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

Percentage Accurate: 99.8% → 99.8%

Time: 6.0s

Alternatives: 14

Speedup: 1.9×

• 71.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (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

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (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, 1.6× speedup

Math

\[\left(\left|\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right)}{2}, 1, \mathsf{fma}\left(0.023809523809523808, \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x, 1\right)\right)\right| \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \left(2 \cdot \left|x\right|\right) \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (*
   (fabs
    (fma
     (/ (* (* x x) (fma (* 0.2 x) x 0.6666666666666666)) 2.0)
     1.0
     (fma 0.023809523809523808 (* (* (* (* (* x x) x) x) x) x) 1.0)))
   (/ 1.0 (sqrt PI)))
  (* 2.0 (fabs x))))

C

double code(double x) {
	return (fabs(fma((((x * x) * fma((0.2 * x), x, 0.6666666666666666)) / 2.0), 1.0, fma(0.023809523809523808, (((((x * x) * x) * x) * x) * x), 1.0))) * (1.0 / sqrt(((double) M_PI)))) * (2.0 * fabs(x));
}

Julia

function code(x)
	return Float64(Float64(abs(fma(Float64(Float64(Float64(x * x) * fma(Float64(0.2 * x), x, 0.6666666666666666)) / 2.0), 1.0, fma(0.023809523809523808, Float64(Float64(Float64(Float64(Float64(x * x) * x) * x) * x) * x), 1.0))) * Float64(1.0 / sqrt(pi))) * Float64(2.0 * abs(x)))
end

Wolfram

code[x_] := N[(N[(N[Abs[N[(N[(N[(N[(x * x), $MachinePrecision] * N[(N[(0.2 * x), $MachinePrecision] * x + 0.6666666666666666), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * 1.0 + N[(0.023809523809523808 * N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Abs[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%

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

5. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\left(\left|\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right)}{2}, 1, \mathsf{fma}\left(0.023809523809523808, \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x, 1\right)\right)\right| \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \left(2 \cdot \left|x\right|\right)} \]
6. Add Preprocessing

Alternative 2: 99.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ 0.5641895835477563 \cdot \left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(t\_0 \cdot t\_0\right) \cdot 0.047619047619047616\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (*
    0.5641895835477563
    (fabs
     (fma
      (fabs x)
      (fma (* 0.2 (* x x)) (* x x) (* (* t_0 t_0) 0.047619047619047616))
      (* (fabs x) (fma (* x x) 0.6666666666666666 2.0)))))))

C

double code(double x) {
	double t_0 = (x * x) * x;
	return 0.5641895835477563 * fabs(fma(fabs(x), fma((0.2 * (x * x)), (x * x), ((t_0 * t_0) * 0.047619047619047616)), (fabs(x) * fma((x * x), 0.6666666666666666, 2.0))));
}

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	return Float64(0.5641895835477563 * abs(fma(abs(x), fma(Float64(0.2 * Float64(x * x)), Float64(x * x), Float64(Float64(t_0 * t_0) * 0.047619047619047616)), Float64(abs(x) * fma(Float64(x * x), 0.6666666666666666, 2.0)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, N[(0.5641895835477563 * N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(N[(0.2 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.047619047619047616), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Evaluated real constant 99.8%

   \[\leadsto \color{blue}{0.5641895835477563} \cdot \left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 0.047619047619047616\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right| \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \left|\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right) \cdot 0.5, x \cdot x, \mathsf{fma}\left(0.023809523809523808 \cdot t\_0, t\_0, 1\right)\right)}{1.772453850905516} \cdot x\right| \cdot 2 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (*
    (fabs
     (*
      (/
       (fma
        (* (fma (* 0.2 x) x 0.6666666666666666) 0.5)
        (* x x)
        (fma (* 0.023809523809523808 t_0) t_0 1.0))
       1.772453850905516)
      x))
    2.0)))

C

double code(double x) {
	double t_0 = (x * x) * x;
	return fabs(((fma((fma((0.2 * x), x, 0.6666666666666666) * 0.5), (x * x), fma((0.023809523809523808 * t_0), t_0, 1.0)) / 1.772453850905516) * x)) * 2.0;
}

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	return Float64(abs(Float64(Float64(fma(Float64(fma(Float64(0.2 * x), x, 0.6666666666666666) * 0.5), Float64(x * x), fma(Float64(0.023809523809523808 * t_0), t_0, 1.0)) / 1.772453850905516) * x)) * 2.0)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, N[(N[Abs[N[(N[(N[(N[(N[(N[(0.2 * x), $MachinePrecision] * x + 0.6666666666666666), $MachinePrecision] * 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(N[(0.023809523809523808 * t$95$0), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] / 1.772453850905516), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\left(\left|\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right)}{2}, 1, \mathsf{fma}\left(0.023809523809523808, \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x, 1\right)\right)\right| \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \left(2 \cdot \left|x\right|\right)} \]

7. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right) \cdot 0.5, x \cdot x, \mathsf{fma}\left(0.023809523809523808 \cdot \left(\left(x \cdot x\right) \cdot x\right), \left(x \cdot x\right) \cdot x, 1\right)\right)}{\sqrt{\pi}} \cdot x\right| \cdot 2} \]

8. Evaluated real constant 99.8%

   \[\leadsto \left|\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right) \cdot 0.5, x \cdot x, \mathsf{fma}\left(0.023809523809523808 \cdot \left(\left(x \cdot x\right) \cdot x\right), \left(x \cdot x\right) \cdot x, 1\right)\right)}{\color{blue}{1.772453850905516}} \cdot x\right| \cdot 2 \]
9. Add Preprocessing

Alternative 4: 99.4% accurate, 1.9× speedup

Math

\[\frac{\left|\mathsf{fma}\left(x, \mathsf{fma}\left(0.5 \cdot \mathsf{fma}\left(x \cdot 0.2, x, 0.6666666666666666\right), x, 0.023809523809523808 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), 1\right) \cdot x\right|}{\sqrt{\pi}} \cdot 2 \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/
   (fabs
    (*
     (fma
      x
      (fma
       (* 0.5 (fma (* x 0.2) x 0.6666666666666666))
       x
       (* 0.023809523809523808 (* (* (* (* x x) x) x) x)))
      1.0)
     x))
   (sqrt PI))
  2.0))

C

double code(double x) {
	return (fabs((fma(x, fma((0.5 * fma((x * 0.2), x, 0.6666666666666666)), x, (0.023809523809523808 * ((((x * x) * x) * x) * x))), 1.0) * x)) / sqrt(((double) M_PI))) * 2.0;
}

Julia

function code(x)
	return Float64(Float64(abs(Float64(fma(x, fma(Float64(0.5 * fma(Float64(x * 0.2), x, 0.6666666666666666)), x, Float64(0.023809523809523808 * Float64(Float64(Float64(Float64(x * x) * x) * x) * x))), 1.0) * x)) / sqrt(pi)) * 2.0)
end

Wolfram

code[x_] := N[(N[(N[Abs[N[(N[(x * N[(N[(0.5 * N[(N[(x * 0.2), $MachinePrecision] * x + 0.6666666666666666), $MachinePrecision]), $MachinePrecision] * x + N[(0.023809523809523808 * N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\left(\left|\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right)}{2}, 1, \mathsf{fma}\left(0.023809523809523808, \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x, 1\right)\right)\right| \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \left(2 \cdot \left|x\right|\right)} \]

7. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right) \cdot 0.5, x \cdot x, \mathsf{fma}\left(0.023809523809523808 \cdot \left(\left(x \cdot x\right) \cdot x\right), \left(x \cdot x\right) \cdot x, 1\right)\right)}{\sqrt{\pi}} \cdot x\right| \cdot 2} \]

9. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\left|\mathsf{fma}\left(x, \mathsf{fma}\left(0.5 \cdot \mathsf{fma}\left(x \cdot 0.2, x, 0.6666666666666666\right), x, 0.023809523809523808 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right)\right), 1\right) \cdot x\right|}{\sqrt{\pi}}} \cdot 2 \]
10. Add Preprocessing

Alternative 5: 99.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 6800:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.2 \cdot \left|x\right|, \left|x\right|, 0.6666666666666666\right) \cdot 0.5, \left|x\right| \cdot \left|x\right|, 1\right)}{\sqrt{\pi}} \cdot \left|x\right|\right| \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{{\left(\left|x\right|\right)}^{6} \cdot \left|\left|x\right|\right|}{\sqrt{\pi}}\right|\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 6800.0)
   (*
    (fabs
     (*
      (/
       (fma
        (* (fma (* 0.2 (fabs x)) (fabs x) 0.6666666666666666) 0.5)
        (* (fabs x) (fabs x))
        1.0)
       (sqrt PI))
      (fabs x)))
    2.0)
   (fabs
    (*
     0.047619047619047616
     (/ (* (pow (fabs x) 6.0) (fabs (fabs x))) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 6800.0) {
		tmp = fabs(((fma((fma((0.2 * fabs(x)), fabs(x), 0.6666666666666666) * 0.5), (fabs(x) * fabs(x)), 1.0) / sqrt(((double) M_PI))) * fabs(x))) * 2.0;
	} else {
		tmp = fabs((0.047619047619047616 * ((pow(fabs(x), 6.0) * fabs(fabs(x))) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 6800.0)
		tmp = Float64(abs(Float64(Float64(fma(Float64(fma(Float64(0.2 * abs(x)), abs(x), 0.6666666666666666) * 0.5), Float64(abs(x) * abs(x)), 1.0) / sqrt(pi)) * abs(x))) * 2.0);
	else
		tmp = abs(Float64(0.047619047619047616 * Float64(Float64((abs(x) ^ 6.0) * abs(abs(x))) / sqrt(pi))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 6800.0], N[(N[Abs[N[(N[(N[(N[(N[(N[(0.2 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[(N[Power[N[Abs[x], $MachinePrecision], 6.0], $MachinePrecision] * N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6800

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(\left|\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right)}{2}, 1, \mathsf{fma}\left(0.023809523809523808, \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x, 1\right)\right)\right| \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \left(2 \cdot \left|x\right|\right)} \]

   7. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right) \cdot 0.5, x \cdot x, \mathsf{fma}\left(0.023809523809523808 \cdot \left(\left(x \cdot x\right) \cdot x\right), \left(x \cdot x\right) \cdot x, 1\right)\right)}{\sqrt{\pi}} \cdot x\right| \cdot 2} \]

   8. Taylor expanded in x around 0

      \[\leadsto \left|\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right) \cdot 0.5, x \cdot x, \color{blue}{1}\right)}{\sqrt{\pi}} \cdot x\right| \cdot 2 \]
   9. Step-by-step derivation

   10. Applied rewrites 93.7%

       \[\leadsto \left|\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right) \cdot 0.5, x \cdot x, \color{blue}{1}\right)}{\sqrt{\pi}} \cdot x\right| \cdot 2 \]

   if 6800 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-PI.f64 37.1%

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

   6. Applied rewrites 37.1%

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

Alternative 6: 99.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} t_0 := \left|\left|x\right|\right|\\ \mathbf{if}\;\left|x\right| \leq 6800:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{t\_0}{\sqrt{\pi}} \cdot \left(\left|x\right| \cdot \left|x\right|\right), 0.6666666666666666, t\_0 \cdot 1.1283791670955126\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{{\left(\left|x\right|\right)}^{6} \cdot t\_0}{\sqrt{\pi}}\right|\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fabs (fabs x))))
   (if (<= (fabs x) 6800.0)
     (fabs
      (fma
       (* (/ t_0 (sqrt PI)) (* (fabs x) (fabs x)))
       0.6666666666666666
       (* t_0 1.1283791670955126)))
     (fabs
      (* 0.047619047619047616 (/ (* (pow (fabs x) 6.0) t_0) (sqrt PI)))))))

C

double code(double x) {
	double t_0 = fabs(fabs(x));
	double tmp;
	if (fabs(x) <= 6800.0) {
		tmp = fabs(fma(((t_0 / sqrt(((double) M_PI))) * (fabs(x) * fabs(x))), 0.6666666666666666, (t_0 * 1.1283791670955126)));
	} else {
		tmp = fabs((0.047619047619047616 * ((pow(fabs(x), 6.0) * t_0) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = abs(abs(x))
	tmp = 0.0
	if (abs(x) <= 6800.0)
		tmp = abs(fma(Float64(Float64(t_0 / sqrt(pi)) * Float64(abs(x) * abs(x))), 0.6666666666666666, Float64(t_0 * 1.1283791670955126)));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64(Float64((abs(x) ^ 6.0) * t_0) / sqrt(pi))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 6800.0], N[Abs[N[(N[(N[(t$95$0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.6666666666666666 + N[(t$95$0 * 1.1283791670955126), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[(N[Power[N[Abs[x], $MachinePrecision], 6.0], $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 6800

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-PI.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-PI.f64 89.1%

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

   6. Applied rewrites 89.1%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 89.1%

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 89.1%

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*r/ N/A

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

      16. *-commutative N/A

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

   8. Applied rewrites 89.5%

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

   9. Evaluated real constant 89.5%

      \[\leadsto \left|\mathsf{fma}\left(\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(x \cdot x\right), 0.6666666666666666, \left|x\right| \cdot 1.1283791670955126\right)\right| \]

   if 6800 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-PI.f64 37.1%

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

   6. Applied rewrites 37.1%

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

Alternative 7: 99.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} t_0 := \left|\left|x\right|\right|\\ \mathbf{if}\;\left|x\right| \leq 6800:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{t\_0}{\sqrt{\pi}} \cdot \left(\left|x\right| \cdot \left|x\right|\right), 0.6666666666666666, t\_0 \cdot 1.1283791670955126\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|{t\_0}^{7} \cdot 0.047619047619047616\right|}{\sqrt{\pi}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fabs (fabs x))))
   (if (<= (fabs x) 6800.0)
     (fabs
      (fma
       (* (/ t_0 (sqrt PI)) (* (fabs x) (fabs x)))
       0.6666666666666666
       (* t_0 1.1283791670955126)))
     (/ (fabs (* (pow t_0 7.0) 0.047619047619047616)) (sqrt PI)))))

C

double code(double x) {
	double t_0 = fabs(fabs(x));
	double tmp;
	if (fabs(x) <= 6800.0) {
		tmp = fabs(fma(((t_0 / sqrt(((double) M_PI))) * (fabs(x) * fabs(x))), 0.6666666666666666, (t_0 * 1.1283791670955126)));
	} else {
		tmp = fabs((pow(t_0, 7.0) * 0.047619047619047616)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = abs(abs(x))
	tmp = 0.0
	if (abs(x) <= 6800.0)
		tmp = abs(fma(Float64(Float64(t_0 / sqrt(pi)) * Float64(abs(x) * abs(x))), 0.6666666666666666, Float64(t_0 * 1.1283791670955126)));
	else
		tmp = Float64(abs(Float64((t_0 ^ 7.0) * 0.047619047619047616)) / sqrt(pi));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 6800.0], N[Abs[N[(N[(N[(t$95$0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.6666666666666666 + N[(t$95$0 * 1.1283791670955126), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(N[Power[t$95$0, 7.0], $MachinePrecision] * 0.047619047619047616), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 6800

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-PI.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-PI.f64 89.1%

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

   6. Applied rewrites 89.1%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 89.1%

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 89.1%

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*r/ N/A

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

      16. *-commutative N/A

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

   8. Applied rewrites 89.5%

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

   9. Evaluated real constant 89.5%

      \[\leadsto \left|\mathsf{fma}\left(\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(x \cdot x\right), 0.6666666666666666, \left|x\right| \cdot 1.1283791670955126\right)\right| \]

   if 6800 < 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%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-fabs.f64 37.1%

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

   6. Applied rewrites 37.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. pow3 N/A

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

      7. pow3 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. swap-sqr N/A

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

      11. pow3 N/A

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

      12. lift-fabs.f64 N/A

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

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

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

      14. pow1/2 N/A

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

      15. pow-prod-up N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. sqrt-pow2 N/A

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

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

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

      20. lift-fabs.f64 N/A

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

      21. lift-pow.f64 37.1%

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

   8. Applied rewrites 37.1%

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

Alternative 8: 99.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} t_0 := \left|\left|x\right|\right|\\ t_1 := \left|x\right| \cdot \left|x\right|\\ t_2 := t\_1 \cdot \left|x\right|\\ \mathbf{if}\;\left|x\right| \leq 6800:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{t\_0}{\sqrt{\pi}} \cdot t\_1, 0.6666666666666666, t\_0 \cdot 1.1283791670955126\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|t\_2 \cdot \left(\left(0.047619047619047616 \cdot t\_2\right) \cdot t\_0\right)\right|}{1.772453850905516}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fabs (fabs x)))
        (t_1 (* (fabs x) (fabs x)))
        (t_2 (* t_1 (fabs x))))
   (if (<= (fabs x) 6800.0)
     (fabs
      (fma
       (* (/ t_0 (sqrt PI)) t_1)
       0.6666666666666666
       (* t_0 1.1283791670955126)))
     (/
      (fabs (* t_2 (* (* 0.047619047619047616 t_2) t_0)))
      1.772453850905516))))

C

double code(double x) {
	double t_0 = fabs(fabs(x));
	double t_1 = fabs(x) * fabs(x);
	double t_2 = t_1 * fabs(x);
	double tmp;
	if (fabs(x) <= 6800.0) {
		tmp = fabs(fma(((t_0 / sqrt(((double) M_PI))) * t_1), 0.6666666666666666, (t_0 * 1.1283791670955126)));
	} else {
		tmp = fabs((t_2 * ((0.047619047619047616 * t_2) * t_0))) / 1.772453850905516;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = abs(abs(x))
	t_1 = Float64(abs(x) * abs(x))
	t_2 = Float64(t_1 * abs(x))
	tmp = 0.0
	if (abs(x) <= 6800.0)
		tmp = abs(fma(Float64(Float64(t_0 / sqrt(pi)) * t_1), 0.6666666666666666, Float64(t_0 * 1.1283791670955126)));
	else
		tmp = Float64(abs(Float64(t_2 * Float64(Float64(0.047619047619047616 * t_2) * t_0))) / 1.772453850905516);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 6800.0], N[Abs[N[(N[(N[(t$95$0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * 0.6666666666666666 + N[(t$95$0 * 1.1283791670955126), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(t$95$2 * N[(N[(0.047619047619047616 * t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 1.772453850905516), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 6800

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-PI.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-PI.f64 89.1%

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

   6. Applied rewrites 89.1%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 89.1%

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 89.1%

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*r/ N/A

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

      16. *-commutative N/A

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

   8. Applied rewrites 89.5%

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

   9. Evaluated real constant 89.5%

      \[\leadsto \left|\mathsf{fma}\left(\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(x \cdot x\right), 0.6666666666666666, \left|x\right| \cdot 1.1283791670955126\right)\right| \]

   if 6800 < 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%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-fabs.f64 37.1%

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

   6. Applied rewrites 37.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow3 N/A

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

      8. pow3 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lift-fabs.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-fabs.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-*r* N/A

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

      20. lower-*.f64 N/A

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

   8. Applied rewrites 37.1%

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

   9. Evaluated real constant 37.1%

      \[\leadsto \frac{\left|\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left|x\right|\right)\right|}{\color{blue}{1.772453850905516}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 99.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \left|\frac{\mathsf{fma}\left(0.3333333333333333, x \cdot x, \mathsf{fma}\left(0.023809523809523808 \cdot t\_0, t\_0, 1\right)\right)}{\sqrt{\pi}} \cdot x\right| \cdot 2 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (*
    (fabs
     (*
      (/
       (fma
        0.3333333333333333
        (* x x)
        (fma (* 0.023809523809523808 t_0) t_0 1.0))
       (sqrt PI))
      x))
    2.0)))

C

double code(double x) {
	double t_0 = (x * x) * x;
	return fabs(((fma(0.3333333333333333, (x * x), fma((0.023809523809523808 * t_0), t_0, 1.0)) / sqrt(((double) M_PI))) * x)) * 2.0;
}

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	return Float64(abs(Float64(Float64(fma(0.3333333333333333, Float64(x * x), fma(Float64(0.023809523809523808 * t_0), t_0, 1.0)) / sqrt(pi)) * x)) * 2.0)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, N[(N[Abs[N[(N[(N[(0.3333333333333333 * N[(x * x), $MachinePrecision] + N[(N[(0.023809523809523808 * t$95$0), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\left(\left|\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right)}{2}, 1, \mathsf{fma}\left(0.023809523809523808, \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x, 1\right)\right)\right| \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \left(2 \cdot \left|x\right|\right)} \]

7. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right) \cdot 0.5, x \cdot x, \mathsf{fma}\left(0.023809523809523808 \cdot \left(\left(x \cdot x\right) \cdot x\right), \left(x \cdot x\right) \cdot x, 1\right)\right)}{\sqrt{\pi}} \cdot x\right| \cdot 2} \]

8. Taylor expanded in x around 0

   \[\leadsto \left|\frac{\mathsf{fma}\left(\color{blue}{\frac{1}{3}}, x \cdot x, \mathsf{fma}\left(0.023809523809523808 \cdot \left(\left(x \cdot x\right) \cdot x\right), \left(x \cdot x\right) \cdot x, 1\right)\right)}{\sqrt{\pi}} \cdot x\right| \cdot 2 \]
9. Step-by-step derivation

10. Applied rewrites 99.2%

    \[\leadsto \left|\frac{\mathsf{fma}\left(\color{blue}{0.3333333333333333}, x \cdot x, \mathsf{fma}\left(0.023809523809523808 \cdot \left(\left(x \cdot x\right) \cdot x\right), \left(x \cdot x\right) \cdot x, 1\right)\right)}{\sqrt{\pi}} \cdot x\right| \cdot 2 \]
11. Add Preprocessing

Alternative 10: 98.4% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-fabs.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-fabs.f64 98.4%

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

6. Applied rewrites 98.4%

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

Alternative 11: 89.5% accurate, 3.9× speedup

Math

\[\left|\mathsf{fma}\left(\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(x \cdot x\right), 0.6666666666666666, \left|x\right| \cdot 1.1283791670955126\right)\right| \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (fma
   (* (/ (fabs x) (sqrt PI)) (* x x))
   0.6666666666666666
   (* (fabs x) 1.1283791670955126))))

C

double code(double x) {
	return fabs(fma(((fabs(x) / sqrt(((double) M_PI))) * (x * x)), 0.6666666666666666, (fabs(x) * 1.1283791670955126)));
}

Julia

function code(x)
	return abs(fma(Float64(Float64(abs(x) / sqrt(pi)) * Float64(x * x)), 0.6666666666666666, Float64(abs(x) * 1.1283791670955126)))
end

Wolfram

code[x_] := N[Abs[N[(N[(N[(N[Abs[x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.6666666666666666 + N[(N[Abs[x], $MachinePrecision] * 1.1283791670955126), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-fabs.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-fabs.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-PI.f64 89.1%

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

6. Applied rewrites 89.1%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 89.1%

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. pow2 N/A

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

   8. associate-/l* N/A

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

   9. lift-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 89.1%

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

   13. lift-*.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*r/ N/A

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

   16. *-commutative N/A

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

8. Applied rewrites 89.5%

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

9. Evaluated real constant 89.5%

   \[\leadsto \left|\mathsf{fma}\left(\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(x \cdot x\right), 0.6666666666666666, \left|x\right| \cdot 1.1283791670955126\right)\right| \]
10. Add Preprocessing

Alternative 12: 89.1% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-fabs.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-fabs.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-PI.f64 89.1%

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

6. Applied rewrites 89.1%

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

   1. lift-fma.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. lift-*.f64 N/A

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

   8. div-add-rev N/A

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

8. Applied rewrites 89.1%

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

Alternative 13: 83.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{if}\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\left|\left|x\right| \cdot 1.1283791670955126\right|\\ \mathbf{else}:\\ \;\;\;\;\left|2 \cdot \sqrt{\frac{x \cdot x}{\pi}}\right|\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (if (<=
        (fabs
         (*
          (/ 1.0 (sqrt PI))
          (+
           (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
           (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))
        2e+306)
     (fabs (* (fabs x) 1.1283791670955126))
     (fabs (* 2.0 (sqrt (/ (* x x) PI)))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	double tmp;
	if (fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x)))))) <= 2e+306) {
		tmp = fabs((fabs(x) * 1.1283791670955126));
	} else {
		tmp = fabs((2.0 * sqrt(((x * x) / ((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	double tmp;
	if (Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x)))))) <= 2e+306) {
		tmp = Math.abs((Math.abs(x) * 1.1283791670955126));
	} else {
		tmp = Math.abs((2.0 * Math.sqrt(((x * x) / Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	tmp = 0
	if math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x)))))) <= 2e+306:
		tmp = math.fabs((math.fabs(x) * 1.1283791670955126))
	else:
		tmp = math.fabs((2.0 * math.sqrt(((x * x) / math.pi))))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	tmp = 0.0
	if (abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x)))))) <= 2e+306)
		tmp = abs(Float64(abs(x) * 1.1283791670955126));
	else
		tmp = abs(Float64(2.0 * sqrt(Float64(Float64(x * x) / pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = 0.0;
	if (abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x)))))) <= 2e+306)
		tmp = abs((abs(x) * 1.1283791670955126));
	else
		tmp = abs((2.0 * sqrt(((x * x) / pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2e+306], N[Abs[N[(N[Abs[x], $MachinePrecision] * 1.1283791670955126), $MachinePrecision]], $MachinePrecision], N[Abs[N[(2.0 * N[Sqrt[N[(N[(x * x), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))))) < 2e306

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 67.1%

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

   6. Applied rewrites 67.1%

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

   7. Evaluated real constant 67.3%

      \[\leadsto \left|2 \cdot \frac{\left|x\right|}{1.772453850905516}\right| \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|2 \cdot \color{blue}{\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      2. count-2-rev N/A

         \[\leadsto \left|\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}} + \color{blue}{\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{\left|x\right|}}{\frac{7982422502469483}{4503599627370496}}\right| \]

      4. mult-flip N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{\left|x\right|}}{\frac{7982422502469483}{4503599627370496}}\right| \]

      5. lift-/.f64 N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\left|x\right|}{\color{blue}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      6. mult-flip N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \left|x\right| \cdot \color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      7. distribute-lft-out N/A

         \[\leadsto \left|\left|x\right| \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\left|x\right| \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      9. metadata-eval N/A

         \[\leadsto \left|\left|x\right| \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{\color{blue}{1}}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      10. metadata-eval N/A

          \[\leadsto \left|\left|x\right| \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{4503599627370496}{7982422502469483}\right)\right| \]

      11. metadata-eval 67.5%

          \[\leadsto \left|\left|x\right| \cdot 1.1283791670955126\right| \]

   9. Applied rewrites 67.5%

      \[\leadsto \left|\left|x\right| \cdot \color{blue}{1.1283791670955126}\right| \]

   if 2e306 < (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))))))

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 67.1%

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

   6. Applied rewrites 67.1%

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

      1. lift-/.f64 N/A

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

      2. lift-fabs.f64 N/A

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

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

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-undiv N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 54.6%

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

   8. Applied rewrites 54.6%

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

Alternative 14: 67.5% accurate, 15.7× speedup

Math

\[\left|\left|x\right| \cdot 1.1283791670955126\right| \]

FPCore

(FPCore (x) :precision binary64 (fabs (* (fabs x) 1.1283791670955126)))

C

double code(double x) {
	return fabs((fabs(x) * 1.1283791670955126));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = abs((abs(x) * 1.1283791670955126d0))
end function

Java

public static double code(double x) {
	return Math.abs((Math.abs(x) * 1.1283791670955126));
}

Python

def code(x):
	return math.fabs((math.fabs(x) * 1.1283791670955126))

Julia

function code(x)
	return abs(Float64(abs(x) * 1.1283791670955126))
end

MATLAB

function tmp = code(x)
	tmp = abs((abs(x) * 1.1283791670955126));
end

Wolfram

code[x_] := N[Abs[N[(N[Abs[x], $MachinePrecision] * 1.1283791670955126), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-fabs.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-PI.f64 67.1%

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

6. Applied rewrites 67.1%

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

7. Evaluated real constant 67.3%

   \[\leadsto \left|2 \cdot \frac{\left|x\right|}{1.772453850905516}\right| \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left|2 \cdot \color{blue}{\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}}\right| \]

   2. count-2-rev N/A

      \[\leadsto \left|\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}} + \color{blue}{\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}}\right| \]

   3. lift-/.f64 N/A

      \[\leadsto \left|\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{\left|x\right|}}{\frac{7982422502469483}{4503599627370496}}\right| \]

   4. mult-flip N/A

      \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{\left|x\right|}}{\frac{7982422502469483}{4503599627370496}}\right| \]

   5. lift-/.f64 N/A

      \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\left|x\right|}{\color{blue}{\frac{7982422502469483}{4503599627370496}}}\right| \]

   6. mult-flip N/A

      \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \left|x\right| \cdot \color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}}\right| \]

   7. distribute-lft-out N/A

      \[\leadsto \left|\left|x\right| \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

   8. lower-*.f64 N/A

      \[\leadsto \left|\left|x\right| \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

   9. metadata-eval N/A

      \[\leadsto \left|\left|x\right| \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{\color{blue}{1}}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

   10. metadata-eval N/A

       \[\leadsto \left|\left|x\right| \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{4503599627370496}{7982422502469483}\right)\right| \]

   11. metadata-eval 67.5%

       \[\leadsto \left|\left|x\right| \cdot 1.1283791670955126\right| \]

9. Applied rewrites 67.5%

   \[\leadsto \left|\left|x\right| \cdot \color{blue}{1.1283791670955126}\right| \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025204 
(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)))))))