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

Percentage Accurate: 99.9% → 99.8%

Time: 13.2s

Alternatives: 16

Speedup: N/A×

• 28.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

Alternative 3: 98.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left|\left|x\right| \cdot \frac{\mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x\right| \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right), 0.6666666666666666\right)\right)\right)}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 2e-5)
   (fabs (* (fabs x) (/ (fma x (* x 0.6666666666666666) 2.0) (sqrt PI))))
   (/
    (*
     (fabs x)
     (*
      x
      (*
       x
       (fma
        x
        (* x (fma (* x x) 0.047619047619047616 0.2))
        0.6666666666666666))))
    (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 2e-5) {
		tmp = fabs((fabs(x) * (fma(x, (x * 0.6666666666666666), 2.0) / sqrt(((double) M_PI)))));
	} else {
		tmp = (fabs(x) * (x * (x * fma(x, (x * fma((x * x), 0.047619047619047616, 0.2)), 0.6666666666666666)))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 2e-5)
		tmp = abs(Float64(abs(x) * Float64(fma(x, Float64(x * 0.6666666666666666), 2.0) / sqrt(pi))));
	else
		tmp = Float64(Float64(abs(x) * Float64(x * Float64(x * fma(x, Float64(x * fma(Float64(x * x), 0.047619047619047616, 0.2)), 0.6666666666666666)))) / sqrt(pi));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 2e-5], N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(N[(x * N[(x * 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[Abs[x], $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.047619047619047616 + 0.2), $MachinePrecision]), $MachinePrecision] + 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2.00000000000000016e-5

   1. Initial program 99.8%

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.8%

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in x around 0

       \[\leadsto \left|\frac{2 + \frac{2}{3} \cdot {x}^{2}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right| \]
   11. Step-by-step derivation

   12. Applied rewrites 99.8%

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

   if 2.00000000000000016e-5 < (fabs.f64 x)

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fabs.f64 5.8

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

   7. Applied rewrites 5.8%

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

      1. lift-fabs.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

   9. Applied rewrites 5.8%

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

   10. Taylor expanded in x around inf

       \[\leadsto \frac{\color{blue}{{x}^{6} \cdot \left(\frac{1}{21} \cdot \left|x\right| + \left(\frac{1}{5} \cdot \frac{\left|x\right|}{{x}^{2}} + \frac{2}{3} \cdot \frac{\left|x\right|}{{x}^{4}}\right)\right)}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   12. Applied rewrites 99.0%

       \[\leadsto \frac{\color{blue}{\left|x\right| \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right), 0.6666666666666666\right)\right)\right)}}{\sqrt{\pi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left|\left|x\right| \cdot \frac{\mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x\right| \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right), 0.6666666666666666\right)\right)\right)}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left|\left|x\right| \cdot \frac{\mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right)\right)}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 2e-5)
   (fabs (* (fabs x) (/ (fma x (* x 0.6666666666666666) 2.0) (sqrt PI))))
   (/
    (* (* x (* x (* x x))) (* (fabs x) (fma (* x x) 0.047619047619047616 0.2)))
    (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 2e-5) {
		tmp = fabs((fabs(x) * (fma(x, (x * 0.6666666666666666), 2.0) / sqrt(((double) M_PI)))));
	} else {
		tmp = ((x * (x * (x * x))) * (fabs(x) * fma((x * x), 0.047619047619047616, 0.2))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 2e-5)
		tmp = abs(Float64(abs(x) * Float64(fma(x, Float64(x * 0.6666666666666666), 2.0) / sqrt(pi))));
	else
		tmp = Float64(Float64(Float64(x * Float64(x * Float64(x * x))) * Float64(abs(x) * fma(Float64(x * x), 0.047619047619047616, 0.2))) / sqrt(pi));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 2e-5], N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(N[(x * N[(x * 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.047619047619047616 + 0.2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2.00000000000000016e-5

   1. Initial program 99.8%

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.8%

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in x around 0

       \[\leadsto \left|\frac{2 + \frac{2}{3} \cdot {x}^{2}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right| \]
   11. Step-by-step derivation

   12. Applied rewrites 99.8%

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

   if 2.00000000000000016e-5 < (fabs.f64 x)

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fabs.f64 5.8

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

   7. Applied rewrites 5.8%

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

      1. lift-fabs.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

   9. Applied rewrites 5.8%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 99.0%

       \[\leadsto \frac{\color{blue}{\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right)\right)}}{\sqrt{\pi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left|\left|x\right| \cdot \frac{\mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right)\right)}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left|\left|x\right| \cdot \frac{\mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left|x\right| \cdot 0.047619047619047616\right)}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 2e-5)
   (fabs (* (fabs x) (/ (fma x (* x 0.6666666666666666) 2.0) (sqrt PI))))
   (/
    (* (* (* x x) (* x (* x (* x x)))) (* (fabs x) 0.047619047619047616))
    (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 2e-5) {
		tmp = fabs((fabs(x) * (fma(x, (x * 0.6666666666666666), 2.0) / sqrt(((double) M_PI)))));
	} else {
		tmp = (((x * x) * (x * (x * (x * x)))) * (fabs(x) * 0.047619047619047616)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 2e-5)
		tmp = abs(Float64(abs(x) * Float64(fma(x, Float64(x * 0.6666666666666666), 2.0) / sqrt(pi))));
	else
		tmp = Float64(Float64(Float64(Float64(x * x) * Float64(x * Float64(x * Float64(x * x)))) * Float64(abs(x) * 0.047619047619047616)) / sqrt(pi));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 2e-5], N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(N[(x * N[(x * 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * 0.047619047619047616), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2.00000000000000016e-5

   1. Initial program 99.8%

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.8%

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in x around 0

       \[\leadsto \left|\frac{2 + \frac{2}{3} \cdot {x}^{2}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right| \]
   11. Step-by-step derivation

   12. Applied rewrites 99.8%

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

   if 2.00000000000000016e-5 < (fabs.f64 x)

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fabs.f64 5.8

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

   7. Applied rewrites 5.8%

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

      1. lift-fabs.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

   9. Applied rewrites 5.8%

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

   10. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. metadata-eval N/A

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

       6. pow-sqr N/A

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

       7. cube-prod N/A

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

       8. unpow2 N/A

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

       9. cube-unmult N/A

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

       10. pow-sqr N/A

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

       11. metadata-eval N/A

           \[\leadsto \frac{\left({x}^{2} \cdot {x}^{\color{blue}{4}}\right) \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

       12. lower-*.f64 N/A

           \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot {x}^{4}\right)} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

       13. unpow2 N/A

           \[\leadsto \frac{\left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{4}\right) \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

       14. lower-*.f64 N/A

           \[\leadsto \frac{\left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{4}\right) \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

       15. metadata-eval N/A

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

       16. pow-plus N/A

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

       17. lower-*.f64 N/A

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

       18. cube-mult N/A

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

       19. unpow2 N/A

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

       20. lower-*.f64 N/A

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

       21. unpow2 N/A

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

       22. lower-*.f64 N/A

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

       23. *-commutative N/A

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

       24. lower-*.f64 N/A

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

       25. lower-fabs.f64 98.8

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

   12. Applied rewrites 98.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left|\left|x\right| \cdot \frac{\mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left|x\right| \cdot 0.047619047619047616\right)}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

8. Applied rewrites 99.8%

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

9. Final simplification 99.8%

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

Alternative 7: 99.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

8. Applied rewrites 99.8%

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

10. Applied rewrites 99.4%

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

11. Final simplification 99.4%

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

Alternative 8: 99.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

8. Applied rewrites 99.4%

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

9. Final simplification 99.4%

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

Alternative 9: 99.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

8. Applied rewrites 99.4%

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

Alternative 10: 93.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left|\left|x\right| \cdot \frac{\mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left|x\right| \cdot \frac{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.2\right)\right)}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 2e-5)
   (fabs (* (fabs x) (/ (fma x (* x 0.6666666666666666) 2.0) (sqrt PI))))
   (fabs (* (fabs x) (/ (* x (* x (* (* x x) 0.2))) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 2e-5) {
		tmp = fabs((fabs(x) * (fma(x, (x * 0.6666666666666666), 2.0) / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((fabs(x) * ((x * (x * ((x * x) * 0.2))) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 2e-5)
		tmp = abs(Float64(abs(x) * Float64(fma(x, Float64(x * 0.6666666666666666), 2.0) / sqrt(pi))));
	else
		tmp = abs(Float64(abs(x) * Float64(Float64(x * Float64(x * Float64(Float64(x * x) * 0.2))) / sqrt(pi))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 2e-5], N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(N[(x * N[(x * 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2.00000000000000016e-5

   1. Initial program 99.8%

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.8%

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in x around 0

       \[\leadsto \left|\frac{2 + \frac{2}{3} \cdot {x}^{2}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right| \]
   11. Step-by-step derivation

   12. Applied rewrites 99.8%

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

   if 2.00000000000000016e-5 < (fabs.f64 x)

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 78.2%

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

   9. Applied rewrites 78.2%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 78.2%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left|\left|x\right| \cdot \frac{\mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left|x\right| \cdot \frac{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 0.2\right)\right)}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 93.9% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 93.1%

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

Alternative 12: 93.4% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-fabs.f64 70.2

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

7. Applied rewrites 70.2%

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

   1. lift-fabs.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

9. Applied rewrites 69.9%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. distribute-lft-in N/A

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

    3. *-commutative N/A

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

    4. associate-*r* N/A

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

    5. *-commutative N/A

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

    6. associate-+r+ N/A

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

    7. associate-*r* N/A

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

    8. distribute-rgt-in N/A

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

    9. associate-*r* N/A

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

    10. associate-*r* N/A

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

12. Applied rewrites 92.7%

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

Alternative 13: 89.6% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 93.1%

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

9. Applied rewrites 93.1%

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

10. Taylor expanded in x around 0

    \[\leadsto \left|\frac{2 + \frac{2}{3} \cdot {x}^{2}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right| \]
11. Step-by-step derivation

12. Applied rewrites 90.4%

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

13. Final simplification 90.4%

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

Alternative 14: 89.2% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 90.4

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

9. Applied rewrites 90.4%

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

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lift-/.f64 N/A

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

    4. un-div-inv N/A

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

    5. lower-/.f64 90.0

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

11. Applied rewrites 90.0%

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

Alternative 15: 68.2% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 93.1%

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

9. Applied rewrites 93.1%

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

10. Taylor expanded in x around 0

    \[\leadsto \left|\frac{2}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right| \]
11. Step-by-step derivation

12. Applied rewrites 70.3%

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

13. Final simplification 70.3%

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

Alternative 16: 67.8% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-fabs.f64 70.2

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

7. Applied rewrites 70.2%

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

   1. lift-fabs.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

9. Applied rewrites 69.9%

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

10. Final simplification 69.9%

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

Reproduce

herbie shell --seed 2024232 
(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)))))))