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

Percentage Accurate: 99.8% → 99.8%

Time: 4.3s

Alternatives: 13

Speedup: 2.5×

• 28.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 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]]]

Derivation

1. Initial program 99.8%

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

Alternative 2: 99.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt PI))
   (fma
    (pow (fabs x) 7.0)
    0.047619047619047616
    (fma
     (* 0.2 (fabs x))
     (* (* (* x x) x) x)
     (* (fabs x) (fma (* x x) 0.6666666666666666 2.0)))))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * fma(pow(fabs(x), 7.0), 0.047619047619047616, fma((0.2 * fabs(x)), (((x * x) * x) * x), (fabs(x) * fma((x * x), 0.6666666666666666, 2.0))))));
}

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * fma((abs(x) ^ 7.0), 0.047619047619047616, fma(Float64(0.2 * abs(x)), Float64(Float64(Float64(x * x) * x) * x), Float64(abs(x) * fma(Float64(x * x), 0.6666666666666666, 2.0))))))
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision] * 0.047619047619047616 + N[(N[(0.2 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)\right)}\right| \]

4. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)}\right| \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left({x}^{4} \cdot \left(\frac{1}{5} + \color{blue}{\frac{1}{21}} \cdot {x}^{2}\right)\right)\right)\right| \]

   2. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left({x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {\color{blue}{x}}^{2}\right)\right)\right)\right| \]

   3. *-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}\right)\right)\right| \]

   4. lower-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}\right)\right)\right| \]

   5. +-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left(\frac{1}{21} \cdot {x}^{2} + \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}\right)\right)\right| \]

   6. *-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left({x}^{2} \cdot \frac{1}{21} + \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

   7. lower-fma.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left({x}^{2}, \frac{1}{21}, \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}\right)\right)\right| \]

   8. pow2 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

   9. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

   10. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

   11. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

   12. metadata-eval N/A

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

   13. pow-prod-up N/A

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

   14. lower-*.f64 N/A

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

   15. pow2 N/A

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

   16. lift-*.f64 N/A

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

   17. pow2 N/A

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

   18. lift-*.f64 99.8

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

6. Applied rewrites 99.8%

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

8. Applied rewrites 99.8%

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

Alternative 4: 99.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left({x}^{4} \cdot \left(\frac{1}{5} + \color{blue}{\frac{1}{21}} \cdot {x}^{2}\right)\right)\right)\right| \]

   2. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left({x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {\color{blue}{x}}^{2}\right)\right)\right)\right| \]

   3. *-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}\right)\right)\right| \]

   4. lower-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}\right)\right)\right| \]

   5. +-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left(\frac{1}{21} \cdot {x}^{2} + \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}\right)\right)\right| \]

   6. *-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left({x}^{2} \cdot \frac{1}{21} + \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

   7. lower-fma.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left({x}^{2}, \frac{1}{21}, \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}\right)\right)\right| \]

   8. pow2 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

   9. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

   10. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

   11. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

   12. metadata-eval N/A

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

   13. pow-prod-up N/A

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

   14. lower-*.f64 N/A

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

   15. pow2 N/A

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

   16. lift-*.f64 N/A

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

   17. pow2 N/A

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

   18. lift-*.f64 99.8

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

6. Applied rewrites 99.8%

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

8. Applied rewrites 99.8%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. pow2 N/A

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

    3. *-commutative N/A

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

    4. lower-fma.f64 N/A

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

11. Applied rewrites 99.8%

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

Alternative 5: 98.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.65:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left(\left|x\right| \cdot \left|\mathsf{fma}\left(\mathsf{fma}\left(x \cdot 0.2, x, 0.6666666666666666\right), x \cdot x, 2\right)\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)}{\sqrt{\pi}} \cdot 0.047619047619047616\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.65)
   (*
    (/ 1.0 (sqrt PI))
    (* (fabs x) (fabs (fma (fma (* x 0.2) x 0.6666666666666666) (* x x) 2.0))))
   (fabs
    (*
     (/ (* (* (* (* (* x x) x) x) x) (* x x)) (sqrt PI))
     0.047619047619047616))))

C

double code(double x) {
	double tmp;
	if (x <= 2.65) {
		tmp = (1.0 / sqrt(((double) M_PI))) * (fabs(x) * fabs(fma(fma((x * 0.2), x, 0.6666666666666666), (x * x), 2.0)));
	} else {
		tmp = fabs((((((((x * x) * x) * x) * x) * (x * x)) / sqrt(((double) M_PI))) * 0.047619047619047616));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.65)
		tmp = Float64(Float64(1.0 / sqrt(pi)) * Float64(abs(x) * abs(fma(fma(Float64(x * 0.2), x, 0.6666666666666666), Float64(x * x), 2.0))));
	else
		tmp = abs(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * x) * x) * x) * x) * Float64(x * x)) / sqrt(pi)) * 0.047619047619047616));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.65], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(N[(N[(x * 0.2), $MachinePrecision] * x + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * 0.047619047619047616), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.64999999999999991

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. metadata-eval N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left({x}^{4} \cdot \left(\frac{1}{5} + \color{blue}{\frac{1}{21}} \cdot {x}^{2}\right)\right)\right)\right| \]

      2. metadata-eval N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left({x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {\color{blue}{x}}^{2}\right)\right)\right)\right| \]

      3. *-commutative N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}\right)\right)\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}\right)\right)\right| \]

      5. +-commutative N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left(\frac{1}{21} \cdot {x}^{2} + \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}\right)\right)\right| \]

      6. *-commutative N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left({x}^{2} \cdot \frac{1}{21} + \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

      7. lower-fma.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left({x}^{2}, \frac{1}{21}, \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}\right)\right)\right| \]

      8. pow2 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

      9. lift-*.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

      10. metadata-eval N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

      11. metadata-eval N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

      12. metadata-eval N/A

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

      13. pow-prod-up N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. pow2 N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

   11. Applied rewrites 93.6%

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

   if 2.64999999999999991 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. metadata-eval N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left({x}^{4} \cdot \left(\frac{1}{5} + \color{blue}{\frac{1}{21}} \cdot {x}^{2}\right)\right)\right)\right| \]

      2. metadata-eval N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left({x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {\color{blue}{x}}^{2}\right)\right)\right)\right| \]

      3. *-commutative N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}\right)\right)\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}\right)\right)\right| \]

      5. +-commutative N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left(\frac{1}{21} \cdot {x}^{2} + \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}\right)\right)\right| \]

      6. *-commutative N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left({x}^{2} \cdot \frac{1}{21} + \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

      7. lower-fma.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left({x}^{2}, \frac{1}{21}, \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}\right)\right)\right| \]

      8. pow2 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

      9. lift-*.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

      10. metadata-eval N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

      11. metadata-eval N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

      12. metadata-eval N/A

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

      13. pow-prod-up N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 36.8%

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

Alternative 6: 98.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \left|x\right| \cdot 2\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt PI))
   (fma (pow (fabs x) 7.0) 0.047619047619047616 (* (fabs x) 2.0)))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * fma(pow(fabs(x), 7.0), 0.047619047619047616, (fabs(x) * 2.0))));
}

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * fma((abs(x) ^ 7.0), 0.047619047619047616, Float64(abs(x) * 2.0))))
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)\right)}\right| \]

4. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)}\right| \]

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, \frac{1}{21}, \left|x\right| \cdot 2\right)\right| \]

   2. lower-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, \frac{1}{21}, \left|x\right| \cdot 2\right)\right| \]

   3. lift-fabs.f64 98.9

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

7. Applied rewrites 98.9%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \left|x\right| \cdot 2\right)\right| \]
8. Add Preprocessing

Alternative 7: 93.6% accurate, 3.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (fabs (* (/ 2.0 (sqrt PI)) x))
   (fabs
    (*
     (/ (* (* (* (* (* x x) x) x) x) (* x x)) (sqrt PI))
     0.047619047619047616))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = fabs(((2.0 / sqrt(((double) M_PI))) * x));
	} else {
		tmp = fabs((((((((x * x) * x) * x) * x) * (x * x)) / sqrt(((double) M_PI))) * 0.047619047619047616));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = Math.abs(((2.0 / Math.sqrt(Math.PI)) * x));
	} else {
		tmp = Math.abs((((((((x * x) * x) * x) * x) * (x * x)) / Math.sqrt(Math.PI)) * 0.047619047619047616));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = abs(((2.0 / sqrt(pi)) * x));
	else
		tmp = abs((((((((x * x) * x) * x) * x) * (x * x)) / sqrt(pi)) * 0.047619047619047616));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[Abs[N[(N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * 0.047619047619047616), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 67.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-fabs.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-sqrt.f64 67.7

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

   8. Applied rewrites 67.7%

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

      1. lift-fabs.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

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

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. metadata-eval N/A

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

      11. sqr-pow N/A

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

      12. unpow1 67.7

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

   10. Applied rewrites 67.7%

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

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. metadata-eval N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left({x}^{4} \cdot \left(\frac{1}{5} + \color{blue}{\frac{1}{21}} \cdot {x}^{2}\right)\right)\right)\right| \]

      2. metadata-eval N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left({x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {\color{blue}{x}}^{2}\right)\right)\right)\right| \]

      3. *-commutative N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}\right)\right)\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}\right)\right)\right| \]

      5. +-commutative N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left(\frac{1}{21} \cdot {x}^{2} + \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}\right)\right)\right| \]

      6. *-commutative N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left({x}^{2} \cdot \frac{1}{21} + \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

      7. lower-fma.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left({x}^{2}, \frac{1}{21}, \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}\right)\right)\right| \]

      8. pow2 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

      9. lift-*.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

      10. metadata-eval N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

      11. metadata-eval N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

      12. metadata-eval N/A

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

      13. pow-prod-up N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 36.8%

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

Alternative 8: 88.8% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0

   \[\leadsto \left|\color{blue}{\frac{\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around inf

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

   1. metadata-eval N/A

      \[\leadsto \left|\frac{1}{5} \cdot \frac{{x}^{4} \cdot \left|x\right|}{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}}\right| \]

   2. *-commutative N/A

      \[\leadsto \left|\frac{{x}^{4} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{1}{\color{blue}{5}}\right| \]

   3. lower-*.f64 N/A

      \[\leadsto \left|\frac{{x}^{4} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{1}{\color{blue}{5}}\right| \]

7. Applied rewrites 31.2%

   \[\leadsto \left|\frac{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)}{\sqrt{\pi}} \cdot \color{blue}{0.2}\right| \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-PI.f64 N/A

      \[\leadsto \left|\frac{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{1}{5}\right| \]

   8. pow3 N/A

      \[\leadsto \left|\frac{{x}^{3} \cdot \left(x \cdot x\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{1}{5}\right| \]

   9. pow2 N/A

      \[\leadsto \left|\frac{{x}^{3} \cdot {x}^{2}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{1}{5}\right| \]

   10. pow-prod-up N/A

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

   11. metadata-eval N/A

       \[\leadsto \left|\frac{{x}^{5}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{1}{5}\right| \]

9. Applied rewrites 31.2%

   \[\leadsto \left|\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \frac{\left|x\right|}{\sqrt{\pi}}\right) \cdot 0.2\right| \]

10. Taylor expanded in x around 0

    \[\leadsto \left|\frac{\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + 2 \cdot \left|x\right|}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]
11. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \left|\frac{\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + 2 \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

12. Applied rewrites 98.4%

    \[\leadsto \left|\frac{\mathsf{fma}\left({x}^{7}, 0.047619047619047616, x + x\right)}{\color{blue}{\sqrt{\pi}}}\right| \]
13. Add Preprocessing

Alternative 9: 83.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8:\\ \;\;\;\;\left|\frac{2}{\sqrt{\pi}} \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{x \cdot x}{\sqrt{\pi}}\right) \cdot 0.2\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.8)
   (fabs (* (/ 2.0 (sqrt PI)) x))
   (fabs (* (* (* (* x x) x) (/ (* x x) (sqrt PI))) 0.2))))

C

double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = fabs(((2.0 / sqrt(((double) M_PI))) * x));
	} else {
		tmp = fabs(((((x * x) * x) * ((x * x) / sqrt(((double) M_PI)))) * 0.2));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = Math.abs(((2.0 / Math.sqrt(Math.PI)) * x));
	} else {
		tmp = Math.abs(((((x * x) * x) * ((x * x) / Math.sqrt(Math.PI))) * 0.2));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.8:
		tmp = math.fabs(((2.0 / math.sqrt(math.pi)) * x))
	else:
		tmp = math.fabs(((((x * x) * x) * ((x * x) / math.sqrt(math.pi))) * 0.2))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.8)
		tmp = abs(Float64(Float64(2.0 / sqrt(pi)) * x));
	else
		tmp = abs(Float64(Float64(Float64(Float64(x * x) * x) * Float64(Float64(x * x) / sqrt(pi))) * 0.2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.8)
		tmp = abs(((2.0 / sqrt(pi)) * x));
	else
		tmp = abs(((((x * x) * x) * ((x * x) / sqrt(pi))) * 0.2));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.8], N[Abs[N[(N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.2), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.80000000000000004

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 67.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-fabs.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-sqrt.f64 67.7

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

   8. Applied rewrites 67.7%

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

      1. lift-fabs.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

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

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. metadata-eval N/A

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

      11. sqr-pow N/A

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

      12. unpow1 67.7

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

   10. Applied rewrites 67.7%

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

   if 1.80000000000000004 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{\frac{\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in x around inf

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

      1. metadata-eval N/A

         \[\leadsto \left|\frac{1}{5} \cdot \frac{{x}^{4} \cdot \left|x\right|}{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}}\right| \]

      2. *-commutative N/A

         \[\leadsto \left|\frac{{x}^{4} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{1}{\color{blue}{5}}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{{x}^{4} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{1}{\color{blue}{5}}\right| \]

   7. Applied rewrites 31.2%

      \[\leadsto \left|\frac{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)}{\sqrt{\pi}} \cdot \color{blue}{0.2}\right| \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. lift-PI.f64 N/A

         \[\leadsto \left|\frac{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{1}{5}\right| \]

      8. associate-/l* N/A

         \[\leadsto \left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{x \cdot x}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{5}\right| \]

      9. pow3 N/A

         \[\leadsto \left|\left({x}^{3} \cdot \frac{x \cdot x}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{5}\right| \]

      10. sqr-pow N/A

          \[\leadsto \left|\left(\left({x}^{\left(\frac{3}{2}\right)} \cdot {x}^{\left(\frac{3}{2}\right)}\right) \cdot \frac{x \cdot x}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{5}\right| \]

      11. pow-prod-down N/A

          \[\leadsto \left|\left({\left(x \cdot x\right)}^{\left(\frac{3}{2}\right)} \cdot \frac{x \cdot x}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{5}\right| \]

      12. sqrt-pow2 N/A

          \[\leadsto \left|\left({\left(\sqrt{x \cdot x}\right)}^{3} \cdot \frac{x \cdot x}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{5}\right| \]

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

          \[\leadsto \left|\left({\left(\left|x\right|\right)}^{3} \cdot \frac{x \cdot x}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{5}\right| \]

      14. pow3 N/A

          \[\leadsto \left|\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \frac{x \cdot x}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{5}\right| \]

      15. lower-*.f64 N/A

          \[\leadsto \left|\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \frac{x \cdot x}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{5}\right| \]

      16. pow3 N/A

          \[\leadsto \left|\left({\left(\left|x\right|\right)}^{3} \cdot \frac{x \cdot x}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{5}\right| \]

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

          \[\leadsto \left|\left({\left(\sqrt{x \cdot x}\right)}^{3} \cdot \frac{x \cdot x}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{5}\right| \]

      18. sqrt-pow2 N/A

          \[\leadsto \left|\left({\left(x \cdot x\right)}^{\left(\frac{3}{2}\right)} \cdot \frac{x \cdot x}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{5}\right| \]

      19. pow-prod-down N/A

          \[\leadsto \left|\left(\left({x}^{\left(\frac{3}{2}\right)} \cdot {x}^{\left(\frac{3}{2}\right)}\right) \cdot \frac{x \cdot x}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{5}\right| \]

      20. sqr-pow N/A

          \[\leadsto \left|\left({x}^{3} \cdot \frac{x \cdot x}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{5}\right| \]

      21. pow3 N/A

          \[\leadsto \left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{x \cdot x}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{5}\right| \]

      22. lift-*.f64 N/A

          \[\leadsto \left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{x \cdot x}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{5}\right| \]

      23. lift-*.f64 N/A

          \[\leadsto \left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{x \cdot x}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{5}\right| \]

   9. Applied rewrites 31.2%

      \[\leadsto \left|\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{x \cdot x}{\sqrt{\pi}}\right) \cdot 0.2\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 67.7% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 88.8%

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

Alternative 11: 67.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{if}\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \leq 10^{-10}:\\ \;\;\;\;\left|\frac{2}{\sqrt{\pi}} \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\sqrt{x \cdot x} \cdot 2}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (if (<=
        (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))))))
        1e-10)
     (fabs (* (/ 2.0 (sqrt PI)) x))
     (fabs (/ (* (sqrt (* x x)) 2.0) (sqrt PI))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	double tmp;
	if (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)))))) <= 1e-10) {
		tmp = fabs(((2.0 / sqrt(((double) M_PI))) * x));
	} else {
		tmp = fabs(((sqrt((x * x)) * 2.0) / sqrt(((double) M_PI))));
	}
	return tmp;
}

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);
	double tmp;
	if (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)))))) <= 1e-10) {
		tmp = Math.abs(((2.0 / Math.sqrt(Math.PI)) * x));
	} else {
		tmp = Math.abs(((Math.sqrt((x * x)) * 2.0) / Math.sqrt(Math.PI)));
	}
	return tmp;
}

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)
	tmp = 0
	if 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)))))) <= 1e-10:
		tmp = math.fabs(((2.0 / math.sqrt(math.pi)) * x))
	else:
		tmp = math.fabs(((math.sqrt((x * x)) * 2.0) / math.sqrt(math.pi)))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	tmp = 0.0
	if (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)))))) <= 1e-10)
		tmp = abs(Float64(Float64(2.0 / sqrt(pi)) * x));
	else
		tmp = abs(Float64(Float64(sqrt(Float64(x * x)) * 2.0) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = 0.0;
	if (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)))))) <= 1e-10)
		tmp = abs(((2.0 / sqrt(pi)) * x));
	else
		tmp = abs(((sqrt((x * x)) * 2.0) / sqrt(pi)));
	end
	tmp_2 = tmp;
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]}, If[LessEqual[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], 1e-10], N[Abs[N[(N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Sqrt[N[(x * x), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))))) < 1.00000000000000004e-10

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 67.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-fabs.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-sqrt.f64 67.7

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

   8. Applied rewrites 67.7%

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

      1. lift-fabs.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

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

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. metadata-eval N/A

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

      11. sqr-pow N/A

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

      12. unpow1 67.7

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

   10. Applied rewrites 67.7%

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

   if 1.00000000000000004e-10 < (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))))))

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 67.3%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. lower-sqrt.f64 N/A

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

      4. lift-*.f64 53.8

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

   8. Applied rewrites 53.8%

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

Alternative 12: 67.7% accurate, 9.2× speedup

Math

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

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)) * 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 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 67.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-sqrt.f64 67.7

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

8. Applied rewrites 67.7%

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

   1. lift-fabs.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

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

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

   6. sqrt-prod N/A

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

   7. pow1/2 N/A

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

   8. metadata-eval N/A

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

   9. pow1/2 N/A

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

   10. metadata-eval N/A

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

   11. sqr-pow N/A

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

   12. unpow1 67.7

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

10. Applied rewrites 67.7%

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

Alternative 13: 67.3% accurate, 9.4× 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) / 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 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 67.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-sqrt.f64 67.7

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

8. Applied rewrites 67.7%

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

10. Applied rewrites 67.3%

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

Reproduce

herbie shell --seed 2025138 
(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)))))))