-step derivation

6. Applied rewrites 98.6%

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

8. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\frac{\sqrt{\pi}}{\pi} \cdot \left|x \cdot \left(x \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \frac{1}{21} + \left(\frac{1}{5} \cdot \left(x \cdot x\right)\right) \cdot x\right) + 2\right)\right|} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 99.1%

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

10. Applied rewrites 99.1%

    \[\leadsto \frac{\sqrt{\pi}}{\pi} \cdot \left|x \cdot \left(x \cdot \left(\color{blue}{\left(\frac{1}{21} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot x} + \left(\frac{1}{5} \cdot \left(x \cdot x\right)\right) \cdot x\right) + 2\right)\right| \]
11. Add Preprocessing

Alternative 8: 98.8% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (*
 (sqrt PI)
 (/
  (fabs
   (*
    (+
     2
     (*
      (- (* (* 1/5 (* x x)) x) (* -1/21 (* (* (* (* x x) x) x) x)))
      x))
    x))
  PI)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[Abs[N[(N[(2 + N[(N[(N[(N[(1/5 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] - N[(-1/21 * N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] / Pi), $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. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

   \[\leadsto \frac{\left|\left|x\right| \cdot \left(\left(\frac{1}{5} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \frac{1}{21}\right) + \left|x\right| \cdot \color{blue}{2}\right|}{\sqrt{\pi}} \]
5. Step-by-step derivation

6. Applied rewrites 98.6%

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

8. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\frac{\sqrt{\pi}}{\pi} \cdot \left|x \cdot \left(x \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \frac{1}{21} + \left(\frac{1}{5} \cdot \left(x \cdot x\right)\right) \cdot x\right) + 2\right)\right|} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 98.8%

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

10. Applied rewrites 98.8%

    \[\leadsto \color{blue}{\sqrt{\pi} \cdot \frac{\left|\left(2 + \left(\left(\frac{1}{5} \cdot \left(x \cdot x\right)\right) \cdot x - \frac{-1}{21} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot x\right|}{\pi}} \]
11. Add Preprocessing

Alternative 9: 98.6% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (/
 (fabs
  (*
   x
   (+
    (* x (+ (* (* (* (* (* x x) x) x) x) 1/21) (* (* 1/5 (* x x)) x)))
    2)))
 (sqrt PI)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Abs[N[(x * N[(N[(x * N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 1/21), $MachinePrecision] + N[(N[(1/5 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $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. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

   \[\leadsto \frac{\left|\left|x\right| \cdot \left(\left(\frac{1}{5} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \frac{1}{21}\right) + \left|x\right| \cdot \color{blue}{2}\right|}{\sqrt{\pi}} \]
5. Step-by-step derivation

6. Applied rewrites 98.6%

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

8. Applied rewrites 98.6%

   \[\leadsto \frac{\color{blue}{\left|x \cdot \left(x \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \frac{1}{21} + \left(\frac{1}{5} \cdot \left(x \cdot x\right)\right) \cdot x\right) + 2\right)\right|}}{\sqrt{\pi}} \]
9. Add Preprocessing

Alternative 10: 83.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq \frac{4722366482869645}{9444732965739290427392}:\\ \;\;\;\;\frac{\sqrt{\pi}}{\pi} \cdot \left|2 \cdot \left|x\right|\right|\\ \mathbf{else}:\\ \;\;\;\;\left|2 \cdot \frac{\sqrt{\left(\left|x\right| \cdot \left|x\right|\right) \cdot \pi}}{\pi}\right|\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= (fabs x) 4722366482869645/9444732965739290427392)
  (* (/ (sqrt PI) PI) (fabs (* 2 (fabs x))))
  (fabs (* 2 (/ (sqrt (* (* (fabs x) (fabs x)) PI)) PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 5e-7) {
		tmp = (sqrt(((double) M_PI)) / ((double) M_PI)) * fabs((2.0 * fabs(x)));
	} else {
		tmp = fabs((2.0 * (sqrt(((fabs(x) * fabs(x)) * ((double) M_PI))) / ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 5e-7) {
		tmp = (Math.sqrt(Math.PI) / Math.PI) * Math.abs((2.0 * Math.abs(x)));
	} else {
		tmp = Math.abs((2.0 * (Math.sqrt(((Math.abs(x) * Math.abs(x)) * Math.PI)) / Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 5e-7:
		tmp = (math.sqrt(math.pi) / math.pi) * math.fabs((2.0 * math.fabs(x)))
	else:
		tmp = math.fabs((2.0 * (math.sqrt(((math.fabs(x) * math.fabs(x)) * math.pi)) / math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 5e-7)
		tmp = Float64(Float64(sqrt(pi) / pi) * abs(Float64(2.0 * abs(x))));
	else
		tmp = abs(Float64(2.0 * Float64(sqrt(Float64(Float64(abs(x) * abs(x)) * pi)) / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 5e-7)
		tmp = (sqrt(pi) / pi) * abs((2.0 * abs(x)));
	else
		tmp = abs((2.0 * (sqrt(((abs(x) * abs(x)) * pi)) / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 4722366482869645/9444732965739290427392], N[(N[(N[Sqrt[Pi], $MachinePrecision] / Pi), $MachinePrecision] * N[Abs[N[(2 * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(2 * N[(N[Sqrt[N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.9999999999999998e-7

   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. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\left|\left|x\right| \cdot \left(\left(\frac{1}{5} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \frac{1}{21}\right) + \left|x\right| \cdot \color{blue}{2}\right|}{\sqrt{\pi}} \]
   5. Step-by-step derivation

   6. Applied rewrites 98.6%

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in x around 0

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

       1. lower-*.f64 67.7%

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

   11. Applied rewrites 67.7%

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

   if 4.9999999999999998e-7 < 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. Applied rewrites 99.6%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto \left|2 \cdot \frac{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}{\mathsf{PI}\left(\right)}\right| \]

      4. lower-fabs.f64 N/A

         \[\leadsto \left|2 \cdot \frac{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}{\mathsf{PI}\left(\right)}\right| \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left|2 \cdot \frac{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}{\mathsf{PI}\left(\right)}\right| \]

      6. lower-PI.f64 N/A

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

      7. lower-PI.f64 67.4%

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

   6. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. rem-sqrt-square-rev N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-*.f64 53.7%

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

   8. Applied rewrites 53.7%

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

Alternative 11: 67.7% accurate, 5.4× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (* (/ (sqrt PI) PI) (fabs (* 2 x))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Sqrt[Pi], $MachinePrecision] / Pi), $MachinePrecision] * N[Abs[N[(2 * x), $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. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

   \[\leadsto \frac{\left|\left|x\right| \cdot \left(\left(\frac{1}{5} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) + \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \frac{1}{21}\right) + \left|x\right| \cdot \color{blue}{2}\right|}{\sqrt{\pi}} \]
5. Step-by-step derivation

6. Applied rewrites 98.6%

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

8. Applied rewrites 99.1%

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

9. Taylor expanded in x around 0

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

    1. lower-*.f64 67.7%

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

11. Applied rewrites 67.7%

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

Alternative 12: 67.2% accurate, 5.9× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (fabs (* 2 (/ (fabs x) (sqrt PI)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(2 * N[(N[Abs[x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. fp-cancel-sign-sub-inv N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) - \left(\mathsf{neg}\left(\frac{1}{5}\right)\right) \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)} + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   4. lower--.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) - \left(\mathsf{neg}\left(\frac{1}{5}\right)\right) \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)} + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{\left(\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{3} + 2\right) - \left(\frac{-1}{5} \cdot \left|x\right|\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right)} + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. mul-fabs N/A

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

   6. lift-*.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. mul-fabs N/A

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

   9. lift-*.f64 N/A

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

   10. rem-sqrt-square-rev N/A

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

   11. sqr-abs-rev N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. mul-fabs N/A

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

   15. lift-*.f64 N/A

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

   16. mul-fabs N/A

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

   17. lift-fabs.f64 N/A

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

   18. lift-fabs.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. lift-fabs.f64 N/A

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

   21. lift-*.f64 N/A

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

   22. pow2 N/A

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-fabs.f64 N/A

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

   4. lower-sqrt.f64 N/A

      \[\leadsto \left|2 \cdot \frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

   5. lower-PI.f64 67.2%

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

8. Applied rewrites 67.2%

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

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  :precision binary64
  :pre (<= x 1/2)
  (fabs (* (/ 1 (sqrt PI)) (+ (+ (+ (* 2 (fabs x)) (* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1 21) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))