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

Percentage Accurate: 99.8% → 99.8%

Time: 5.7s

Alternatives: 14

Speedup: 1.6×

• 28.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

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, 1.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt PI))
   (fma
    (pow (fabs x) 7.0)
    0.047619047619047616
    (fma
     (* 0.2 (fabs x))
     (* (* (* x x) x) x)
     (* (fabs x) (fma (* x x) 0.6666666666666666 2.0)))))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * fma(pow(fabs(x), 7.0), 0.047619047619047616, fma((0.2 * fabs(x)), (((x * x) * x) * x), (fabs(x) * fma((x * x), 0.6666666666666666, 2.0))))));
}

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * fma((abs(x) ^ 7.0), 0.047619047619047616, fma(Float64(0.2 * abs(x)), Float64(Float64(Float64(x * x) * x) * x), Float64(abs(x) * fma(Float64(x * x), 0.6666666666666666, 2.0))))))
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision] * 0.047619047619047616 + N[(N[(0.2 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision]), $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%

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

Alternative 2: 99.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(t\_0 \cdot t\_0\right) \cdot 0.047619047619047616\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (*
    (/ 1.0 (sqrt PI))
    (fabs
     (fma
      (fabs x)
      (fma (* 0.2 (* x x)) (* x x) (* (* t_0 t_0) 0.047619047619047616))
      (* (fabs x) (fma (* x x) 0.6666666666666666 2.0)))))))

C

double code(double x) {
	double t_0 = (x * x) * x;
	return (1.0 / sqrt(((double) M_PI))) * fabs(fma(fabs(x), fma((0.2 * (x * x)), (x * x), ((t_0 * t_0) * 0.047619047619047616)), (fabs(x) * fma((x * x), 0.6666666666666666, 2.0))));
}

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	return Float64(Float64(1.0 / sqrt(pi)) * abs(fma(abs(x), fma(Float64(0.2 * Float64(x * x)), Float64(x * x), Float64(Float64(t_0 * t_0) * 0.047619047619047616)), Float64(abs(x) * fma(Float64(x * x), 0.6666666666666666, 2.0)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(N[(0.2 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.047619047619047616), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.6666666666666666 + 2.0), $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%

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

Alternative 3: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Applied rewrites 99.6%

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

Alternative 4: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.6%

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

Alternative 5: 99.4% accurate, 2.0× speedup

Math

\[\frac{1.772453850905516 \cdot \left|\mathsf{fma}\left(0.047619047619047616 \cdot \left(x \cdot x\right), \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right), 2\right)\right) \cdot x\right|}{\pi} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (*
   1.772453850905516
   (fabs
    (*
     (fma
      (* 0.047619047619047616 (* x x))
      (* (* (* x x) x) x)
      (fma (* x x) (fma (* 0.2 x) x 0.6666666666666666) 2.0))
     x)))
  PI))

C

double code(double x) {
	return (1.772453850905516 * fabs((fma((0.047619047619047616 * (x * x)), (((x * x) * x) * x), fma((x * x), fma((0.2 * x), x, 0.6666666666666666), 2.0)) * x))) / ((double) M_PI);
}

Julia

function code(x)
	return Float64(Float64(1.772453850905516 * abs(Float64(fma(Float64(0.047619047619047616 * Float64(x * x)), Float64(Float64(Float64(x * x) * x) * x), fma(Float64(x * x), fma(Float64(0.2 * x), x, 0.6666666666666666), 2.0)) * x))) / pi)
end

Wolfram

code[x_] := N[(N[(1.772453850905516 * N[Abs[N[(N[(N[(0.047619047619047616 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(N[(0.2 * x), $MachinePrecision] * x + 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $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%

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.6%

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

8. Evaluated real constant 99.4%

   \[\leadsto \frac{\color{blue}{\frac{7982422502469483}{4503599627370496}} \cdot \left|\mathsf{fma}\left(\frac{1}{21} \cdot \left(x \cdot x\right), \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{5} \cdot x, x, \frac{2}{3}\right), 2\right)\right) \cdot x\right|}{\pi} \]
9. Add Preprocessing

Alternative 6: 98.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 240000:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|2 \cdot \left|x\right|\right|\\ \mathbf{else}:\\ \;\;\;\;0.5641895835477563 \cdot \left|0.047619047619047616 \cdot \left({\left(\left|x\right|\right)}^{6} \cdot \left|\left|x\right|\right|\right)\right|\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 240000.0)
   (* (/ 1.0 (sqrt PI)) (fabs (* 2.0 (fabs x))))
   (*
    0.5641895835477563
    (fabs (* 0.047619047619047616 (* (pow (fabs x) 6.0) (fabs (fabs x))))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 240000.0) {
		tmp = (1.0 / sqrt(((double) M_PI))) * fabs((2.0 * fabs(x)));
	} else {
		tmp = 0.5641895835477563 * fabs((0.047619047619047616 * (pow(fabs(x), 6.0) * fabs(fabs(x)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 240000.0) {
		tmp = (1.0 / Math.sqrt(Math.PI)) * Math.abs((2.0 * Math.abs(x)));
	} else {
		tmp = 0.5641895835477563 * Math.abs((0.047619047619047616 * (Math.pow(Math.abs(x), 6.0) * Math.abs(Math.abs(x)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 240000.0:
		tmp = (1.0 / math.sqrt(math.pi)) * math.fabs((2.0 * math.fabs(x)))
	else:
		tmp = 0.5641895835477563 * math.fabs((0.047619047619047616 * (math.pow(math.fabs(x), 6.0) * math.fabs(math.fabs(x)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 240000.0)
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(2.0 * abs(x))));
	else
		tmp = Float64(0.5641895835477563 * abs(Float64(0.047619047619047616 * Float64((abs(x) ^ 6.0) * abs(abs(x))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 240000.0)
		tmp = (1.0 / sqrt(pi)) * abs((2.0 * abs(x)));
	else
		tmp = 0.5641895835477563 * abs((0.047619047619047616 * ((abs(x) ^ 6.0) * abs(abs(x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 240000.0], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5641895835477563 * N[Abs[N[(0.047619047619047616 * N[(N[Power[N[Abs[x], $MachinePrecision], 6.0], $MachinePrecision] * N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.4e5

   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%

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. pow1/2 N/A

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

      6. unpow1 N/A

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

      7. pow-div N/A

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

      8. metadata-eval N/A

         \[\leadsto \left|x \cdot 2\right| \cdot {\pi}^{\color{blue}{\frac{-1}{2}}} \]

      9. metadata-eval N/A

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

      10. pow-flip N/A

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

      11. pow1/2 N/A

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

      12. lift-PI.f64 N/A

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

   10. Applied rewrites 67.9%

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

   if 2.4e5 < 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%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-fabs.f64 36.6

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

   6. Applied rewrites 36.6%

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

   7. Evaluated real constant 36.6%

      \[\leadsto \color{blue}{\frac{5081767996463981}{9007199254740992}} \cdot \left|\frac{1}{21} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.5% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt PI))
   (fma (pow (fabs x) 7.0) 0.047619047619047616 (* 2.0 (fabs x))))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * fma(pow(fabs(x), 7.0), 0.047619047619047616, (2.0 * fabs(x)))));
}

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * fma((abs(x) ^ 7.0), 0.047619047619047616, Float64(2.0 * abs(x)))))
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision] * 0.047619047619047616 + N[(2.0 * N[Abs[x], $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%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-fabs.f64 98.8

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

6. Applied rewrites 98.8%

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

Alternative 8: 98.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 240000:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|2 \cdot \left|x\right|\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|{\left(\left|\left|x\right|\right|\right)}^{7} \cdot 0.047619047619047616\right|}{\sqrt{\pi}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 240000.0)
   (* (/ 1.0 (sqrt PI)) (fabs (* 2.0 (fabs x))))
   (/ (fabs (* (pow (fabs (fabs x)) 7.0) 0.047619047619047616)) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 240000.0) {
		tmp = (1.0 / sqrt(((double) M_PI))) * fabs((2.0 * fabs(x)));
	} else {
		tmp = fabs((pow(fabs(fabs(x)), 7.0) * 0.047619047619047616)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 240000.0) {
		tmp = (1.0 / Math.sqrt(Math.PI)) * Math.abs((2.0 * Math.abs(x)));
	} else {
		tmp = Math.abs((Math.pow(Math.abs(Math.abs(x)), 7.0) * 0.047619047619047616)) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 240000.0:
		tmp = (1.0 / math.sqrt(math.pi)) * math.fabs((2.0 * math.fabs(x)))
	else:
		tmp = math.fabs((math.pow(math.fabs(math.fabs(x)), 7.0) * 0.047619047619047616)) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 240000.0)
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(2.0 * abs(x))));
	else
		tmp = Float64(abs(Float64((abs(abs(x)) ^ 7.0) * 0.047619047619047616)) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 240000.0)
		tmp = (1.0 / sqrt(pi)) * abs((2.0 * abs(x)));
	else
		tmp = abs(((abs(abs(x)) ^ 7.0) * 0.047619047619047616)) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 240000.0], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(N[Power[N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision], 7.0], $MachinePrecision] * 0.047619047619047616), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.4e5

   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%

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. pow1/2 N/A

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

      6. unpow1 N/A

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

      7. pow-div N/A

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

      8. metadata-eval N/A

         \[\leadsto \left|x \cdot 2\right| \cdot {\pi}^{\color{blue}{\frac{-1}{2}}} \]

      9. metadata-eval N/A

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

      10. pow-flip N/A

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

      11. pow1/2 N/A

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

      12. lift-PI.f64 N/A

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

   10. Applied rewrites 67.9%

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

   if 2.4e5 < 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%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-fabs.f64 36.6

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

   6. Applied rewrites 36.6%

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

   8. Applied rewrites 36.6%

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

Alternative 9: 98.3% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return abs(Float64(fma(0.047619047619047616, (abs(x) ^ 7.0), Float64(2.0 * abs(x))) / sqrt(pi)))
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] / 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%

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

4. Taylor expanded in x around 0

   \[\leadsto \left|\color{blue}{\frac{\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + 2 \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]
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.f64 N/A

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

   4. lower-fabs.f64 N/A

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

   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.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-PI.f64 98.3

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

6. Applied rewrites 98.3%

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

Alternative 10: 93.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 240000:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|2 \cdot \left|x\right|\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\left(\left|\left|x\right|\right| \cdot \left|x\right|\right) \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) \cdot 0.11283791670955126\right|\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 240000.0)
   (* (/ 1.0 (sqrt PI)) (fabs (* 2.0 (fabs x))))
   (fabs
    (*
     (* (* (fabs (fabs x)) (fabs x)) (* (* (fabs x) (fabs x)) (fabs x)))
     0.11283791670955126))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 240000.0) {
		tmp = (1.0 / sqrt(((double) M_PI))) * fabs((2.0 * fabs(x)));
	} else {
		tmp = fabs((((fabs(fabs(x)) * fabs(x)) * ((fabs(x) * fabs(x)) * fabs(x))) * 0.11283791670955126));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 240000.0) {
		tmp = (1.0 / Math.sqrt(Math.PI)) * Math.abs((2.0 * Math.abs(x)));
	} else {
		tmp = Math.abs((((Math.abs(Math.abs(x)) * Math.abs(x)) * ((Math.abs(x) * Math.abs(x)) * Math.abs(x))) * 0.11283791670955126));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 240000.0:
		tmp = (1.0 / math.sqrt(math.pi)) * math.fabs((2.0 * math.fabs(x)))
	else:
		tmp = math.fabs((((math.fabs(math.fabs(x)) * math.fabs(x)) * ((math.fabs(x) * math.fabs(x)) * math.fabs(x))) * 0.11283791670955126))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 240000.0)
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(2.0 * abs(x))));
	else
		tmp = abs(Float64(Float64(Float64(abs(abs(x)) * abs(x)) * Float64(Float64(abs(x) * abs(x)) * abs(x))) * 0.11283791670955126));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 240000.0)
		tmp = (1.0 / sqrt(pi)) * abs((2.0 * abs(x)));
	else
		tmp = abs((((abs(abs(x)) * abs(x)) * ((abs(x) * abs(x)) * abs(x))) * 0.11283791670955126));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 240000.0], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[(N[(N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.11283791670955126), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.4e5

   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%

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. pow1/2 N/A

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

      6. unpow1 N/A

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

      7. pow-div N/A

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

      8. metadata-eval N/A

         \[\leadsto \left|x \cdot 2\right| \cdot {\pi}^{\color{blue}{\frac{-1}{2}}} \]

      9. metadata-eval N/A

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

      10. pow-flip N/A

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

      11. pow1/2 N/A

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

      12. lift-PI.f64 N/A

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

   10. Applied rewrites 67.9%

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

   if 2.4e5 < 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%

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

   4. Taylor expanded in x around inf

      \[\leadsto \left|\color{blue}{\frac{1}{5} \cdot \frac{{x}^{4} \cdot \left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-PI.f64 31.2

         \[\leadsto \left|0.2 \cdot \frac{{x}^{4} \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

   6. Applied rewrites 31.2%

      \[\leadsto \left|\color{blue}{0.2 \cdot \frac{{x}^{4} \cdot \left|x\right|}{\sqrt{\pi}}}\right| \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|\frac{1}{5} \cdot \color{blue}{\frac{{x}^{4} \cdot \left|x\right|}{\sqrt{\pi}}}\right| \]

      2. lift-/.f64 N/A

         \[\leadsto \left|\frac{1}{5} \cdot \frac{{x}^{4} \cdot \left|x\right|}{\color{blue}{\sqrt{\pi}}}\right| \]

      3. associate-*r/ N/A

         \[\leadsto \left|\frac{\frac{1}{5} \cdot \left({x}^{4} \cdot \left|x\right|\right)}{\color{blue}{\sqrt{\pi}}}\right| \]

      4. *-commutative N/A

         \[\leadsto \left|\frac{\left({x}^{4} \cdot \left|x\right|\right) \cdot \frac{1}{5}}{\sqrt{\color{blue}{\pi}}}\right| \]

      5. associate-/l* N/A

         \[\leadsto \left|\left({x}^{4} \cdot \left|x\right|\right) \cdot \color{blue}{\frac{\frac{1}{5}}{\sqrt{\pi}}}\right| \]

      6. lower-*.f64 N/A

         \[\leadsto \left|\left({x}^{4} \cdot \left|x\right|\right) \cdot \color{blue}{\frac{\frac{1}{5}}{\sqrt{\pi}}}\right| \]

   8. Applied rewrites 31.2%

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

   9. Evaluated real constant 31.2%

      \[\leadsto \left|\left(\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \frac{4065414397171185}{36028797018963968}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 83.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \frac{1}{\sqrt{\pi}}\\ t_2 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{if}\;\left|t\_1 \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_2\right) + \frac{1}{21} \cdot \left(\left(t\_2 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \leq 2 \cdot 10^{-8}:\\ \;\;\;\;t\_1 \cdot \left|2 \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(2 \cdot x\right) \cdot x\right) \cdot 2\right) \cdot \pi}}{\pi}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (/ 1.0 (sqrt PI)))
        (t_2 (* (* t_0 (fabs x)) (fabs x))))
   (if (<=
        (fabs
         (*
          t_1
          (+
           (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_2))
           (* (/ 1.0 21.0) (* (* t_2 (fabs x)) (fabs x))))))
        2e-8)
     (* t_1 (fabs (* 2.0 x)))
     (/ (sqrt (* (* (* (* 2.0 x) x) 2.0) PI)) PI))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = 1.0 / sqrt(((double) M_PI));
	double t_2 = (t_0 * fabs(x)) * fabs(x);
	double tmp;
	if (fabs((t_1 * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * ((t_2 * fabs(x)) * fabs(x)))))) <= 2e-8) {
		tmp = t_1 * fabs((2.0 * x));
	} else {
		tmp = sqrt(((((2.0 * x) * x) * 2.0) * ((double) M_PI))) / ((double) M_PI);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = 1.0 / Math.sqrt(Math.PI);
	double t_2 = (t_0 * Math.abs(x)) * Math.abs(x);
	double tmp;
	if (Math.abs((t_1 * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * ((t_2 * Math.abs(x)) * Math.abs(x)))))) <= 2e-8) {
		tmp = t_1 * Math.abs((2.0 * x));
	} else {
		tmp = Math.sqrt(((((2.0 * x) * x) * 2.0) * Math.PI)) / Math.PI;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = 1.0 / math.sqrt(math.pi)
	t_2 = (t_0 * math.fabs(x)) * math.fabs(x)
	tmp = 0
	if math.fabs((t_1 * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * ((t_2 * math.fabs(x)) * math.fabs(x)))))) <= 2e-8:
		tmp = t_1 * math.fabs((2.0 * x))
	else:
		tmp = math.sqrt(((((2.0 * x) * x) * 2.0) * math.pi)) / math.pi
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(1.0 / sqrt(pi))
	t_2 = Float64(Float64(t_0 * abs(x)) * abs(x))
	tmp = 0.0
	if (abs(Float64(t_1 * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_2)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_2 * abs(x)) * abs(x)))))) <= 2e-8)
		tmp = Float64(t_1 * abs(Float64(2.0 * x)));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(2.0 * x) * x) * 2.0) * pi)) / pi);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = 1.0 / sqrt(pi);
	t_2 = (t_0 * abs(x)) * abs(x);
	tmp = 0.0;
	if (abs((t_1 * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * ((t_2 * abs(x)) * abs(x)))))) <= 2e-8)
		tmp = t_1 * abs((2.0 * x));
	else
		tmp = sqrt(((((2.0 * x) * x) * 2.0) * pi)) / pi;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[N[(t$95$1 * 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$2), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$2 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2e-8], N[(t$95$1 * N[Abs[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(2.0 * x), $MachinePrecision] * x), $MachinePrecision] * 2.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-sqrt.f64 N/A

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

      5. pow1/2 N/A

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

      6. unpow1 N/A

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

      7. pow-div N/A

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

      8. metadata-eval N/A

         \[\leadsto \left|x \cdot 2\right| \cdot {\pi}^{\color{blue}{\frac{-1}{2}}} \]

      9. metadata-eval N/A

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

      10. pow-flip N/A

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

      11. pow1/2 N/A

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

      12. lift-PI.f64 N/A

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

   10. Applied rewrites 67.9%

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

   if 2e-8 < (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))))))

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.7%

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

      1. lift-*.f64 N/A

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

      2. lift-fabs.f64 N/A

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

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

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

   10. Applied rewrites 52.9%

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

Alternative 12: 67.9% accurate, 7.2× speedup

Math

\[\frac{1}{\sqrt{\pi}} \cdot \left|2 \cdot x\right| \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 67.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-sqrt.f64 N/A

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

   5. pow1/2 N/A

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

   6. unpow1 N/A

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

   7. pow-div N/A

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

   8. metadata-eval N/A

      \[\leadsto \left|x \cdot 2\right| \cdot {\pi}^{\color{blue}{\frac{-1}{2}}} \]

   9. metadata-eval N/A

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

   10. pow-flip N/A

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

   11. pow1/2 N/A

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

   12. lift-PI.f64 N/A

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

10. Applied rewrites 67.9%

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

Alternative 13: 67.5% accurate, 8.3× speedup

Math

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

FPCore

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

C

double code(double x) {
	return (fabs((x * 2.0)) * 1.772453850905516) / ((double) M_PI);
}

Java

public static double code(double x) {
	return (Math.abs((x * 2.0)) * 1.772453850905516) / Math.PI;
}

Python

def code(x):
	return (math.fabs((x * 2.0)) * 1.772453850905516) / math.pi

Julia

function code(x)
	return Float64(Float64(abs(Float64(x * 2.0)) * 1.772453850905516) / pi)
end

MATLAB

function tmp = code(x)
	tmp = (abs((x * 2.0)) * 1.772453850905516) / pi;
end

Wolfram

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

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 67.7%

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

9. Evaluated real constant 67.5%

   \[\leadsto \frac{\left|x \cdot 2\right| \cdot \color{blue}{\frac{7982422502469483}{4503599627370496}}}{\pi} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025176 
(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)))))))