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

Percentage Accurate: 99.8% → 99.8%

Time: 13.8s

Alternatives: 16

Speedup: 3.0×

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

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.9%

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

8. Final simplification 99.9%

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

Alternative 3: 99.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2

   1. Initial program 99.9%

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

   7. Applied rewrites 99.5%

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

   if 2 < (fabs.f64 x)

   1. Initial program 99.9%

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

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   9. Applied rewrites 99.9%

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

   11. Applied rewrites 99.9%

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

4. Final simplification 99.6%

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

Alternative 4: 99.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

6. Applied rewrites 99.3%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 99.9%

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

10. Final simplification 99.9%

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

Alternative 5: 99.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

6. Applied rewrites 99.3%

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

8. Applied rewrites 99.8%

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

Alternative 6: 98.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

6. Applied rewrites 99.3%

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

7. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow3 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 99.1

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

9. Applied rewrites 99.1%

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

10. Final simplification 99.1%

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

Alternative 7: 93.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-+l+ N/A

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

7. Applied rewrites 94.2%

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

Alternative 8: 93.4% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-+l+ N/A

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

7. Applied rewrites 94.2%

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

9. Applied rewrites 94.2%

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

Alternative 9: 93.0% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

6. Applied rewrites 99.3%

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

7. Taylor expanded in x around 0

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

   1. lower-*.f64 93.7

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

9. Applied rewrites 93.7%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. associate-*r/ N/A

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

    4. lower-/.f64 N/A

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

11. Applied rewrites 93.7%

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

Alternative 10: 93.0% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

6. Applied rewrites 99.3%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 93.7

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

9. Applied rewrites 93.7%

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

Alternative 11: 89.1% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

6. Applied rewrites 99.3%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-in N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-in N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

9. Applied rewrites 89.0%

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

10. Final simplification 89.0%

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

Alternative 12: 89.1% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 89.0

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

7. Applied rewrites 89.0%

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

   1. lift-fabs.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. fabs-div N/A

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

9. Applied rewrites 88.4%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. *-commutative N/A

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

    4. associate-/l* N/A

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

    5. lower-*.f64 N/A

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

11. Applied rewrites 88.9%

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

12. Final simplification 88.9%

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

Alternative 13: 88.7% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 89.0

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

7. Applied rewrites 89.0%

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

   1. lift-fabs.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. fabs-div N/A

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

9. Applied rewrites 88.4%

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

11. Applied rewrites 88.4%

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

Alternative 14: 88.6% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

6. Applied rewrites 99.3%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 88.4

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

9. Applied rewrites 88.4%

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

Alternative 15: 67.7% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

6. Applied rewrites 99.3%

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

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-fabs.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-PI.f64 73.3

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

9. Applied rewrites 73.3%

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

Alternative 16: 67.2% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

6. Applied rewrites 99.3%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 72.7%

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

Reproduce

herbie shell --seed 2024237 
(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)))))))