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

Percentage Accurate: 99.8% → 99.8%

Time: 6.1s

Alternatives: 12

Speedup: 2.1×

• 71.6% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

Alternative 2: 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 3: 99.8% accurate, 1.5× 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 4: 99.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Abs[N[(N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 0.047619047619047616), $MachinePrecision] * x + N[(N[(x * x), $MachinePrecision] * N[(N[(0.2 * x), $MachinePrecision] * x + 0.6666666666666666), $MachinePrecision] + 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 99.8%

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

5. Applied rewrites 99.4%

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

6. Evaluated real constant 99.6%

   \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, 2 + \left(x \cdot x\right) \cdot \mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right), 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right|}{\color{blue}{1.772453850905516}} \]

8. Applied rewrites 99.8%

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

Alternative 5: 99.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.65:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|\left|x\right| + \left|x\right|\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\pi}}{\left|{\left(\left|\left|x\right|\right|\right)}^{7} \cdot 0.047619047619047616\right|}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.65)
   (* (/ 1.0 (sqrt PI)) (fabs (+ (fabs x) (fabs x))))
   (/
    1.0
    (/ (sqrt PI) (fabs (* (pow (fabs (fabs x)) 7.0) 0.047619047619047616))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.65) {
		tmp = (1.0 / sqrt(((double) M_PI))) * fabs((fabs(x) + fabs(x)));
	} else {
		tmp = 1.0 / (sqrt(((double) M_PI)) / fabs((pow(fabs(fabs(x)), 7.0) * 0.047619047619047616)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.65) {
		tmp = (1.0 / Math.sqrt(Math.PI)) * Math.abs((Math.abs(x) + Math.abs(x)));
	} else {
		tmp = 1.0 / (Math.sqrt(Math.PI) / Math.abs((Math.pow(Math.abs(Math.abs(x)), 7.0) * 0.047619047619047616)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.65:
		tmp = (1.0 / math.sqrt(math.pi)) * math.fabs((math.fabs(x) + math.fabs(x)))
	else:
		tmp = 1.0 / (math.sqrt(math.pi) / math.fabs((math.pow(math.fabs(math.fabs(x)), 7.0) * 0.047619047619047616)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.65)
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(abs(x) + abs(x))));
	else
		tmp = Float64(1.0 / Float64(sqrt(pi) / abs(Float64((abs(abs(x)) ^ 7.0) * 0.047619047619047616))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.65)
		tmp = (1.0 / sqrt(pi)) * abs((abs(x) + abs(x)));
	else
		tmp = 1.0 / (sqrt(pi) / abs(((abs(abs(x)) ^ 7.0) * 0.047619047619047616)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.65], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(N[Abs[x], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[Pi], $MachinePrecision] / N[Abs[N[(N[Power[N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision], 7.0], $MachinePrecision] * 0.047619047619047616), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.65000000000000002

   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. Step-by-step derivation

      1. lift-fabs.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. fabs-mul N/A

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

      6. lift-fabs.f64 N/A

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

      7. fabs-fabs N/A

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

      8. mul-fabs N/A

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

      9. lower-fabs.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 67.5%

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

   8. Applied rewrites 67.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 67.9%

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

   10. Applied rewrites 67.9%

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

   if 0.65000000000000002 < 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 36.7%

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

   6. Applied rewrites 36.7%

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\color{blue}{0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)}\right| \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. div-flip N/A

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

      5. lower-unsound-/.f64 N/A

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

   8. Applied rewrites 36.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\pi}}{\left|{\left(\left|x\right|\right)}^{7} \cdot 0.047619047619047616\right|}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.65:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|\left|x\right| + \left|x\right|\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right|}{\sqrt{\pi}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.65)
   (* (/ 1.0 (sqrt PI)) (fabs (+ (fabs x) (fabs x))))
   (/ (fabs (* 0.047619047619047616 (pow (fabs x) 7.0))) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.65) {
		tmp = (1.0 / sqrt(((double) M_PI))) * fabs((fabs(x) + fabs(x)));
	} else {
		tmp = fabs((0.047619047619047616 * pow(fabs(x), 7.0))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.65) {
		tmp = (1.0 / Math.sqrt(Math.PI)) * Math.abs((Math.abs(x) + Math.abs(x)));
	} else {
		tmp = Math.abs((0.047619047619047616 * Math.pow(Math.abs(x), 7.0))) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.65:
		tmp = (1.0 / math.sqrt(math.pi)) * math.fabs((math.fabs(x) + math.fabs(x)))
	else:
		tmp = math.fabs((0.047619047619047616 * math.pow(math.fabs(x), 7.0))) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.65)
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(abs(x) + abs(x))));
	else
		tmp = Float64(abs(Float64(0.047619047619047616 * (abs(x) ^ 7.0))) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.65)
		tmp = (1.0 / sqrt(pi)) * abs((abs(x) + abs(x)));
	else
		tmp = abs((0.047619047619047616 * (abs(x) ^ 7.0))) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.65], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(N[Abs[x], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(0.047619047619047616 * N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.65000000000000002

   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. Step-by-step derivation

      1. lift-fabs.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. fabs-mul N/A

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

      6. lift-fabs.f64 N/A

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

      7. fabs-fabs N/A

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

      8. mul-fabs N/A

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

      9. lower-fabs.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 67.5%

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

   8. Applied rewrites 67.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 67.9%

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

   10. Applied rewrites 67.9%

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

   if 0.65000000000000002 < 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. Step-by-step derivation

      1. lift-fabs.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. fabs-mul N/A

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

      6. lift-fabs.f64 N/A

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

      7. fabs-fabs N/A

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

      8. mul-fabs N/A

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

      9. lower-fabs.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 36.7%

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

   8. Applied rewrites 36.7%

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

Alternative 7: 98.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(abs(fma(abs(x), Float64(2.0 + Float64(Float64(x * x) * 0.6666666666666666)), Float64(0.047619047619047616 * (abs(x) ^ 7.0)))) / 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[(0.047619047619047616 * N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision]), $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 99.8%

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

5. Applied rewrites 99.4%

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

6. Evaluated real constant 99.6%

   \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, 2 + \left(x \cdot x\right) \cdot \mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right), 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right|}{\color{blue}{1.772453850905516}} \]

7. Taylor expanded in x around 0

   \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, 2 + \left(x \cdot x\right) \cdot \color{blue}{\frac{2}{3}}, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right|}{1.772453850905516} \]
8. Step-by-step derivation

9. Applied rewrites 99.0%

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

Alternative 8: 98.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Abs[N[(N[Abs[x], $MachinePrecision] * 2.0 + N[(0.047619047619047616 * N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision]), $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 99.8%

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

5. Applied rewrites 99.4%

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

6. Evaluated real constant 99.6%

   \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, 2 + \left(x \cdot x\right) \cdot \mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right), 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right|}{\color{blue}{1.772453850905516}} \]

7. Taylor expanded in x around 0

   \[\leadsto \frac{\left|\mathsf{fma}\left(\left|x\right|, \color{blue}{2}, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right|}{1.772453850905516} \]
8. Step-by-step derivation

9. Applied rewrites 98.7%

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

Alternative 9: 93.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 18:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|\left|x\right| + \left|x\right|\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\left(\left|\left|x\right|\right| \cdot \left|x\right|\right) \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) \cdot -0.2}{-1.772453850905516}\right|\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 18.0)
   (* (/ 1.0 (sqrt PI)) (fabs (+ (fabs x) (fabs x))))
   (fabs
    (/
     (*
      (* (* (fabs (fabs x)) (fabs x)) (* (* (fabs x) (fabs x)) (fabs x)))
      -0.2)
     -1.772453850905516))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 18.0) {
		tmp = (1.0 / sqrt(((double) M_PI))) * fabs((fabs(x) + fabs(x)));
	} else {
		tmp = fabs(((((fabs(fabs(x)) * fabs(x)) * ((fabs(x) * fabs(x)) * fabs(x))) * -0.2) / -1.772453850905516));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 18.0) {
		tmp = (1.0 / Math.sqrt(Math.PI)) * Math.abs((Math.abs(x) + Math.abs(x)));
	} else {
		tmp = Math.abs(((((Math.abs(Math.abs(x)) * Math.abs(x)) * ((Math.abs(x) * Math.abs(x)) * Math.abs(x))) * -0.2) / -1.772453850905516));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 18.0:
		tmp = (1.0 / math.sqrt(math.pi)) * math.fabs((math.fabs(x) + math.fabs(x)))
	else:
		tmp = math.fabs(((((math.fabs(math.fabs(x)) * math.fabs(x)) * ((math.fabs(x) * math.fabs(x)) * math.fabs(x))) * -0.2) / -1.772453850905516))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 18.0)
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(abs(x) + abs(x))));
	else
		tmp = abs(Float64(Float64(Float64(Float64(abs(abs(x)) * abs(x)) * Float64(Float64(abs(x) * abs(x)) * abs(x))) * -0.2) / -1.772453850905516));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 18.0)
		tmp = (1.0 / sqrt(pi)) * abs((abs(x) + abs(x)));
	else
		tmp = abs(((((abs(abs(x)) * abs(x)) * ((abs(x) * abs(x)) * abs(x))) * -0.2) / -1.772453850905516));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 18.0], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(N[Abs[x], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(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] * -0.2), $MachinePrecision] / -1.772453850905516), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 18

   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. Step-by-step derivation

      1. lift-fabs.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. fabs-mul N/A

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

      6. lift-fabs.f64 N/A

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

      7. fabs-fabs N/A

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

      8. mul-fabs N/A

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

      9. lower-fabs.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 67.5%

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

   8. Applied rewrites 67.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 67.9%

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

   10. Applied rewrites 67.9%

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

   if 18 < 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%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-fabs.f64 31.3%

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

   6. Applied rewrites 31.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto \left|\frac{\left(\left|x\right| \cdot {x}^{\left(3 + 1\right)}\right) \cdot \frac{-1}{5}}{-\sqrt{\pi}}\right| \]

      8. pow-plus N/A

         \[\leadsto \left|\frac{\left(\left|x\right| \cdot \left({x}^{3} \cdot x\right)\right) \cdot \frac{-1}{5}}{-\sqrt{\pi}}\right| \]

      9. pow3 N/A

         \[\leadsto \left|\frac{\left(\left|x\right| \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot \frac{-1}{5}}{-\sqrt{\pi}}\right| \]

      10. lower-*.f64 N/A

          \[\leadsto \left|\frac{\left(\left|x\right| \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot \frac{-1}{5}}{-\sqrt{\pi}}\right| \]

      11. lift-*.f64 N/A

          \[\leadsto \left|\frac{\left(\left|x\right| \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot \frac{-1}{5}}{-\sqrt{\pi}}\right| \]

      12. lift-*.f64 N/A

          \[\leadsto \left|\frac{\left(\left|x\right| \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot \frac{-1}{5}}{-\sqrt{\pi}}\right| \]

      13. *-commutative N/A

          \[\leadsto \left|\frac{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left|x\right|\right) \cdot \frac{-1}{5}}{-\sqrt{\pi}}\right| \]

      14. lower-*.f64 N/A

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

   8. Applied rewrites 31.3%

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

   9. Evaluated real constant 31.3%

      \[\leadsto \left|\frac{\left(\left(\left|x\right| \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot -0.2}{\color{blue}{-1.772453850905516}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 83.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| + \left|x\right|\\ \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-94}:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|t\_0\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{t\_0 \cdot t\_0}{\pi}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (fabs x) (fabs x))))
   (if (<= (fabs x) 5e-94)
     (* (/ 1.0 (sqrt PI)) (fabs t_0))
     (sqrt (/ (* t_0 t_0) PI)))))

C

double code(double x) {
	double t_0 = fabs(x) + fabs(x);
	double tmp;
	if (fabs(x) <= 5e-94) {
		tmp = (1.0 / sqrt(((double) M_PI))) * fabs(t_0);
	} else {
		tmp = sqrt(((t_0 * t_0) / ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.abs(x) + Math.abs(x);
	double tmp;
	if (Math.abs(x) <= 5e-94) {
		tmp = (1.0 / Math.sqrt(Math.PI)) * Math.abs(t_0);
	} else {
		tmp = Math.sqrt(((t_0 * t_0) / Math.PI));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.fabs(x) + math.fabs(x)
	tmp = 0
	if math.fabs(x) <= 5e-94:
		tmp = (1.0 / math.sqrt(math.pi)) * math.fabs(t_0)
	else:
		tmp = math.sqrt(((t_0 * t_0) / math.pi))
	return tmp

Julia

function code(x)
	t_0 = Float64(abs(x) + abs(x))
	tmp = 0.0
	if (abs(x) <= 5e-94)
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(t_0));
	else
		tmp = sqrt(Float64(Float64(t_0 * t_0) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = abs(x) + abs(x);
	tmp = 0.0;
	if (abs(x) <= 5e-94)
		tmp = (1.0 / sqrt(pi)) * abs(t_0);
	else
		tmp = sqrt(((t_0 * t_0) / pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 5e-94], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.9999999999999995e-94

   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. Step-by-step derivation

      1. lift-fabs.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. fabs-mul N/A

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

      6. lift-fabs.f64 N/A

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

      7. fabs-fabs N/A

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

      8. mul-fabs N/A

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

      9. lower-fabs.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 67.5%

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

   8. Applied rewrites 67.5%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 67.9%

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

   10. Applied rewrites 67.9%

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

   if 4.9999999999999995e-94 < 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. Step-by-step derivation

      1. lift-fabs.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. fabs-mul N/A

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

      6. lift-fabs.f64 N/A

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

      7. fabs-fabs N/A

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

      8. mul-fabs N/A

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

      9. lower-fabs.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 67.5%

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

   8. Applied rewrites 67.5%

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

      1. lift-/.f64 N/A

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

      2. lift-fabs.f64 N/A

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

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

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-undiv N/A

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

   10. Applied rewrites 53.6%

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

Alternative 11: 67.9% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. lift-fabs.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. fabs-mul N/A

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

   6. lift-fabs.f64 N/A

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

   7. fabs-fabs N/A

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

   8. mul-fabs N/A

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

   9. lower-fabs.f64 N/A

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

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

   1. lower-*.f64 67.5%

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

8. Applied rewrites 67.5%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 67.9%

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

10. Applied rewrites 67.9%

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

Alternative 12: 67.5% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Abs[N[(x + x), $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%

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

   1. lift-fabs.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. fabs-mul N/A

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

   6. lift-fabs.f64 N/A

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

   7. fabs-fabs N/A

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

   8. mul-fabs N/A

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

   9. lower-fabs.f64 N/A

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

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

   1. lower-*.f64 67.5%

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

8. Applied rewrites 67.5%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 67.5%

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

10. Applied rewrites 67.5%

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

Reproduce

herbie shell --seed 2025206 
(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)))))))