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

Percentage Accurate: 99.8% → 99.8%

Time: 5.0s

Alternatives: 13

Speedup: 1.9×

• 28.6% 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, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

   1. lift-fabs.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. fabs-mul N/A

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

   6. lift-fabs.f64 N/A

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

   7. fabs-fabs N/A

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

   8. mul-fabs N/A

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

   9. lower-fabs.f64 N/A

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

5. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.8

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

7. Applied rewrites 99.8%

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

Alternative 3: 99.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

   1. lift-fabs.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. fabs-mul N/A

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

   6. lift-fabs.f64 N/A

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

   7. fabs-fabs N/A

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

   8. mul-fabs N/A

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

   9. lower-fabs.f64 N/A

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

5. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.8

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.2%

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

Alternative 4: 98.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{\pi}}\\ \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;t\_0 \cdot \left|x \cdot 2\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left|0.047619047619047616 \cdot {x}^{7}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt PI))))
   (if (<= x 1.85)
     (* t_0 (fabs (* x 2.0)))
     (* t_0 (fabs (* 0.047619047619047616 (pow x 7.0)))))))

C

double code(double x) {
	double t_0 = 1.0 / sqrt(((double) M_PI));
	double tmp;
	if (x <= 1.85) {
		tmp = t_0 * fabs((x * 2.0));
	} else {
		tmp = t_0 * fabs((0.047619047619047616 * pow(x, 7.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.sqrt(Math.PI);
	double tmp;
	if (x <= 1.85) {
		tmp = t_0 * Math.abs((x * 2.0));
	} else {
		tmp = t_0 * Math.abs((0.047619047619047616 * Math.pow(x, 7.0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 / math.sqrt(math.pi)
	tmp = 0
	if x <= 1.85:
		tmp = t_0 * math.fabs((x * 2.0))
	else:
		tmp = t_0 * math.fabs((0.047619047619047616 * math.pow(x, 7.0)))
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 / sqrt(pi))
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64(t_0 * abs(Float64(x * 2.0)));
	else
		tmp = Float64(t_0 * abs(Float64(0.047619047619047616 * (x ^ 7.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 / sqrt(pi);
	tmp = 0.0;
	if (x <= 1.85)
		tmp = t_0 * abs((x * 2.0));
	else
		tmp = t_0 * abs((0.047619047619047616 * (x ^ 7.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.85], N[(t$95$0 * N[Abs[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Abs[N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]], $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%

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

      1. lift-fabs.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. fabs-mul N/A

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

      6. lift-fabs.f64 N/A

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

      7. fabs-fabs N/A

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

      8. mul-fabs N/A

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

      9. lower-fabs.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.7%

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

   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%

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

      1. lift-fabs.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. fabs-mul N/A

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

      6. lift-fabs.f64 N/A

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

      7. fabs-fabs N/A

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

      8. mul-fabs N/A

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

      9. lower-fabs.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\color{blue}{\frac{1}{21} \cdot {x}^{7}}\right| \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\frac{1}{21} \cdot \color{blue}{{x}^{7}}\right| \]

      2. lower-pow.f64 36.9

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

   8. Applied rewrites 36.9%

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

Alternative 5: 98.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|x \cdot 2\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.047619047619047616\right) \cdot x\right) \cdot x\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (* (/ 1.0 (sqrt PI)) (fabs (* x 2.0)))
   (/
    (fabs (* (* (* (* (* (* (* x x) x) x) x) 0.047619047619047616) x) x))
    (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = (1.0 / sqrt(((double) M_PI))) * fabs((x * 2.0));
	} else {
		tmp = fabs((((((((x * x) * x) * x) * x) * 0.047619047619047616) * x) * x)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = (1.0 / Math.sqrt(Math.PI)) * Math.abs((x * 2.0));
	} else {
		tmp = Math.abs((((((((x * x) * x) * x) * x) * 0.047619047619047616) * x) * x)) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = (1.0 / math.sqrt(math.pi)) * math.fabs((x * 2.0))
	else:
		tmp = math.fabs((((((((x * x) * x) * x) * x) * 0.047619047619047616) * x) * x)) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(x * 2.0)));
	else
		tmp = Float64(abs(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * x) * x) * x) * x) * 0.047619047619047616) * x) * x)) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = (1.0 / sqrt(pi)) * abs((x * 2.0));
	else
		tmp = abs((((((((x * x) * x) * x) * x) * 0.047619047619047616) * x) * x)) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 0.047619047619047616), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $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%

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

      1. lift-fabs.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. fabs-mul N/A

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

      6. lift-fabs.f64 N/A

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

      7. fabs-fabs N/A

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

      8. mul-fabs N/A

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

      9. lower-fabs.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.7%

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

   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%

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

      1. lift-fabs.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. fabs-mul N/A

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

      6. lift-fabs.f64 N/A

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

      7. fabs-fabs N/A

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

      8. mul-fabs N/A

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

      9. lower-fabs.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 36.9

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

   8. Applied rewrites 36.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 36.9

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

   10. Applied rewrites 36.9%

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

Alternative 6: 67.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|x \cdot 2\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{\sqrt{\pi}} \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.047619047619047616\right) \cdot x\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (* (/ 1.0 (sqrt PI)) (fabs (* x 2.0)))
   (fabs
    (*
     (/ x (sqrt PI))
     (* (* (* (* (* (* x x) x) x) x) 0.047619047619047616) x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = (1.0 / sqrt(((double) M_PI))) * fabs((x * 2.0));
	} else {
		tmp = fabs(((x / sqrt(((double) M_PI))) * ((((((x * x) * x) * x) * x) * 0.047619047619047616) * x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = (1.0 / Math.sqrt(Math.PI)) * Math.abs((x * 2.0));
	} else {
		tmp = Math.abs(((x / Math.sqrt(Math.PI)) * ((((((x * x) * x) * x) * x) * 0.047619047619047616) * x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = (1.0 / math.sqrt(math.pi)) * math.fabs((x * 2.0))
	else:
		tmp = math.fabs(((x / math.sqrt(math.pi)) * ((((((x * x) * x) * x) * x) * 0.047619047619047616) * x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(x * 2.0)));
	else
		tmp = abs(Float64(Float64(x / sqrt(pi)) * Float64(Float64(Float64(Float64(Float64(Float64(x * x) * x) * x) * x) * 0.047619047619047616) * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = (1.0 / sqrt(pi)) * abs((x * 2.0));
	else
		tmp = abs(((x / sqrt(pi)) * ((((((x * x) * x) * x) * x) * 0.047619047619047616) * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 0.047619047619047616), $MachinePrecision] * x), $MachinePrecision]), $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%

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

      1. lift-fabs.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. fabs-mul N/A

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

      6. lift-fabs.f64 N/A

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

      7. fabs-fabs N/A

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

      8. mul-fabs N/A

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

      9. lower-fabs.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.7%

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

   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%

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

      1. lift-fabs.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. fabs-mul N/A

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

      6. lift-fabs.f64 N/A

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

      7. fabs-fabs N/A

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

      8. mul-fabs N/A

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

      9. lower-fabs.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 36.9

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

   8. Applied rewrites 36.9%

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

   10. Applied rewrites 36.9%

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

Alternative 7: 67.7% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[Abs[N[(N[(N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision] * -0.047619047619047616 + (-N[(N[Abs[x], $MachinePrecision] * 2.0), $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 99.4%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-fabs.f64 98.4

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

6. Applied rewrites 98.4%

   \[\leadsto \left|\frac{{\left(\left|x\right|\right)}^{7} \cdot -0.047619047619047616 - \color{blue}{2 \cdot \left|x\right|}}{-\sqrt{\pi}}\right| \]
7. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-neg.f64 98.4

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 98.4

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

8. Applied rewrites 98.4%

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

Alternative 8: 67.7% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return Math.abs(((Math.abs(x) * 2.0) - (Math.pow(Math.abs(x), 7.0) * -0.047619047619047616))) / Math.sqrt(Math.PI);
}

Python

def code(x):
	return math.fabs(((math.fabs(x) * 2.0) - (math.pow(math.fabs(x), 7.0) * -0.047619047619047616))) / math.sqrt(math.pi)

Julia

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

MATLAB

function tmp = code(x)
	tmp = abs(((abs(x) * 2.0) - ((abs(x) ^ 7.0) * -0.047619047619047616))) / sqrt(pi);
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-fabs.f64 98.4

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

6. Applied rewrites 98.4%

   \[\leadsto \left|\frac{{\left(\left|x\right|\right)}^{7} \cdot -0.047619047619047616 - \color{blue}{2 \cdot \left|x\right|}}{-\sqrt{\pi}}\right| \]
7. Step-by-step derivation

   1. lift-fabs.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. fabs-div N/A

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

   4. lift-neg.f64 N/A

      \[\leadsto \frac{\left|{\left(\left|x\right|\right)}^{7} \cdot \frac{-1}{21} - 2 \cdot \left|x\right|\right|}{\left|\color{blue}{\mathsf{neg}\left(\sqrt{\pi}\right)}\right|} \]

   5. neg-fabs N/A

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

   6. lift-sqrt.f64 N/A

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

   7. sqrt-fabs-rev N/A

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

   8. lift-sqrt.f64 N/A

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

8. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\frac{\left|\left|x\right| \cdot 2 - {\left(\left|x\right|\right)}^{7} \cdot -0.047619047619047616\right|}{\sqrt{\pi}}} \]
9. Add Preprocessing

Alternative 9: 67.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|x \cdot 2\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(0.2 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right) \cdot x\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.8)
   (* (/ 1.0 (sqrt PI)) (fabs (* x 2.0)))
   (fabs (* (* (/ (fabs x) (sqrt PI)) (* 0.2 (* (* x x) x))) x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = (1.0 / sqrt(((double) M_PI))) * fabs((x * 2.0));
	} else {
		tmp = fabs((((fabs(x) / sqrt(((double) M_PI))) * (0.2 * ((x * x) * x))) * x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = (1.0 / Math.sqrt(Math.PI)) * Math.abs((x * 2.0));
	} else {
		tmp = Math.abs((((Math.abs(x) / Math.sqrt(Math.PI)) * (0.2 * ((x * x) * x))) * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.8:
		tmp = (1.0 / math.sqrt(math.pi)) * math.fabs((x * 2.0))
	else:
		tmp = math.fabs((((math.fabs(x) / math.sqrt(math.pi)) * (0.2 * ((x * x) * x))) * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.8)
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(x * 2.0)));
	else
		tmp = abs(Float64(Float64(Float64(abs(x) / sqrt(pi)) * Float64(0.2 * Float64(Float64(x * x) * x))) * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.8)
		tmp = (1.0 / sqrt(pi)) * abs((x * 2.0));
	else
		tmp = abs((((abs(x) / sqrt(pi)) * (0.2 * ((x * x) * x))) * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.8], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[(N[(N[Abs[x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(0.2 * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $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%

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

      1. lift-fabs.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. fabs-mul N/A

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

      6. lift-fabs.f64 N/A

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

      7. fabs-fabs N/A

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

      8. mul-fabs N/A

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

      9. lower-fabs.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.7%

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

   if 1.80000000000000004 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-PI.f64 31.2

         \[\leadsto \left|0.2 \cdot \frac{{x}^{4} \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

   6. Applied rewrites 31.2%

      \[\leadsto \left|\color{blue}{0.2 \cdot \frac{{x}^{4} \cdot \left|x\right|}{\sqrt{\pi}}}\right| \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|\frac{1}{5} \cdot \color{blue}{\frac{{x}^{4} \cdot \left|x\right|}{\sqrt{\pi}}}\right| \]

      2. lift-/.f64 N/A

         \[\leadsto \left|\frac{1}{5} \cdot \frac{{x}^{4} \cdot \left|x\right|}{\color{blue}{\sqrt{\pi}}}\right| \]

      3. lift-*.f64 N/A

         \[\leadsto \left|\frac{1}{5} \cdot \frac{{x}^{4} \cdot \left|x\right|}{\sqrt{\color{blue}{\pi}}}\right| \]

      4. associate-/l* N/A

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-plus N/A

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

      8. pow3 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

   8. Applied rewrites 31.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   10. Applied rewrites 31.1%

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

Alternative 10: 67.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|x \cdot 2\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(\left(2 \cdot x\right) \cdot x\right) \cdot 2}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e-21)
   (* (/ 1.0 (sqrt PI)) (fabs (* x 2.0)))
   (sqrt (/ (* (* (* 2.0 x) x) 2.0) PI))))

C

double code(double x) {
	double tmp;
	if (x <= 2e-21) {
		tmp = (1.0 / sqrt(((double) M_PI))) * fabs((x * 2.0));
	} else {
		tmp = sqrt(((((2.0 * x) * x) * 2.0) / ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2e-21) {
		tmp = (1.0 / Math.sqrt(Math.PI)) * Math.abs((x * 2.0));
	} else {
		tmp = Math.sqrt(((((2.0 * x) * x) * 2.0) / Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2e-21:
		tmp = (1.0 / math.sqrt(math.pi)) * math.fabs((x * 2.0))
	else:
		tmp = math.sqrt(((((2.0 * x) * x) * 2.0) / math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e-21)
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(x * 2.0)));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(2.0 * x) * x) * 2.0) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2e-21)
		tmp = (1.0 / sqrt(pi)) * abs((x * 2.0));
	else
		tmp = sqrt(((((2.0 * x) * x) * 2.0) / pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2e-21], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[(N[(2.0 * x), $MachinePrecision] * x), $MachinePrecision] * 2.0), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999999999982e-21

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

      1. lift-fabs.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. fabs-mul N/A

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

      6. lift-fabs.f64 N/A

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

      7. fabs-fabs N/A

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

      8. mul-fabs N/A

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

      9. lower-fabs.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.7%

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

   if 1.99999999999999982e-21 < 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%

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

      1. lift-fabs.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. fabs-mul N/A

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

      6. lift-fabs.f64 N/A

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

      7. fabs-fabs N/A

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

      8. mul-fabs N/A

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

      9. lower-fabs.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-fabs.f64 N/A

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

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

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

      7. lift-sqrt.f64 N/A

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

   10. Applied rewrites 52.8%

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

Alternative 11: 67.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{\pi}}\\ \mathbf{if}\;x \leq 100000000:\\ \;\;\;\;t\_0 \cdot \left|x \cdot 2\right|\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{\left(2 \cdot x\right) \cdot x} \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt PI))))
   (if (<= x 100000000.0)
     (* t_0 (fabs (* x 2.0)))
     (* t_0 (* (sqrt (* (* 2.0 x) x)) (sqrt 2.0))))))

C

double code(double x) {
	double t_0 = 1.0 / sqrt(((double) M_PI));
	double tmp;
	if (x <= 100000000.0) {
		tmp = t_0 * fabs((x * 2.0));
	} else {
		tmp = t_0 * (sqrt(((2.0 * x) * x)) * sqrt(2.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.sqrt(Math.PI);
	double tmp;
	if (x <= 100000000.0) {
		tmp = t_0 * Math.abs((x * 2.0));
	} else {
		tmp = t_0 * (Math.sqrt(((2.0 * x) * x)) * Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 / math.sqrt(math.pi)
	tmp = 0
	if x <= 100000000.0:
		tmp = t_0 * math.fabs((x * 2.0))
	else:
		tmp = t_0 * (math.sqrt(((2.0 * x) * x)) * math.sqrt(2.0))
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 / sqrt(pi))
	tmp = 0.0
	if (x <= 100000000.0)
		tmp = Float64(t_0 * abs(Float64(x * 2.0)));
	else
		tmp = Float64(t_0 * Float64(sqrt(Float64(Float64(2.0 * x) * x)) * sqrt(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 / sqrt(pi);
	tmp = 0.0;
	if (x <= 100000000.0)
		tmp = t_0 * abs((x * 2.0));
	else
		tmp = t_0 * (sqrt(((2.0 * x) * x)) * sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 100000000.0], N[(t$95$0 * N[Abs[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[Sqrt[N[(N[(2.0 * x), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1e8

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

      1. lift-fabs.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. fabs-mul N/A

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

      6. lift-fabs.f64 N/A

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

      7. fabs-fabs N/A

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

      8. mul-fabs N/A

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

      9. lower-fabs.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.7%

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

   if 1e8 < 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%

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

      1. lift-fabs.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. fabs-mul N/A

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

      6. lift-fabs.f64 N/A

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

      7. fabs-fabs N/A

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

      8. mul-fabs N/A

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

      9. lower-fabs.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.7%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

   10. Applied rewrites 52.7%

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

Alternative 12: 67.7% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

   1. lift-fabs.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. fabs-mul N/A

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

   6. lift-fabs.f64 N/A

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

   7. fabs-fabs N/A

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

   8. mul-fabs N/A

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

   9. lower-fabs.f64 N/A

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 67.7%

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

Alternative 13: 67.2% accurate, 9.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

   1. lift-fabs.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. fabs-mul N/A

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

   6. lift-fabs.f64 N/A

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

   7. fabs-fabs N/A

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

   8. mul-fabs N/A

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

   9. lower-fabs.f64 N/A

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 67.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip-rev N/A

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

   5. lower-/.f64 67.2

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

10. Applied rewrites 67.2%

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

Reproduce

herbie shell --seed 2025159 
(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)))))))