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

Percentage Accurate: 99.8% → 99.9%

Time: 4.4s

Alternatives: 11

Speedup: 2.1×

• 28.2% 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: 99.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Abs[N[(N[(N[(0.047619047619047616 * N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x + N[(N[(N[(N[(0.2 * x), $MachinePrecision] * x + 0.6666666666666666), $MachinePrecision] * x), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5641895835477563), $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 99.4%

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

5. Applied rewrites 99.4%

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

6. Evaluated real constant 99.6%

   \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, 2 + \left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{5} \cdot x, x, \frac{2}{3}\right), {\left(\left|x\right|\right)}^{7} \cdot \frac{1}{21}\right)\right|}{\color{blue}{\frac{7982422502469483}{4503599627370496}}} \]

8. Applied rewrites 99.9%

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

Alternative 2: 99.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} t_0 := \left|\left|x\right|\right|\\ \mathbf{if}\;\left|x\right| \leq 0.26:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\left|x\right| \cdot \left|x\right|, 0.6666666666666666, 2\right) \cdot t\_0\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|0.047619047619047616 \cdot \left({\left(\left|x\right|\right)}^{6} \cdot t\_0\right)\right|}{\sqrt{\pi}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fabs (fabs x))))
   (if (<= (fabs x) 0.26)
     (*
      (/ 1.0 (sqrt PI))
      (fabs (* (fma (* (fabs x) (fabs x)) 0.6666666666666666 2.0) t_0)))
     (/
      (fabs (* 0.047619047619047616 (* (pow (fabs x) 6.0) t_0)))
      (sqrt PI)))))

C

double code(double x) {
	double t_0 = fabs(fabs(x));
	double tmp;
	if (fabs(x) <= 0.26) {
		tmp = (1.0 / sqrt(((double) M_PI))) * fabs((fma((fabs(x) * fabs(x)), 0.6666666666666666, 2.0) * t_0));
	} else {
		tmp = fabs((0.047619047619047616 * (pow(fabs(x), 6.0) * t_0))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = abs(abs(x))
	tmp = 0.0
	if (abs(x) <= 0.26)
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(fma(Float64(abs(x) * abs(x)), 0.6666666666666666, 2.0) * t_0)));
	else
		tmp = Float64(abs(Float64(0.047619047619047616 * Float64((abs(x) ^ 6.0) * t_0))) / sqrt(pi));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.26], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(0.047619047619047616 * N[(N[Power[N[Abs[x], $MachinePrecision], 6.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.26000000000000001

   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 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-fabs.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fabs.f64 89.0

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

   6. Applied rewrites 89.0%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

   8. Applied rewrites 89.4%

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

   if 0.26000000000000001 < 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 99.4%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-fabs.f64 36.3

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

   6. Applied rewrites 36.3%

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

Alternative 3: 99.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} t_0 := \left|\left|x\right|\right|\\ \mathbf{if}\;\left|x\right| \leq 0.26:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\left|x\right| \cdot \left|x\right|, 0.6666666666666666, 2\right) \cdot t\_0\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|{t\_0}^{7} \cdot 0.047619047619047616\right|}{\sqrt{\pi}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fabs (fabs x))))
   (if (<= (fabs x) 0.26)
     (*
      (/ 1.0 (sqrt PI))
      (fabs (* (fma (* (fabs x) (fabs x)) 0.6666666666666666 2.0) t_0)))
     (/ (fabs (* (pow t_0 7.0) 0.047619047619047616)) (sqrt PI)))))

C

double code(double x) {
	double t_0 = fabs(fabs(x));
	double tmp;
	if (fabs(x) <= 0.26) {
		tmp = (1.0 / sqrt(((double) M_PI))) * fabs((fma((fabs(x) * fabs(x)), 0.6666666666666666, 2.0) * t_0));
	} else {
		tmp = fabs((pow(t_0, 7.0) * 0.047619047619047616)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = abs(abs(x))
	tmp = 0.0
	if (abs(x) <= 0.26)
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(fma(Float64(abs(x) * abs(x)), 0.6666666666666666, 2.0) * t_0)));
	else
		tmp = Float64(abs(Float64((t_0 ^ 7.0) * 0.047619047619047616)) / sqrt(pi));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.26], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(N[Power[t$95$0, 7.0], $MachinePrecision] * 0.047619047619047616), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.26000000000000001

   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 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-fabs.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fabs.f64 89.0

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

   6. Applied rewrites 89.0%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

   8. Applied rewrites 89.4%

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

   if 0.26000000000000001 < 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 99.4%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-fabs.f64 36.3

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

   6. Applied rewrites 36.3%

      \[\leadsto \frac{\left|\color{blue}{0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)}\right|}{\sqrt{\pi}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow-prod-down N/A

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

      8. lift-fabs.f64 N/A

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

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

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

      10. pow1/2 N/A

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

      11. pow-prod-up N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. sqrt-pow2 N/A

          \[\leadsto \frac{\left|{\left(\sqrt{x \cdot x}\right)}^{7} \cdot \frac{1}{21}\right|}{\sqrt{\pi}} \]

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

          \[\leadsto \frac{\left|{\left(\left|x\right|\right)}^{7} \cdot \frac{1}{21}\right|}{\sqrt{\pi}} \]

      16. lift-fabs.f64 N/A

          \[\leadsto \frac{\left|{\left(\left|x\right|\right)}^{7} \cdot \frac{1}{21}\right|}{\sqrt{\pi}} \]

      17. lift-pow.f64 N/A

          \[\leadsto \frac{\left|{\left(\left|x\right|\right)}^{7} \cdot \frac{1}{21}\right|}{\sqrt{\pi}} \]

      18. lift-*.f64 36.3

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

   8. Applied rewrites 36.3%

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

Alternative 4: 99.0% accurate, 2.4× speedup

Math

\[\frac{\left|\mathsf{fma}\left(\left|x\right|, 2 + \left(x \cdot x\right) \cdot 0.6666666666666666, {\left(\left|x\right|\right)}^{7} \cdot 0.047619047619047616\right)\right|}{1.772453850905516} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fabs
   (fma
    (fabs x)
    (+ 2.0 (* (* x x) 0.6666666666666666))
    (* (pow (fabs x) 7.0) 0.047619047619047616)))
  1.772453850905516))

C

double code(double x) {
	return fabs(fma(fabs(x), (2.0 + ((x * x) * 0.6666666666666666)), (pow(fabs(x), 7.0) * 0.047619047619047616))) / 1.772453850905516;
}

Julia

function code(x)
	return Float64(abs(fma(abs(x), Float64(2.0 + Float64(Float64(x * x) * 0.6666666666666666)), Float64((abs(x) ^ 7.0) * 0.047619047619047616))) / 1.772453850905516)
end

Wolfram

code[x_] := N[(N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(2.0 + N[(N[(x * x), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision] * 0.047619047619047616), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 1.772453850905516), $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 99.4%

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

5. Applied rewrites 99.4%

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

6. Evaluated real constant 99.6%

   \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, 2 + \left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{5} \cdot x, x, \frac{2}{3}\right), {\left(\left|x\right|\right)}^{7} \cdot \frac{1}{21}\right)\right|}{\color{blue}{\frac{7982422502469483}{4503599627370496}}} \]

7. Taylor expanded in x around 0

   \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, 2 + \left(x \cdot x\right) \cdot \color{blue}{\frac{2}{3}}, {\left(\left|x\right|\right)}^{7} \cdot \frac{1}{21}\right)\right|}{\frac{7982422502469483}{4503599627370496}} \]
8. Step-by-step derivation

9. Applied rewrites 99.0%

   \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, 2 + \left(x \cdot x\right) \cdot \color{blue}{0.6666666666666666}, {\left(\left|x\right|\right)}^{7} \cdot 0.047619047619047616\right)\right|}{1.772453850905516} \]
10. Add Preprocessing

Alternative 5: 98.8% accurate, 2.5× speedup

Math

\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left({\left(\left|x\right|\right)}^{7} \cdot 0.047619047619047616 + \left|x\right|\right) + \left|x\right|\right)\right| \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt PI))
   (+ (+ (* (pow (fabs x) 7.0) 0.047619047619047616) (fabs x)) (fabs x)))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * (((pow(fabs(x), 7.0) * 0.047619047619047616) + fabs(x)) + fabs(x))));
}

Java

public static double code(double x) {
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * (((Math.pow(Math.abs(x), 7.0) * 0.047619047619047616) + Math.abs(x)) + Math.abs(x))));
}

Python

def code(x):
	return math.fabs(((1.0 / math.sqrt(math.pi)) * (((math.pow(math.fabs(x), 7.0) * 0.047619047619047616) + math.fabs(x)) + math.fabs(x))))

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64((abs(x) ^ 7.0) * 0.047619047619047616) + abs(x)) + abs(x))))
end

MATLAB

function tmp = code(x)
	tmp = abs(((1.0 / sqrt(pi)) * ((((abs(x) ^ 7.0) * 0.047619047619047616) + abs(x)) + abs(x))));
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision] * 0.047619047619047616), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[Abs[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%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-fabs.f64 98.8

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

6. Applied rewrites 98.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \color{blue}{0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}}\right)\right| \]
7. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. count-2-rev N/A

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

   5. associate-+r+ N/A

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

   6. lower-+.f64 N/A

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

8. Applied rewrites 98.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left({\left(\left|x\right|\right)}^{7} \cdot 0.047619047619047616 + \left|x\right|\right) + \left|x\right|\right)}\right| \]
9. Add Preprocessing

Alternative 6: 98.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-fabs.f64 98.8

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

6. Applied rewrites 98.8%

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

7. Evaluated real constant 98.8%

   \[\leadsto \left|\color{blue}{\frac{5081767996463981}{9007199254740992}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7}\right)\right| \]
8. Add Preprocessing

Alternative 7: 89.4% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(N[(N[(x * x), $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision] * N[Abs[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 99.4%

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

4. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-fabs.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-fabs.f64 89.0

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

6. Applied rewrites 89.0%

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

   1. lift-fabs.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

8. Applied rewrites 89.4%

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

Alternative 8: 89.0% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-fabs.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-fabs.f64 89.0

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

6. Applied rewrites 89.0%

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

   1. lift-fabs.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

8. Applied rewrites 89.0%

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

Alternative 9: 83.6% accurate, 4.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\left|2 \cdot \left|\left|x\right|\right|\right| \cdot 0.5641895835477563\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|2 \cdot \sqrt{\left|x\right| \cdot \left|x\right|}\right|}{\sqrt{\pi}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 5e-15)
   (* (fabs (* 2.0 (fabs (fabs x)))) 0.5641895835477563)
   (/ (fabs (* 2.0 (sqrt (* (fabs x) (fabs x))))) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 5e-15) {
		tmp = fabs((2.0 * fabs(fabs(x)))) * 0.5641895835477563;
	} else {
		tmp = fabs((2.0 * sqrt((fabs(x) * fabs(x))))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 5e-15) {
		tmp = Math.abs((2.0 * Math.abs(Math.abs(x)))) * 0.5641895835477563;
	} else {
		tmp = Math.abs((2.0 * Math.sqrt((Math.abs(x) * Math.abs(x))))) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 5e-15:
		tmp = math.fabs((2.0 * math.fabs(math.fabs(x)))) * 0.5641895835477563
	else:
		tmp = math.fabs((2.0 * math.sqrt((math.fabs(x) * math.fabs(x))))) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 5e-15)
		tmp = Float64(abs(Float64(2.0 * abs(abs(x)))) * 0.5641895835477563);
	else
		tmp = Float64(abs(Float64(2.0 * sqrt(Float64(abs(x) * abs(x))))) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 5e-15)
		tmp = abs((2.0 * abs(abs(x)))) * 0.5641895835477563;
	else
		tmp = abs((2.0 * sqrt((abs(x) * abs(x))))) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 5e-15], N[(N[Abs[N[(2.0 * N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5641895835477563), $MachinePrecision], N[(N[Abs[N[(2.0 * N[Sqrt[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999999999999e-15

   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 99.4%

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

   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. lower-*.f64 N/A

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

      2. lower-fabs.f64 67.8

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

   6. Applied rewrites 67.8%

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

   7. Evaluated real constant 68.1%

      \[\leadsto \frac{\left|2 \cdot \left|x\right|\right|}{\color{blue}{\frac{7982422502469483}{4503599627370496}}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

         \[\leadsto \color{blue}{\left|2 \cdot \left|x\right|\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left|2 \cdot \left|x\right|\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}}} \]

      4. metadata-eval 68.3

         \[\leadsto \left|2 \cdot \left|x\right|\right| \cdot \color{blue}{0.5641895835477563} \]

   9. Applied rewrites 68.3%

      \[\leadsto \color{blue}{\left|2 \cdot \left|x\right|\right| \cdot 0.5641895835477563} \]

   if 4.99999999999999999e-15 < 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 99.4%

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

   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. lower-*.f64 N/A

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

      2. lower-fabs.f64 67.8

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

   6. Applied rewrites 67.8%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. lower-sqrt.f64 N/A

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

      4. lift-*.f64 53.4

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

   8. Applied rewrites 53.4%

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

Alternative 10: 83.6% accurate, 4.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-47}:\\ \;\;\;\;\left|2 \cdot \left|\left|x\right|\right|\right| \cdot 0.5641895835477563\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|2 \cdot \sqrt{\left|x\right| \cdot \left|x\right|}\right|}{1.772453850905516}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 5e-47)
   (* (fabs (* 2.0 (fabs (fabs x)))) 0.5641895835477563)
   (/ (fabs (* 2.0 (sqrt (* (fabs x) (fabs x))))) 1.772453850905516)))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 5e-47) {
		tmp = fabs((2.0 * fabs(fabs(x)))) * 0.5641895835477563;
	} else {
		tmp = fabs((2.0 * sqrt((fabs(x) * fabs(x))))) / 1.772453850905516;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (abs(x) <= 5d-47) then
        tmp = abs((2.0d0 * abs(abs(x)))) * 0.5641895835477563d0
    else
        tmp = abs((2.0d0 * sqrt((abs(x) * abs(x))))) / 1.772453850905516d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 5e-47) {
		tmp = Math.abs((2.0 * Math.abs(Math.abs(x)))) * 0.5641895835477563;
	} else {
		tmp = Math.abs((2.0 * Math.sqrt((Math.abs(x) * Math.abs(x))))) / 1.772453850905516;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 5e-47:
		tmp = math.fabs((2.0 * math.fabs(math.fabs(x)))) * 0.5641895835477563
	else:
		tmp = math.fabs((2.0 * math.sqrt((math.fabs(x) * math.fabs(x))))) / 1.772453850905516
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 5e-47)
		tmp = Float64(abs(Float64(2.0 * abs(abs(x)))) * 0.5641895835477563);
	else
		tmp = Float64(abs(Float64(2.0 * sqrt(Float64(abs(x) * abs(x))))) / 1.772453850905516);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 5e-47)
		tmp = abs((2.0 * abs(abs(x)))) * 0.5641895835477563;
	else
		tmp = abs((2.0 * sqrt((abs(x) * abs(x))))) / 1.772453850905516;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 5e-47], N[(N[Abs[N[(2.0 * N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5641895835477563), $MachinePrecision], N[(N[Abs[N[(2.0 * N[Sqrt[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 1.772453850905516), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000011e-47

   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 99.4%

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

   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. lower-*.f64 N/A

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

      2. lower-fabs.f64 67.8

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

   6. Applied rewrites 67.8%

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

   7. Evaluated real constant 68.1%

      \[\leadsto \frac{\left|2 \cdot \left|x\right|\right|}{\color{blue}{\frac{7982422502469483}{4503599627370496}}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

         \[\leadsto \color{blue}{\left|2 \cdot \left|x\right|\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left|2 \cdot \left|x\right|\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}}} \]

      4. metadata-eval 68.3

         \[\leadsto \left|2 \cdot \left|x\right|\right| \cdot \color{blue}{0.5641895835477563} \]

   9. Applied rewrites 68.3%

      \[\leadsto \color{blue}{\left|2 \cdot \left|x\right|\right| \cdot 0.5641895835477563} \]

   if 5.00000000000000011e-47 < 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 99.4%

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

   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. lower-*.f64 N/A

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

      2. lower-fabs.f64 67.8

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

   6. Applied rewrites 67.8%

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

   7. Evaluated real constant 68.1%

      \[\leadsto \frac{\left|2 \cdot \left|x\right|\right|}{\color{blue}{\frac{7982422502469483}{4503599627370496}}} \]
   8. Step-by-step derivation

      1. lift-fabs.f64 N/A

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

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

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

      3. lower-sqrt.f64 N/A

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

      4. lift-*.f64 53.5

         \[\leadsto \frac{\left|2 \cdot \sqrt{x \cdot x}\right|}{1.772453850905516} \]

   9. Applied rewrites 53.5%

      \[\leadsto \frac{\left|2 \cdot \sqrt{x \cdot x}\right|}{1.772453850905516} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 68.3% accurate, 10.2× speedup

Math

\[\left|2 \cdot \left|x\right|\right| \cdot 0.5641895835477563 \]

FPCore

(FPCore (x) :precision binary64 (* (fabs (* 2.0 (fabs x))) 0.5641895835477563))

C

double code(double x) {
	return fabs((2.0 * fabs(x))) * 0.5641895835477563;
}

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((2.0d0 * abs(x))) * 0.5641895835477563d0
end function

Java

public static double code(double x) {
	return Math.abs((2.0 * Math.abs(x))) * 0.5641895835477563;
}

Python

def code(x):
	return math.fabs((2.0 * math.fabs(x))) * 0.5641895835477563

Julia

function code(x)
	return Float64(abs(Float64(2.0 * abs(x))) * 0.5641895835477563)
end

MATLAB

function tmp = code(x)
	tmp = abs((2.0 * abs(x))) * 0.5641895835477563;
end

Wolfram

code[x_] := N[(N[Abs[N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5641895835477563), $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 99.4%

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

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. lower-*.f64 N/A

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

   2. lower-fabs.f64 67.8

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

6. Applied rewrites 67.8%

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

7. Evaluated real constant 68.1%

   \[\leadsto \frac{\left|2 \cdot \left|x\right|\right|}{\color{blue}{\frac{7982422502469483}{4503599627370496}}} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

      \[\leadsto \color{blue}{\left|2 \cdot \left|x\right|\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}}} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left|2 \cdot \left|x\right|\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}}} \]

   4. metadata-eval 68.3

      \[\leadsto \left|2 \cdot \left|x\right|\right| \cdot \color{blue}{0.5641895835477563} \]

9. Applied rewrites 68.3%

   \[\leadsto \color{blue}{\left|2 \cdot \left|x\right|\right| \cdot 0.5641895835477563} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025170 
(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)))))))