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

Percentage Accurate: 99.8% → 99.8%

Time: 7.6s

Alternatives: 11

Speedup: 2.1×

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

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 2.1× speedup

Math

\[0.5641895835477563 \cdot \left|\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right), 2\right) \cdot x\right)\right| \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  0.5641895835477563
  (fabs
   (fma
    0.047619047619047616
    (pow x 7.0)
    (* (fma (* x x) (fma (* 0.2 x) x 0.6666666666666666) 2.0) x)))))

C

double code(double x) {
	return 0.5641895835477563 * fabs(fma(0.047619047619047616, pow(x, 7.0), (fma((x * x), fma((0.2 * x), x, 0.6666666666666666), 2.0) * x)));
}

Julia

function code(x)
	return Float64(0.5641895835477563 * abs(fma(0.047619047619047616, (x ^ 7.0), Float64(fma(Float64(x * x), fma(Float64(0.2 * x), x, 0.6666666666666666), 2.0) * x))))
end

Wolfram

code[x_] := N[(0.5641895835477563 * N[Abs[N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision] + N[(N[(N[(x * x), $MachinePrecision] * N[(N[(0.2 * x), $MachinePrecision] * x + 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision] * x), $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%

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.8%

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

8. Evaluated real constant 99.8%

   \[\leadsto \color{blue}{0.5641895835477563} \cdot \left|\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right), 2\right) \cdot x\right)\right| \]
9. Add Preprocessing

Alternative 2: 99.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_0 := \left|\left|x\right|\right|\\ \mathbf{if}\;\left|x\right| \leq 0.0063:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{1}{\sqrt{\pi}}, t\_0 \cdot 2, \left(0.6666666666666666 \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left|\frac{\left|x\right|}{\sqrt{\pi}}\right|\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.5641895835477563 \cdot \left(0.047619047619047616 \cdot \left({\left(\left|x\right|\right)}^{6} \cdot t\_0\right)\right)\right|\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fabs (fabs x))))
   (if (<= (fabs x) 0.0063)
     (fabs
      (fma
       (/ 1.0 (sqrt PI))
       (* t_0 2.0)
       (*
        (* 0.6666666666666666 (* (fabs x) (fabs x)))
        (fabs (/ (fabs x) (sqrt PI))))))
     (fabs
      (*
       0.5641895835477563
       (* 0.047619047619047616 (* (pow (fabs x) 6.0) t_0)))))))

C

double code(double x) {
	double t_0 = fabs(fabs(x));
	double tmp;
	if (fabs(x) <= 0.0063) {
		tmp = fabs(fma((1.0 / sqrt(((double) M_PI))), (t_0 * 2.0), ((0.6666666666666666 * (fabs(x) * fabs(x))) * fabs((fabs(x) / sqrt(((double) M_PI)))))));
	} else {
		tmp = fabs((0.5641895835477563 * (0.047619047619047616 * (pow(fabs(x), 6.0) * t_0))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = abs(abs(x))
	tmp = 0.0
	if (abs(x) <= 0.0063)
		tmp = abs(fma(Float64(1.0 / sqrt(pi)), Float64(t_0 * 2.0), Float64(Float64(0.6666666666666666 * Float64(abs(x) * abs(x))) * abs(Float64(abs(x) / sqrt(pi))))));
	else
		tmp = abs(Float64(0.5641895835477563 * Float64(0.047619047619047616 * Float64((abs(x) ^ 6.0) * t_0))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.0063], N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * 2.0), $MachinePrecision] + N[(N[(0.6666666666666666 * N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[N[(N[Abs[x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.5641895835477563 * N[(0.047619047619047616 * N[(N[Power[N[Abs[x], $MachinePrecision], 6.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0063

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.64 N/A

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

      5. lower-fabs.64 N/A

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

      6. lower-sqrt.64 N/A

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

      7. lower-PI.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fabs.64 N/A

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

      11. lower-sqrt.64 N/A

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

      12. lower-PI.f64 88.7%

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

   6. Applied rewrites 88.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. count-2-rev N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. lift-/.f64 N/A

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

      11. distribute-rgt-out N/A

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

      12. count-2-rev N/A

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

      13. lower-fma.f64 N/A

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

   8. Applied rewrites 89.1%

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

   if 0.0063 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.64 N/A

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

      4. lower-fabs.64 36.8%

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

   6. Applied rewrites 36.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)}\right| \]

   7. Evaluated real constant 36.8%

      \[\leadsto \left|\color{blue}{0.5641895835477563} \cdot \left(0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} t_0 := \left|\left|x\right|\right|\\ \mathbf{if}\;\left|x\right| \leq 0.0063:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(0.6666666666666666, \left|x\right| \cdot \left|x\right|, 2\right)}{\sqrt{\pi}} \cdot t\_0\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.5641895835477563 \cdot \left(0.047619047619047616 \cdot \left({\left(\left|x\right|\right)}^{6} \cdot t\_0\right)\right)\right|\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fabs (fabs x))))
   (if (<= (fabs x) 0.0063)
     (fabs
      (* (/ (fma 0.6666666666666666 (* (fabs x) (fabs x)) 2.0) (sqrt PI)) t_0))
     (fabs
      (*
       0.5641895835477563
       (* 0.047619047619047616 (* (pow (fabs x) 6.0) t_0)))))))

C

double code(double x) {
	double t_0 = fabs(fabs(x));
	double tmp;
	if (fabs(x) <= 0.0063) {
		tmp = fabs(((fma(0.6666666666666666, (fabs(x) * fabs(x)), 2.0) / sqrt(((double) M_PI))) * t_0));
	} else {
		tmp = fabs((0.5641895835477563 * (0.047619047619047616 * (pow(fabs(x), 6.0) * t_0))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = abs(abs(x))
	tmp = 0.0
	if (abs(x) <= 0.0063)
		tmp = abs(Float64(Float64(fma(0.6666666666666666, Float64(abs(x) * abs(x)), 2.0) / sqrt(pi)) * t_0));
	else
		tmp = abs(Float64(0.5641895835477563 * Float64(0.047619047619047616 * Float64((abs(x) ^ 6.0) * t_0))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.0063], N[Abs[N[(N[(N[(0.6666666666666666 * N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.5641895835477563 * N[(0.047619047619047616 * N[(N[Power[N[Abs[x], $MachinePrecision], 6.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0063

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.64 N/A

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

      5. lower-fabs.64 N/A

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

      6. lower-sqrt.64 N/A

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

      7. lower-PI.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fabs.64 N/A

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

      11. lower-sqrt.64 N/A

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

      12. lower-PI.f64 88.7%

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

   6. Applied rewrites 88.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

   8. Applied rewrites 88.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 89.1%

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

   10. Applied rewrites 89.1%

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

   if 0.0063 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.64 N/A

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

      4. lower-fabs.64 36.8%

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

   6. Applied rewrites 36.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)}\right| \]

   7. Evaluated real constant 36.8%

      \[\leadsto \left|\color{blue}{0.5641895835477563} \cdot \left(0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0063:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(0.6666666666666666, \left|x\right| \cdot \left|x\right|, 2\right)}{\sqrt{\pi}} \cdot \left|\left|x\right|\right|\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left|{\left(\left|x\right|\right)}^{7}\right| \cdot 0.047619047619047616\right|}{\sqrt{\pi}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.0063)
   (fabs
    (*
     (/ (fma 0.6666666666666666 (* (fabs x) (fabs x)) 2.0) (sqrt PI))
     (fabs (fabs x))))
   (/ (fabs (* (fabs (pow (fabs x) 7.0)) 0.047619047619047616)) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.0063) {
		tmp = fabs(((fma(0.6666666666666666, (fabs(x) * fabs(x)), 2.0) / sqrt(((double) M_PI))) * fabs(fabs(x))));
	} else {
		tmp = fabs((fabs(pow(fabs(x), 7.0)) * 0.047619047619047616)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.0063)
		tmp = abs(Float64(Float64(fma(0.6666666666666666, Float64(abs(x) * abs(x)), 2.0) / sqrt(pi)) * abs(abs(x))));
	else
		tmp = Float64(abs(Float64(abs((abs(x) ^ 7.0)) * 0.047619047619047616)) / sqrt(pi));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.0063], N[Abs[N[(N[(N[(0.6666666666666666 * N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(N[Abs[N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision]], $MachinePrecision] * 0.047619047619047616), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0063

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.64 N/A

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

      5. lower-fabs.64 N/A

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

      6. lower-sqrt.64 N/A

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

      7. lower-PI.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fabs.64 N/A

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

      11. lower-sqrt.64 N/A

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

      12. lower-PI.f64 88.7%

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

   6. Applied rewrites 88.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

   8. Applied rewrites 88.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 89.1%

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

   10. Applied rewrites 89.1%

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

   if 0.0063 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.64 N/A

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

      4. lower-fabs.64 36.8%

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

   6. Applied rewrites 36.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)}\right| \]

   8. Applied rewrites 36.8%

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

Alternative 5: 98.6% accurate, 3.0× speedup

Math

\[\left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)}{1.772453850905516}\right| \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (/
   (fma 0.047619047619047616 (pow (fabs x) 7.0) (* 2.0 (fabs x)))
   1.772453850905516)))

C

double code(double x) {
	return fabs((fma(0.047619047619047616, pow(fabs(x), 7.0), (2.0 * fabs(x))) / 1.772453850905516));
}

Julia

function code(x)
	return abs(Float64(fma(0.047619047619047616, (abs(x) ^ 7.0), Float64(2.0 * abs(x))) / 1.772453850905516))
end

Wolfram

code[x_] := N[Abs[N[(N[(0.047619047619047616 * N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision] + N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.772453850905516), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-pow.64 N/A

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

   4. lower-fabs.64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-fabs.64 N/A

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

   7. lower-sqrt.64 N/A

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

   8. lower-PI.f64 98.4%

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

6. Applied rewrites 98.4%

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

7. Evaluated real constant 98.6%

   \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)}{1.772453850905516}\right| \]
8. Add Preprocessing

Alternative 6: 91.4% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.64 N/A

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

   5. lower-fabs.64 N/A

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

   6. lower-sqrt.64 N/A

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

   7. lower-PI.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-fabs.64 N/A

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

   11. lower-sqrt.64 N/A

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

   12. lower-PI.f64 88.7%

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

6. Applied rewrites 88.7%

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

   1. lift-fma.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. div-add-rev N/A

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

8. Applied rewrites 88.7%

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

   1. rem-square-sqrt N/A

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

   2. sqrt-unprod N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. lower-sqrt.64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lower-*.f64 91.4%

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

10. Applied rewrites 91.4%

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

Alternative 7: 89.1% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[Abs[N[(N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $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 99.8%

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

4. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.64 N/A

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

   5. lower-fabs.64 N/A

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

   6. lower-sqrt.64 N/A

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

   7. lower-PI.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-fabs.64 N/A

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

   11. lower-sqrt.64 N/A

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

   12. lower-PI.f64 88.7%

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

6. Applied rewrites 88.7%

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

   1. lift-fma.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. div-add-rev N/A

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

8. Applied rewrites 88.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 89.1%

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

10. Applied rewrites 89.1%

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

Alternative 8: 88.9% accurate, 5.4× speedup

Math

\[\left|\frac{\left|x\right| \cdot \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{1.772453850905516}\right| \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (/ (* (fabs x) (fma 0.6666666666666666 (* x x) 2.0)) 1.772453850905516)))

C

double code(double x) {
	return fabs(((fabs(x) * fma(0.6666666666666666, (x * x), 2.0)) / 1.772453850905516));
}

Julia

function code(x)
	return abs(Float64(Float64(abs(x) * fma(0.6666666666666666, Float64(x * x), 2.0)) / 1.772453850905516))
end

Wolfram

code[x_] := N[Abs[N[(N[(N[Abs[x], $MachinePrecision] * N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / 1.772453850905516), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.64 N/A

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

   5. lower-fabs.64 N/A

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

   6. lower-sqrt.64 N/A

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

   7. lower-PI.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-fabs.64 N/A

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

   11. lower-sqrt.64 N/A

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

   12. lower-PI.f64 88.7%

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

6. Applied rewrites 88.7%

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

   1. lift-fma.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. div-add-rev N/A

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

8. Applied rewrites 88.7%

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

9. Evaluated real constant 88.9%

   \[\leadsto \left|\frac{\left|x\right| \cdot \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{1.772453850905516}\right| \]
10. Add Preprocessing

Alternative 9: 83.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-7}:\\ \;\;\;\;\left|\left|\left|x\right|\right| \cdot 1.1283791670955126\right|\\ \mathbf{else}:\\ \;\;\;\;\left|2 \cdot \sqrt{\frac{\left|x\right| \cdot \left|x\right|}{\pi}}\right|\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 1e-7)
   (fabs (* (fabs (fabs x)) 1.1283791670955126))
   (fabs (* 2.0 (sqrt (/ (* (fabs x) (fabs x)) PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 1e-7) {
		tmp = fabs((fabs(fabs(x)) * 1.1283791670955126));
	} else {
		tmp = fabs((2.0 * sqrt(((fabs(x) * fabs(x)) / ((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 1e-7) {
		tmp = Math.abs((Math.abs(Math.abs(x)) * 1.1283791670955126));
	} else {
		tmp = Math.abs((2.0 * Math.sqrt(((Math.abs(x) * Math.abs(x)) / Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 1e-7:
		tmp = math.fabs((math.fabs(math.fabs(x)) * 1.1283791670955126))
	else:
		tmp = math.fabs((2.0 * math.sqrt(((math.fabs(x) * math.fabs(x)) / math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 1e-7)
		tmp = abs(Float64(abs(abs(x)) * 1.1283791670955126));
	else
		tmp = abs(Float64(2.0 * sqrt(Float64(Float64(abs(x) * abs(x)) / pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 1e-7)
		tmp = abs((abs(abs(x)) * 1.1283791670955126));
	else
		tmp = abs((2.0 * sqrt(((abs(x) * abs(x)) / pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 1e-7], N[Abs[N[(N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision] * 1.1283791670955126), $MachinePrecision]], $MachinePrecision], N[Abs[N[(2.0 * N[Sqrt[N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.9999999999999995e-8

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.64 N/A

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

      4. lower-sqrt.64 N/A

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

      5. lower-PI.f64 67.3%

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

   6. Applied rewrites 67.3%

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

   7. Evaluated real constant 67.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval 67.7%

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

   9. Applied rewrites 67.7%

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

   if 9.9999999999999995e-8 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.64 N/A

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

      4. lower-sqrt.64 N/A

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

      5. lower-PI.f64 67.3%

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

   6. Applied rewrites 67.3%

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

      1. lift-/.f64 N/A

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

      2. lift-fabs.64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. lift-sqrt.64 N/A

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

      6. sqrt-undiv N/A

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

      7. lower-sqrt.64 N/A

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

      8. lower-/.f64 52.6%

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

   8. Applied rewrites 52.6%

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

Alternative 10: 67.7% accurate, 15.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-fabs.64 N/A

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

   4. lower-sqrt.64 N/A

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

   5. lower-PI.f64 67.3%

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

6. Applied rewrites 67.3%

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

7. Evaluated real constant 67.5%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. lift-/.f64 N/A

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

   6. mult-flip N/A

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

   7. distribute-lft-out N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval 67.7%

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

9. Applied rewrites 67.7%

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

Reproduce

herbie shell --seed 2025183 
(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)))))))