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

Percentage Accurate: 99.8% → 99.8%

Time: 14.7s

Alternatives: 7

Speedup: 3.0×

• 28.9% 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, 3.0× speedup

Math

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

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

Wolfram

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

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.9%

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

9. Applied rewrites 99.9%

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

11. Applied rewrites 99.9%

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

Alternative 2: 99.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1

   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%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. lower-*.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

   7. Applied rewrites 99.1%

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

   9. Applied rewrites 99.1%

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

   if 1 < (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%

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

   5. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. pow-plus N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. associate-*l/ N/A

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

      13. *-lft-identity N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. pow-plus N/A

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

      17. pow-plus N/A

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

      18. associate-*r* N/A

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

      19. unpow2 N/A

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

      20. associate-*l/ N/A

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

   7. Applied rewrites 99.6%

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

      1. lift-PI.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

   9. Applied rewrites 99.6%

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

   11. Applied rewrites 99.6%

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

Alternative 3: 93.5% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1

   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%

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

   5. Taylor expanded in x around 0

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   7. Applied rewrites 98.7%

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

      1. lift-fabs.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. fabs-mul N/A

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

      10. lift-fabs.f64 N/A

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

      11. fabs-fabs N/A

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

      12. mul-fabs N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-*.f64 98.7

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

   9. Applied rewrites 98.7%

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

   if 1 < (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%

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

   5. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. pow-plus N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. associate-*l/ N/A

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

      13. *-lft-identity N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. pow-plus N/A

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

      17. pow-plus N/A

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

      18. associate-*r* N/A

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

      19. unpow2 N/A

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

      20. associate-*l/ N/A

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

   7. Applied rewrites 99.6%

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

      1. lift-PI.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

   9. Applied rewrites 99.6%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 79.5%

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

4. Final simplification 92.3%

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

Alternative 4: 93.6% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-out N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt-in N/A

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

   9. lower-*.f64 N/A

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

   10. lower-fabs.f64 N/A

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

   11. +-commutative N/A

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

   12. unpow2 N/A

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

   13. associate-*l* N/A

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

7. Applied rewrites 92.5%

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

9. Applied rewrites 92.5%

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

Alternative 5: 93.2% accurate, 3.6× speedup

Math

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

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

Wolfram

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

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-out N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt-in N/A

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

   9. lower-*.f64 N/A

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

   10. lower-fabs.f64 N/A

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

   11. +-commutative N/A

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

   12. unpow2 N/A

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

   13. associate-*l* N/A

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

7. Applied rewrites 92.5%

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

9. Applied rewrites 92.1%

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

    1. lift-fabs.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-fma.f64 N/A

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

    5. lift-fma.f64 N/A

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

    6. lift-fabs.f64 N/A

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

    7. lift-PI.f64 N/A

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

    8. lift-sqrt.f64 N/A

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

    9. lift-*.f64 N/A

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

    10. lift-/.f64 92.1

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

11. Applied rewrites 92.1%

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

Alternative 6: 89.3% accurate, 4.6× speedup

Math

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

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

Wolfram

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

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

5. Taylor expanded in x around 0

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-in N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-fabs.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

7. Applied rewrites 88.1%

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

   1. lift-fabs.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. fabs-mul N/A

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

   10. lift-fabs.f64 N/A

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

   11. fabs-fabs N/A

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

   12. mul-fabs N/A

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

   13. lower-fabs.f64 N/A

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

   14. lower-*.f64 88.1

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

9. Applied rewrites 88.1%

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

10. Final simplification 88.1%

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

Alternative 7: 67.6% accurate, 6.3× speedup

Math

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

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(abs(x) * Float64(2.0 / sqrt(pi)))
end

MATLAB

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

Wolfram

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

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 67.3%

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

   1. lift-PI.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. inv-pow N/A

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

   4. sqr-pow N/A

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

   5. fabs-sqr N/A

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

   6. sqr-pow N/A

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

   7. inv-pow N/A

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

   8. lift-/.f64 N/A

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

   9. fabs-fabs N/A

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

   10. lift-fabs.f64 N/A

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

   11. metadata-eval N/A

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

   12. fabs-mul N/A

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

   13. lift-*.f64 N/A

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

   14. fabs-mul N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*r* N/A

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

   17. fabs-mul N/A

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

9. Applied rewrites 67.3%

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

Reproduce

herbie shell --seed 2024219 
(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)))))))