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

Percentage Accurate: 99.9% → 99.9%

Time: 5.8s

Alternatives: 15

Speedup: 1.5×

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

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot \left(\left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left({\left(\left|x\right|\right)}^{6} \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+
       (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0))
       (* (/ 1.0 5.0) (* (* t_0 (fabs x)) (fabs x))))
      (* (/ 1.0 21.0) (* (pow (fabs x) 6.0) (fabs x))))))))

C

double code(double x) {
	double t_0 = (fabs(x) * 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_0 * fabs(x)) * fabs(x)))) + ((1.0 / 21.0) * (pow(fabs(x), 6.0) * fabs(x))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * 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_0 * Math.abs(x)) * Math.abs(x)))) + ((1.0 / 21.0) * (Math.pow(Math.abs(x), 6.0) * Math.abs(x))))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * 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_0 * math.fabs(x)) * math.fabs(x)))) + ((1.0 / 21.0) * (math.pow(math.fabs(x), 6.0) * math.fabs(x))))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * 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) * Float64(Float64(t_0 * abs(x)) * abs(x)))) + Float64(Float64(1.0 / 21.0) * Float64((abs(x) ^ 6.0) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * 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_0 * abs(x)) * abs(x)))) + ((1.0 / 21.0) * ((abs(x) ^ 6.0) * 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]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[Power[N[Abs[x], $MachinePrecision], 6.0], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\color{blue}{{\left(\left|x\right|\right)}^{6}} \cdot \left|x\right|\right)\right)\right| \]
3. Step-by-step derivation

   1. lower-pow.f64 N/A

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

   2. lower-fabs.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 99.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x \cdot x\right) \cdot x\right) \cdot x\\ \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(t\_0 \cdot \left|x\right|, \left(x \cdot x\right) \cdot 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, t\_0, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (* x x) x) x)))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (fma
      (* t_0 (fabs x))
      (* (* x x) 0.047619047619047616)
      (fma
       (* 0.2 (fabs x))
       t_0
       (* (fabs x) (fma (* x x) 0.6666666666666666 2.0))))))))

C

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

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.047619047619047616), $MachinePrecision] + N[(N[(0.2 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.9%

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

3. Applied rewrites 99.9%

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

Alternative 3: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Applied rewrites 99.9%

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

Alternative 4: 99.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Applied rewrites 99.9%

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

5. Applied rewrites 99.9%

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

Alternative 5: 99.1% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ 1.0 (sqrt PI))
  (fabs
   (-
    (* (* (fma (* 0.047619047619047616 x) x 0.2) (* (* (* (- x) x) x) x)) x)
    (* x 2.0)))))

C

double code(double x) {
	return (1.0 / sqrt(((double) M_PI))) * fabs((((fma((0.047619047619047616 * x), x, 0.2) * (((-x * x) * x) * x)) * x) - (x * 2.0)));
}

Julia

function code(x)
	return Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(Float64(Float64(fma(Float64(0.047619047619047616 * x), x, 0.2) * Float64(Float64(Float64(Float64(-x) * x) * x) * x)) * x) - Float64(x * 2.0))))
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Applied rewrites 99.9%

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.1%

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

10. Applied rewrites 99.1%

    \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\color{blue}{\left(\mathsf{fma}\left(0.047619047619047616 \cdot x, x, 0.2\right) \cdot \left(\left(\left(\left(-x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot x - x \cdot 2}\right| \]
11. Add Preprocessing

Alternative 6: 98.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\left|\frac{2}{\sqrt{\pi}} \cdot \left(-x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{{x}^{6} \cdot \left(-x\right)}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.9)
   (fabs (* (/ 2.0 (sqrt PI)) (- x)))
   (fabs (* 0.047619047619047616 (/ (* (pow x 6.0) (- x)) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = fabs(((2.0 / sqrt(((double) M_PI))) * -x));
	} else {
		tmp = fabs((0.047619047619047616 * ((pow(x, 6.0) * -x) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = Math.abs(((2.0 / Math.sqrt(Math.PI)) * -x));
	} else {
		tmp = Math.abs((0.047619047619047616 * ((Math.pow(x, 6.0) * -x) / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.9:
		tmp = math.fabs(((2.0 / math.sqrt(math.pi)) * -x))
	else:
		tmp = math.fabs((0.047619047619047616 * ((math.pow(x, 6.0) * -x) / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.9)
		tmp = abs(Float64(Float64(2.0 / sqrt(pi)) * Float64(-x)));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64(Float64((x ^ 6.0) * Float64(-x)) / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.9)
		tmp = abs(((2.0 / sqrt(pi)) * -x));
	else
		tmp = abs((0.047619047619047616 * (((x ^ 6.0) * -x) / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.9], N[Abs[N[(N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * (-x)), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[(N[Power[x, 6.0], $MachinePrecision] * (-x)), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   1. Initial program 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]
   5. 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 67.1

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

   6. Applied rewrites 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 67.5

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

   8. Applied rewrites 67.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.5

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

      4. lift-fabs.f64 N/A

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

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

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

      6. sqr-neg-rev N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sqrt-unprod N/A

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

      10. rem-square-sqrt 67.5

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

   10. Applied rewrites 67.5%

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

   if 1.8999999999999999 < x

   1. Initial program 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around inf

      \[\leadsto \left|\color{blue}{\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{\color{blue}{{x}^{6} \cdot \left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \color{blue}{\frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

      3. metadata-eval N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{\color{blue}{{x}^{6} \cdot \left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      4. lower-/.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

      5. lower-*.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}}\right| \]

      6. lower-pow.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      7. lower-fabs.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      9. lower-PI.f64 37.1

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

   6. Applied rewrites 37.1%

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

      1. lift-fabs.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

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

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \sqrt{x \cdot x}}{\sqrt{\pi}}\right| \]

      3. sqr-neg-rev N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \sqrt{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)}}{\sqrt{\pi}}\right| \]

      4. lift-neg.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \sqrt{\left(-x\right) \cdot \left(\mathsf{neg}\left(x\right)\right)}}{\sqrt{\pi}}\right| \]

      5. lift-neg.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \sqrt{\left(-x\right) \cdot \left(-x\right)}}{\sqrt{\pi}}\right| \]

      6. sqrt-unprod N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left(\sqrt{-x} \cdot \sqrt{-x}\right)}{\sqrt{\pi}}\right| \]

      7. rem-square-sqrt 37.1

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

   8. Applied rewrites 37.1%

      \[\leadsto \left|0.047619047619047616 \cdot \frac{{x}^{6} \cdot \left(-x\right)}{\sqrt{\pi}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 88.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\left|\frac{2}{\sqrt{\pi}} \cdot \left(-x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{0.047619047619047616}{\frac{-\sqrt{\pi}}{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.9)
   (fabs (* (/ 2.0 (sqrt PI)) (- x)))
   (fabs
    (/
     0.047619047619047616
     (/ (- (sqrt PI)) (* (* (* (* (* x x) x) x) x) (* x x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = fabs(((2.0 / sqrt(((double) M_PI))) * -x));
	} else {
		tmp = fabs((0.047619047619047616 / (-sqrt(((double) M_PI)) / (((((x * x) * x) * x) * x) * (x * x)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = Math.abs(((2.0 / Math.sqrt(Math.PI)) * -x));
	} else {
		tmp = Math.abs((0.047619047619047616 / (-Math.sqrt(Math.PI) / (((((x * x) * x) * x) * x) * (x * x)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.9:
		tmp = math.fabs(((2.0 / math.sqrt(math.pi)) * -x))
	else:
		tmp = math.fabs((0.047619047619047616 / (-math.sqrt(math.pi) / (((((x * x) * x) * x) * x) * (x * x)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.9)
		tmp = abs(Float64(Float64(2.0 / sqrt(pi)) * Float64(-x)));
	else
		tmp = abs(Float64(0.047619047619047616 / Float64(Float64(-sqrt(pi)) / Float64(Float64(Float64(Float64(Float64(x * x) * x) * x) * x) * Float64(x * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.9)
		tmp = abs(((2.0 / sqrt(pi)) * -x));
	else
		tmp = abs((0.047619047619047616 / (-sqrt(pi) / (((((x * x) * x) * x) * x) * (x * x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.9], N[Abs[N[(N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * (-x)), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 / N[((-N[Sqrt[Pi], $MachinePrecision]) / N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   1. Initial program 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]
   5. 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 67.1

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

   6. Applied rewrites 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 67.5

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

   8. Applied rewrites 67.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.5

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

      4. lift-fabs.f64 N/A

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

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

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

      6. sqr-neg-rev N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sqrt-unprod N/A

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

      10. rem-square-sqrt 67.5

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

   10. Applied rewrites 67.5%

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

   if 1.8999999999999999 < x

   1. Initial program 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around inf

      \[\leadsto \left|\color{blue}{\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{\color{blue}{{x}^{6} \cdot \left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \color{blue}{\frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

      3. metadata-eval N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{\color{blue}{{x}^{6} \cdot \left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      4. lower-/.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

      5. lower-*.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}}\right| \]

      6. lower-pow.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      7. lower-fabs.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      9. lower-PI.f64 37.1

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

   6. Applied rewrites 37.1%

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

   8. Applied rewrites 37.1%

      \[\leadsto \left|\frac{0.047619047619047616}{\color{blue}{\frac{-\sqrt{\pi}}{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)}}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 67.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\left|\frac{2}{\sqrt{\pi}} \cdot \left(-x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-x}{\sqrt{\pi}}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.9)
   (fabs (* (/ 2.0 (sqrt PI)) (- x)))
   (fabs
    (*
     0.047619047619047616
     (* (* (* (* x x) x) x) (* (* x x) (/ (- x) (sqrt PI))))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = fabs(((2.0 / sqrt(((double) M_PI))) * -x));
	} else {
		tmp = fabs((0.047619047619047616 * ((((x * x) * x) * x) * ((x * x) * (-x / sqrt(((double) M_PI)))))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = Math.abs(((2.0 / Math.sqrt(Math.PI)) * -x));
	} else {
		tmp = Math.abs((0.047619047619047616 * ((((x * x) * x) * x) * ((x * x) * (-x / Math.sqrt(Math.PI))))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.9:
		tmp = math.fabs(((2.0 / math.sqrt(math.pi)) * -x))
	else:
		tmp = math.fabs((0.047619047619047616 * ((((x * x) * x) * x) * ((x * x) * (-x / math.sqrt(math.pi))))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.9)
		tmp = abs(Float64(Float64(2.0 / sqrt(pi)) * Float64(-x)));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64(Float64(Float64(Float64(x * x) * x) * x) * Float64(Float64(x * x) * Float64(Float64(-x) / sqrt(pi))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.9)
		tmp = abs(((2.0 / sqrt(pi)) * -x));
	else
		tmp = abs((0.047619047619047616 * ((((x * x) * x) * x) * ((x * x) * (-x / sqrt(pi))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.9], N[Abs[N[(N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * (-x)), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[((-x) / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   1. Initial program 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]
   5. 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 67.1

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

   6. Applied rewrites 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 67.5

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

   8. Applied rewrites 67.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.5

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

      4. lift-fabs.f64 N/A

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

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

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

      6. sqr-neg-rev N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sqrt-unprod N/A

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

      10. rem-square-sqrt 67.5

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

   10. Applied rewrites 67.5%

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

   if 1.8999999999999999 < x

   1. Initial program 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around inf

      \[\leadsto \left|\color{blue}{\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{\color{blue}{{x}^{6} \cdot \left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \color{blue}{\frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

      3. metadata-eval N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{\color{blue}{{x}^{6} \cdot \left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      4. lower-/.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

      5. lower-*.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}}\right| \]

      6. lower-pow.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      7. lower-fabs.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      9. lower-PI.f64 37.1

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

   6. Applied rewrites 37.1%

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

      1. lift-/.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\color{blue}{\sqrt{\pi}}}\right| \]

      2. lift-*.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\color{blue}{\pi}}}\right| \]

      3. associate-/l* N/A

         \[\leadsto \left|\frac{1}{21} \cdot \left({x}^{6} \cdot \color{blue}{\frac{\left|x\right|}{\sqrt{\pi}}}\right)\right| \]

      4. lift-pow.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \left({x}^{6} \cdot \frac{\color{blue}{\left|x\right|}}{\sqrt{\pi}}\right)\right| \]

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow-prod-down N/A

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

      8. pow3 N/A

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

      9. associate-*l* N/A

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

      10. lift-fabs.f64 N/A

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

      11. neg-fabs N/A

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

      12. lift-neg.f64 N/A

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

      13. rem-square-sqrt N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. fabs-sqr N/A

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

      17. lift-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. associate-*l* N/A

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

   8. Applied rewrites 37.1%

      \[\leadsto \left|0.047619047619047616 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \frac{-x}{\sqrt{\pi}}\right)}\right)\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 67.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\left|\frac{2}{\sqrt{\pi}} \cdot \left(-x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\left(0.047619047619047616 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot \frac{-x}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.9)
   (fabs (* (/ 2.0 (sqrt PI)) (- x)))
   (fabs
    (*
     (* (* 0.047619047619047616 (* (* (* (* x x) x) x) x)) x)
     (/ (- x) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = fabs(((2.0 / sqrt(((double) M_PI))) * -x));
	} else {
		tmp = fabs((((0.047619047619047616 * ((((x * x) * x) * x) * x)) * x) * (-x / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = Math.abs(((2.0 / Math.sqrt(Math.PI)) * -x));
	} else {
		tmp = Math.abs((((0.047619047619047616 * ((((x * x) * x) * x) * x)) * x) * (-x / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.9:
		tmp = math.fabs(((2.0 / math.sqrt(math.pi)) * -x))
	else:
		tmp = math.fabs((((0.047619047619047616 * ((((x * x) * x) * x) * x)) * x) * (-x / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.9)
		tmp = abs(Float64(Float64(2.0 / sqrt(pi)) * Float64(-x)));
	else
		tmp = abs(Float64(Float64(Float64(0.047619047619047616 * Float64(Float64(Float64(Float64(x * x) * x) * x) * x)) * x) * Float64(Float64(-x) / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.9)
		tmp = abs(((2.0 / sqrt(pi)) * -x));
	else
		tmp = abs((((0.047619047619047616 * ((((x * x) * x) * x) * x)) * x) * (-x / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.9], N[Abs[N[(N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * (-x)), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(0.047619047619047616 * N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * N[((-x) / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   1. Initial program 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]
   5. 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 67.1

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

   6. Applied rewrites 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 67.5

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

   8. Applied rewrites 67.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.5

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

      4. lift-fabs.f64 N/A

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

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

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

      6. sqr-neg-rev N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sqrt-unprod N/A

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

      10. rem-square-sqrt 67.5

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

   10. Applied rewrites 67.5%

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

   if 1.8999999999999999 < x

   1. Initial program 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around inf

      \[\leadsto \left|\color{blue}{\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{\color{blue}{{x}^{6} \cdot \left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \color{blue}{\frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

      3. metadata-eval N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{\color{blue}{{x}^{6} \cdot \left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      4. lower-/.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

      5. lower-*.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}}\right| \]

      6. lower-pow.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      7. lower-fabs.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      9. lower-PI.f64 37.1

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

   6. Applied rewrites 37.1%

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

   8. Applied rewrites 37.1%

      \[\leadsto \left|\left(\left(0.047619047619047616 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot x\right) \cdot \color{blue}{\frac{-x}{\sqrt{\pi}}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 67.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\left|\frac{2}{\sqrt{\pi}} \cdot \left(-x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(-0.047619047619047616 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot x}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.9)
   (fabs (* (/ 2.0 (sqrt PI)) (- x)))
   (fabs
    (/
     (* (* -0.047619047619047616 (* (* (* (* (* x x) x) x) x) x)) x)
     (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = fabs(((2.0 / sqrt(((double) M_PI))) * -x));
	} else {
		tmp = fabs((((-0.047619047619047616 * (((((x * x) * x) * x) * x) * x)) * x) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = Math.abs(((2.0 / Math.sqrt(Math.PI)) * -x));
	} else {
		tmp = Math.abs((((-0.047619047619047616 * (((((x * x) * x) * x) * x) * x)) * x) / Math.sqrt(Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.9:
		tmp = math.fabs(((2.0 / math.sqrt(math.pi)) * -x))
	else:
		tmp = math.fabs((((-0.047619047619047616 * (((((x * x) * x) * x) * x) * x)) * x) / math.sqrt(math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.9)
		tmp = abs(Float64(Float64(2.0 / sqrt(pi)) * Float64(-x)));
	else
		tmp = abs(Float64(Float64(Float64(-0.047619047619047616 * Float64(Float64(Float64(Float64(Float64(x * x) * x) * x) * x) * x)) * x) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.9)
		tmp = abs(((2.0 / sqrt(pi)) * -x));
	else
		tmp = abs((((-0.047619047619047616 * (((((x * x) * x) * x) * x) * x)) * x) / sqrt(pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.9], N[Abs[N[(N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * (-x)), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(-0.047619047619047616 * N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   1. Initial program 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]
   5. 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 67.1

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

   6. Applied rewrites 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 67.5

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

   8. Applied rewrites 67.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.5

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

      4. lift-fabs.f64 N/A

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

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

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

      6. sqr-neg-rev N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sqrt-unprod N/A

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

      10. rem-square-sqrt 67.5

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

   10. Applied rewrites 67.5%

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

   if 1.8999999999999999 < x

   1. Initial program 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around inf

      \[\leadsto \left|\color{blue}{\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{\color{blue}{{x}^{6} \cdot \left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \color{blue}{\frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

      3. metadata-eval N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{\color{blue}{{x}^{6} \cdot \left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      4. lower-/.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

      5. lower-*.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}}\right| \]

      6. lower-pow.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      7. lower-fabs.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{1}{21} \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

      9. lower-PI.f64 37.1

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

   6. Applied rewrites 37.1%

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

   8. Applied rewrites 37.1%

      \[\leadsto \color{blue}{\left|\frac{\left(-0.047619047619047616 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot x}{\sqrt{\pi}}\right|} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 67.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{\left|\left(-x\right) \cdot 2 - \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \mathsf{fma}\left(0.047619047619047616 \cdot x, x, 0.2\right)\right|}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fabs
   (-
    (* (- x) 2.0)
    (* (* (* (* (* x x) x) x) x) (fma (* 0.047619047619047616 x) x 0.2))))
  (sqrt PI)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Applied rewrites 99.9%

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip-rev N/A

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

   5. lower-/.f64 98.6

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

10. Applied rewrites 98.6%

    \[\leadsto \color{blue}{\frac{\left|\left(-x\right) \cdot 2 - \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \mathsf{fma}\left(0.047619047619047616 \cdot x, x, 0.2\right)\right|}{\sqrt{\pi}}} \]
11. Add Preprocessing

Alternative 12: 67.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left|\mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) \cdot \left(-x\right)\right|}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fabs (* (fma (* x x) 0.6666666666666666 2.0) (- x))) (sqrt PI)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. lower-fma.f64 N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-fabs.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-fabs.f64 88.8

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

6. Applied rewrites 88.8%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-fabs.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-*r* N/A

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

   15. lift-*.f64 N/A

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

   16. lower-fma.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lift-fabs.f64 N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right| \cdot x, x \cdot \frac{2}{3}, \left|x\right| \cdot 2\right)\right|}{\sqrt{\pi}} \]

   20. lower-*.f64 88.8

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

8. Applied rewrites 88.8%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. distribute-lft-in N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-*.f64 88.8

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

   12. lift-fabs.f64 N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right) \cdot \left|x\right|\right|}{\sqrt{\pi}} \]

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

       \[\leadsto \frac{\left|\mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right) \cdot \sqrt{x \cdot x}\right|}{\sqrt{\pi}} \]

   14. sqr-neg-rev N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right) \cdot \sqrt{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)}\right|}{\sqrt{\pi}} \]

   15. lift-neg.f64 N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right) \cdot \sqrt{\left(-x\right) \cdot \left(\mathsf{neg}\left(x\right)\right)}\right|}{\sqrt{\pi}} \]

   16. lift-neg.f64 N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right) \cdot \sqrt{\left(-x\right) \cdot \left(-x\right)}\right|}{\sqrt{\pi}} \]

   17. sqrt-unprod N/A

       \[\leadsto \frac{\left|\mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right) \cdot \left(\sqrt{-x} \cdot \color{blue}{\sqrt{-x}}\right)\right|}{\sqrt{\pi}} \]

   18. rem-square-sqrt 88.8

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

10. Applied rewrites 88.8%

    \[\leadsto \frac{\left|\color{blue}{\mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) \cdot \left(-x\right)}\right|}{\sqrt{\pi}} \]
11. Add Preprocessing

Alternative 13: 67.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-25}:\\ \;\;\;\;\left|\frac{2}{\sqrt{\pi}} \cdot \left(-x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|2 \cdot \sqrt{\frac{x \cdot x}{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e-25)
   (fabs (* (/ 2.0 (sqrt PI)) (- x)))
   (fabs (* 2.0 (sqrt (/ (* x x) PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 1e-25) {
		tmp = fabs(((2.0 / sqrt(((double) M_PI))) * -x));
	} else {
		tmp = fabs((2.0 * sqrt(((x * x) / ((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1e-25) {
		tmp = Math.abs(((2.0 / Math.sqrt(Math.PI)) * -x));
	} else {
		tmp = Math.abs((2.0 * Math.sqrt(((x * x) / Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1e-25:
		tmp = math.fabs(((2.0 / math.sqrt(math.pi)) * -x))
	else:
		tmp = math.fabs((2.0 * math.sqrt(((x * x) / math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e-25)
		tmp = abs(Float64(Float64(2.0 / sqrt(pi)) * Float64(-x)));
	else
		tmp = abs(Float64(2.0 * sqrt(Float64(Float64(x * x) / pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1e-25)
		tmp = abs(((2.0 / sqrt(pi)) * -x));
	else
		tmp = abs((2.0 * sqrt(((x * x) / pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1e-25], N[Abs[N[(N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * (-x)), $MachinePrecision]], $MachinePrecision], N[Abs[N[(2.0 * N[Sqrt[N[(N[(x * x), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000004e-25

   1. Initial program 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]
   5. 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 67.1

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

   6. Applied rewrites 67.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 67.5

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

   8. Applied rewrites 67.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 67.5

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

      4. lift-fabs.f64 N/A

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

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

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

      6. sqr-neg-rev N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. sqrt-unprod N/A

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

      10. rem-square-sqrt 67.5

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

   10. Applied rewrites 67.5%

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

   if 1.00000000000000004e-25 < x

   1. Initial program 99.9%

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]
   5. 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 67.1

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

   6. Applied rewrites 67.1%

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

      1. lift-/.f64 N/A

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

      2. lift-fabs.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-undiv N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 53.5

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

   8. Applied rewrites 53.5%

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

Alternative 14: 67.5% accurate, 8.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Applied rewrites 99.9%

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

4. Taylor expanded in x around 0

   \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]
5. 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 67.1

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

6. Applied rewrites 67.1%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 67.5

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

8. Applied rewrites 67.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 67.5

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

   4. lift-fabs.f64 N/A

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

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

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

   6. sqr-neg-rev N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-neg.f64 N/A

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

   9. sqrt-unprod N/A

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

   10. rem-square-sqrt 67.5

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

10. Applied rewrites 67.5%

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

Alternative 15: 67.1% accurate, 8.6× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (fabs (/ (+ x x) (- (sqrt PI)))))

C

double code(double x) {
	return fabs(((x + x) / -sqrt(((double) M_PI))));
}

Java

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

Python

def code(x):
	return math.fabs(((x + x) / -math.sqrt(math.pi)))

Julia

function code(x)
	return abs(Float64(Float64(x + x) / Float64(-sqrt(pi))))
end

MATLAB

function tmp = code(x)
	tmp = abs(((x + x) / -sqrt(pi)));
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Applied rewrites 99.9%

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

4. Taylor expanded in x around 0

   \[\leadsto \left|\color{blue}{2 \cdot \frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]
5. 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 67.1

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

6. Applied rewrites 67.1%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 67.5

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

8. Applied rewrites 67.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. associate-/l* N/A

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

   6. lift-fabs.f64 N/A

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

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

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

   8. sqr-neg-rev N/A

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

   9. lift-neg.f64 N/A

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

   10. lift-neg.f64 N/A

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

   11. sqrt-unprod N/A

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

   12. rem-square-sqrt N/A

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

   13. div-flip N/A

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

   14. mult-flip-rev N/A

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

   15. metadata-eval N/A

       \[\leadsto \left|\frac{1 + 1}{\frac{\color{blue}{\sqrt{\pi}}}{-x}}\right| \]

10. Applied rewrites 67.1%

    \[\leadsto \left|\color{blue}{\frac{x + x}{-\sqrt{\pi}}}\right| \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025144 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  :precision binary64
  :pre (<= x 0.5)
  (fabs (* (/ 1.0 (sqrt PI)) (+ (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1.0 5.0) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1.0 21.0) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))