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

Percentage Accurate: 99.8% → 99.9%

Time: 10.1s

Alternatives: 11

Speedup: 2.8×

• 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: 99.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. pow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 99.9

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

9. Applied rewrites 99.9%

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

Alternative 2: 99.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.86:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{0.6666666666666666}{\sqrt{\pi}} \cdot x, x, \frac{2}{\sqrt{\pi}}\right) \cdot \left|x\right|\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\left(\left(\frac{\mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right)}{\sqrt{\pi}} \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x\right) \cdot x\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.86)
   (fabs
    (*
     (fma (* (/ 0.6666666666666666 (sqrt PI)) x) x (/ 2.0 (sqrt PI)))
     (fabs x)))
   (fabs
    (*
     (*
      (* (* (/ (fma (* x x) 0.047619047619047616 0.2) (sqrt PI)) (* x x)) x)
      x)
     x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.86) {
		tmp = fabs((fma(((0.6666666666666666 / sqrt(((double) M_PI))) * x), x, (2.0 / sqrt(((double) M_PI)))) * fabs(x)));
	} else {
		tmp = fabs((((((fma((x * x), 0.047619047619047616, 0.2) / sqrt(((double) M_PI))) * (x * x)) * x) * x) * x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.86)
		tmp = abs(Float64(fma(Float64(Float64(0.6666666666666666 / sqrt(pi)) * x), x, Float64(2.0 / sqrt(pi))) * abs(x)));
	else
		tmp = abs(Float64(Float64(Float64(Float64(Float64(fma(Float64(x * x), 0.047619047619047616, 0.2) / sqrt(pi)) * Float64(x * x)) * x) * x) * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.86], N[Abs[N[(N[(N[(N[(0.6666666666666666 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x + N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.047619047619047616 + 0.2), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8600000000000001

   1. Initial program 99.8%

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

   3. Applied rewrites 82.9%

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

   4. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

   6. Applied rewrites 89.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. div-add N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

      9. associate-*l/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. associate-*r/ N/A

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

      16. lower-fma.f64 N/A

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

   8. Applied rewrites 89.1%

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

   if 1.8600000000000001 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 82.9%

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

   4. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-fabs.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-sqrt.f64 67.9

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

   6. Applied rewrites 67.9%

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

   7. Taylor expanded in x around inf

      \[\leadsto \left|\color{blue}{{x}^{7} \cdot \left(\frac{1}{21} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}} + \frac{1}{5} \cdot \frac{1}{{x}^{2} \cdot \sqrt{\mathsf{PI}\left(\right)}}\right)}\right| \]

   9. Applied rewrites 36.9%

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

Alternative 3: 89.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{0.6666666666666666}{\sqrt{\pi}} \cdot x, x, \frac{2}{\sqrt{\pi}}\right) \cdot \left|x\right|\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{{x}^{7}}{\sqrt{\pi}} \cdot 0.047619047619047616\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (fabs
    (*
     (fma (* (/ 0.6666666666666666 (sqrt PI)) x) x (/ 2.0 (sqrt PI)))
     (fabs x)))
   (fabs (* (/ (pow x 7.0) (sqrt PI)) 0.047619047619047616))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = fabs((fma(((0.6666666666666666 / sqrt(((double) M_PI))) * x), x, (2.0 / sqrt(((double) M_PI)))) * fabs(x)));
	} else {
		tmp = fabs(((pow(x, 7.0) / sqrt(((double) M_PI))) * 0.047619047619047616));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = abs(Float64(fma(Float64(Float64(0.6666666666666666 / sqrt(pi)) * x), x, Float64(2.0 / sqrt(pi))) * abs(x)));
	else
		tmp = abs(Float64(Float64((x ^ 7.0) / sqrt(pi)) * 0.047619047619047616));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[Abs[N[(N[(N[(N[(0.6666666666666666 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x + N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Power[x, 7.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * 0.047619047619047616), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

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

   3. Applied rewrites 82.9%

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

   4. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

   6. Applied rewrites 89.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. div-add N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

      9. associate-*l/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. pow2 N/A

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

      13. associate-*r* N/A

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

      14. metadata-eval N/A

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

      15. associate-*r/ N/A

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

      16. lower-fma.f64 N/A

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

   8. Applied rewrites 89.1%

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

   if 2.2000000000000002 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. sqr-pow N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 N/A

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

      11. pow3 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. pow3 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

   6. Applied rewrites 36.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow3 N/A

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

      13. pow-pow N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. associate-/l* N/A

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

   8. Applied rewrites 36.6%

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

Alternative 4: 89.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in x around inf

   \[\leadsto \left|x \cdot \frac{\mathsf{fma}\left(\color{blue}{\left({x}^{4} \cdot \left(\frac{1}{21} + \frac{1}{5} \cdot \frac{1}{{x}^{2}}\right)\right)} \cdot x, x, 2\right)}{\sqrt{\pi}}\right| \]
9. Step-by-step derivation

   1. distribute-rgt-in N/A

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

   2. associate-*l* N/A

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

   3. pow-flip N/A

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

   4. pow-prod-up N/A

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

   5. metadata-eval N/A

      \[\leadsto \left|x \cdot \frac{\mathsf{fma}\left(\left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{\left(-2 + 4\right)}\right) \cdot x, x, 2\right)}{\sqrt{\pi}}\right| \]

   6. metadata-eval N/A

      \[\leadsto \left|x \cdot \frac{\mathsf{fma}\left(\left(\frac{1}{21} \cdot {x}^{4} + \frac{1}{5} \cdot {x}^{2}\right) \cdot x, x, 2\right)}{\sqrt{\pi}}\right| \]

   7. metadata-eval N/A

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

   8. pow-prod-up N/A

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

   9. associate-*l* N/A

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

   10. distribute-rgt-in N/A

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

   11. pow2 N/A

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

   12. +-commutative N/A

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

   13. associate-*l* N/A

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

   14. +-commutative N/A

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

   15. pow2 N/A

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

   16. *-commutative N/A

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

10. Applied rewrites 99.0%

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

Alternative 5: 89.1% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 82.9%

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

4. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. associate-*r/ N/A

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

   3. div-add-rev N/A

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

6. Applied rewrites 89.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. div-add N/A

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

   7. pow2 N/A

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

   8. *-commutative N/A

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

   9. associate-*l/ N/A

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

   10. metadata-eval N/A

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

   11. associate-*r/ N/A

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

   12. pow2 N/A

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

   13. associate-*r* N/A

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

   14. metadata-eval N/A

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

   15. associate-*r/ N/A

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

   16. lower-fma.f64 N/A

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

8. Applied rewrites 89.1%

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

Alternative 6: 89.1% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 89.1%

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

Alternative 7: 88.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{if}\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(x \cdot \left(x \cdot \frac{x}{\sqrt{\pi}}\right)\right) \cdot 0.6666666666666666\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (if (<=
        (fabs
         (*
          (/ 1.0 (sqrt PI))
          (+
           (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
           (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))
        2e-5)
     (fabs (* x (/ 2.0 (sqrt PI))))
     (fabs (* (* x (* x (/ x (sqrt PI)))) 0.6666666666666666)))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	double tmp;
	if (fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x)))))) <= 2e-5) {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs(((x * (x * (x / sqrt(((double) M_PI))))) * 0.6666666666666666));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	double tmp;
	if (Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x)))))) <= 2e-5) {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs(((x * (x * (x / Math.sqrt(Math.PI)))) * 0.6666666666666666));
	}
	return tmp;
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	tmp = 0
	if math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x)))))) <= 2e-5:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs(((x * (x * (x / math.sqrt(math.pi)))) * 0.6666666666666666))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	tmp = 0.0
	if (abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x)))))) <= 2e-5)
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(Float64(Float64(x * Float64(x * Float64(x / sqrt(pi)))) * 0.6666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = 0.0;
	if (abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x)))))) <= 2e-5)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = abs(((x * (x * (x / sqrt(pi)))) * 0.6666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2e-5], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x * N[(x * N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))))) < 2.00000000000000016e-5

   1. Initial program 99.9%

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

   3. 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. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.7%

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

   if 2.00000000000000016e-5 < (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))))))

   1. Initial program 99.7%

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

   3. Applied rewrites 50.4%

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

   4. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r/ N/A

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

      3. div-add-rev N/A

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

   6. Applied rewrites 68.4%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 66.7%

      \[\leadsto \left|\left(x \cdot \left(x \cdot \frac{x}{\sqrt{\pi}}\right)\right) \cdot \color{blue}{0.6666666666666666}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 83.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{if}\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\sqrt{x \cdot x} \cdot 2\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (if (<=
        (fabs
         (*
          (/ 1.0 (sqrt PI))
          (+
           (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
           (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))
        5e-9)
     (fabs (* x (/ 2.0 (sqrt PI))))
     (/ (fabs (* (sqrt (* x x)) 2.0)) (sqrt PI)))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	double tmp;
	if (fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x)))))) <= 5e-9) {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((sqrt((x * x)) * 2.0)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	double tmp;
	if (Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x)))))) <= 5e-9) {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs((Math.sqrt((x * x)) * 2.0)) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	tmp = 0
	if math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x)))))) <= 5e-9:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs((math.sqrt((x * x)) * 2.0)) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	tmp = 0.0
	if (abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x)))))) <= 5e-9)
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	else
		tmp = Float64(abs(Float64(sqrt(Float64(x * x)) * 2.0)) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = 0.0;
	if (abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x)))))) <= 5e-9)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = abs((sqrt((x * x)) * 2.0)) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 5e-9], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(N[Sqrt[N[(x * x), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))))) < 5.0000000000000001e-9

   1. Initial program 99.9%

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

   3. 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. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.9%

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

   if 5.0000000000000001e-9 < (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))))))

   1. Initial program 99.7%

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

   3. Applied rewrites 36.2%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-fabs.f64 8.9

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

   6. Applied rewrites 8.9%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. pow2 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 52.8

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

   8. Applied rewrites 52.8%

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

Alternative 9: 83.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{\sqrt{\pi}}\\ t_1 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_2 := \left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{if}\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_1\right) + \frac{1}{5} \cdot t\_2\right) + \frac{1}{21} \cdot \left(\left(t\_2 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \leq 10^{-157}:\\ \;\;\;\;\left|x \cdot t\_0\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{x \cdot x} \cdot t\_0\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 2.0 (sqrt PI)))
        (t_1 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_2 (* (* t_1 (fabs x)) (fabs x))))
   (if (<=
        (fabs
         (*
          (/ 1.0 (sqrt PI))
          (+
           (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_1)) (* (/ 1.0 5.0) t_2))
           (* (/ 1.0 21.0) (* (* t_2 (fabs x)) (fabs x))))))
        1e-157)
     (fabs (* x t_0))
     (fabs (* (sqrt (* x x)) t_0)))))

C

double code(double x) {
	double t_0 = 2.0 / sqrt(((double) M_PI));
	double t_1 = (fabs(x) * fabs(x)) * fabs(x);
	double t_2 = (t_1 * fabs(x)) * fabs(x);
	double tmp;
	if (fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_1)) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * ((t_2 * fabs(x)) * fabs(x)))))) <= 1e-157) {
		tmp = fabs((x * t_0));
	} else {
		tmp = fabs((sqrt((x * x)) * t_0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 2.0 / Math.sqrt(Math.PI);
	double t_1 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_2 = (t_1 * Math.abs(x)) * Math.abs(x);
	double tmp;
	if (Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_1)) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * ((t_2 * Math.abs(x)) * Math.abs(x)))))) <= 1e-157) {
		tmp = Math.abs((x * t_0));
	} else {
		tmp = Math.abs((Math.sqrt((x * x)) * t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 2.0 / math.sqrt(math.pi)
	t_1 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_2 = (t_1 * math.fabs(x)) * math.fabs(x)
	tmp = 0
	if math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_1)) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * ((t_2 * math.fabs(x)) * math.fabs(x)))))) <= 1e-157:
		tmp = math.fabs((x * t_0))
	else:
		tmp = math.fabs((math.sqrt((x * x)) * t_0))
	return tmp

Julia

function code(x)
	t_0 = Float64(2.0 / sqrt(pi))
	t_1 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_2 = Float64(Float64(t_1 * abs(x)) * abs(x))
	tmp = 0.0
	if (abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_1)) + Float64(Float64(1.0 / 5.0) * t_2)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_2 * abs(x)) * abs(x)))))) <= 1e-157)
		tmp = abs(Float64(x * t_0));
	else
		tmp = abs(Float64(sqrt(Float64(x * x)) * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 2.0 / sqrt(pi);
	t_1 = (abs(x) * abs(x)) * abs(x);
	t_2 = (t_1 * abs(x)) * abs(x);
	tmp = 0.0;
	if (abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_1)) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * ((t_2 * abs(x)) * abs(x)))))) <= 1e-157)
		tmp = abs((x * t_0));
	else
		tmp = abs((sqrt((x * x)) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$2 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1e-157], N[Abs[N[(x * t$95$0), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sqrt[N[(x * x), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))))) < 9.99999999999999943e-158

   1. Initial program 99.9%

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

   3. 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. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.9%

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

   if 9.99999999999999943e-158 < (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))))))

   1. Initial program 99.8%

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

   3. Applied rewrites 74.7%

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

   4. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-fabs.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-sqrt.f64 52.6

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

   6. Applied rewrites 52.6%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. lower-sqrt.f64 N/A

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

      4. lift-*.f64 75.3

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

   8. Applied rewrites 75.3%

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

Alternative 10: 67.9% accurate, 9.2× speedup

Math

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

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 67.9%

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

Alternative 11: 67.5% accurate, 9.4× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (fabs (/ (+ x x) (sqrt PI))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. count-2-rev N/A

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

   6. lower-+.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-sqrt.f64 67.5

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

10. Applied rewrites 67.5%

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

Reproduce

herbie shell --seed 2025134 
(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)))))))