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

Percentage Accurate: 99.9% → 99.9%

Time: 11.9s

Alternatives: 11

Speedup: 3.6×

• 29.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \mathsf{fma}\left(0.047619047619047616, x\_m \cdot {x\_m}^{6}, \mathsf{fma}\left(x\_m, 2, \mathsf{fma}\left(0.6666666666666666, {x\_m}^{3}, 0.2 \cdot {x\_m}^{5}\right)\right)\right) \cdot {\pi}^{-0.5} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (fma
   0.047619047619047616
   (* x_m (pow x_m 6.0))
   (fma x_m 2.0 (fma 0.6666666666666666 (pow x_m 3.0) (* 0.2 (pow x_m 5.0)))))
  (pow PI -0.5)))

C

x_m = fabs(x);
double code(double x_m) {
	return fma(0.047619047619047616, (x_m * pow(x_m, 6.0)), fma(x_m, 2.0, fma(0.6666666666666666, pow(x_m, 3.0), (0.2 * pow(x_m, 5.0))))) * pow(((double) M_PI), -0.5);
}

Julia

x_m = abs(x)
function code(x_m)
	return Float64(fma(0.047619047619047616, Float64(x_m * (x_m ^ 6.0)), fma(x_m, 2.0, fma(0.6666666666666666, (x_m ^ 3.0), Float64(0.2 * (x_m ^ 5.0))))) * (pi ^ -0.5))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(0.047619047619047616 * N[(x$95$m * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 2.0 + N[(0.6666666666666666 * N[Power[x$95$m, 3.0], $MachinePrecision] + N[(0.2 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 34.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.047619047619047616, x \cdot {x}^{6}, \mathsf{fma}\left(x, 2, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right) \cdot {\pi}^{-0.5}} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x\_m}^{3}, 0.2 \cdot {x\_m}^{5}\right) + \left(0.047619047619047616 \cdot {x\_m}^{7} + x\_m \cdot 2\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (pow PI -0.5)
  (+
   (fma 0.6666666666666666 (pow x_m 3.0) (* 0.2 (pow x_m 5.0)))
   (+ (* 0.047619047619047616 (pow x_m 7.0)) (* x_m 2.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(((double) M_PI), -0.5) * (fma(0.6666666666666666, pow(x_m, 3.0), (0.2 * pow(x_m, 5.0))) + ((0.047619047619047616 * pow(x_m, 7.0)) + (x_m * 2.0)));
}

Julia

x_m = abs(x)
function code(x_m)
	return Float64((pi ^ -0.5) * Float64(fma(0.6666666666666666, (x_m ^ 3.0), Float64(0.2 * (x_m ^ 5.0))) + Float64(Float64(0.047619047619047616 * (x_m ^ 7.0)) + Float64(x_m * 2.0))))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(N[(0.6666666666666666 * N[Power[x$95$m, 3.0], $MachinePrecision] + N[(0.2 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.047619047619047616 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 34.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.047619047619047616, x \cdot {x}^{6}, \mathsf{fma}\left(x, 2, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right) \cdot {\pi}^{-0.5}} \]
5. Step-by-step derivation

   1. fma-undefine 34.3%

      \[\leadsto \color{blue}{\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right) + \mathsf{fma}\left(x, 2, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right)} \cdot {\pi}^{-0.5} \]

   2. fma-undefine 34.3%

      \[\leadsto \left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right) + \color{blue}{\left(x \cdot 2 + \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)}\right) \cdot {\pi}^{-0.5} \]

   3. associate-+r+ 34.3%

      \[\leadsto \color{blue}{\left(\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right) + x \cdot 2\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)} \cdot {\pi}^{-0.5} \]

   4. pow1 34.3%

      \[\leadsto \left(\left(0.047619047619047616 \cdot \left(\color{blue}{{x}^{1}} \cdot {x}^{6}\right) + x \cdot 2\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) \cdot {\pi}^{-0.5} \]

   5. pow-prod-up 34.3%

      \[\leadsto \left(\left(0.047619047619047616 \cdot \color{blue}{{x}^{\left(1 + 6\right)}} + x \cdot 2\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) \cdot {\pi}^{-0.5} \]

   6. metadata-eval 34.3%

      \[\leadsto \left(\left(0.047619047619047616 \cdot {x}^{\color{blue}{7}} + x \cdot 2\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) \cdot {\pi}^{-0.5} \]

6. Applied egg-rr 34.3%

   \[\leadsto \color{blue}{\left(\left(0.047619047619047616 \cdot {x}^{7} + x \cdot 2\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)} \cdot {\pi}^{-0.5} \]

7. Final simplification 34.3%

   \[\leadsto {\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right) + \left(0.047619047619047616 \cdot {x}^{7} + x \cdot 2\right)\right) \]
8. Add Preprocessing

Alternative 3: 99.8% accurate, 4.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {\pi}^{-0.5} \cdot \left(x\_m \cdot \left(2 + {x\_m}^{2} \cdot \left(0.6666666666666666 + {x\_m}^{2} \cdot \left(0.2 + 0.047619047619047616 \cdot {x\_m}^{2}\right)\right)\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (pow PI -0.5)
  (*
   x_m
   (+
    2.0
    (*
     (pow x_m 2.0)
     (+
      0.6666666666666666
      (* (pow x_m 2.0) (+ 0.2 (* 0.047619047619047616 (pow x_m 2.0))))))))))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(((double) M_PI), -0.5) * (x_m * (2.0 + (pow(x_m, 2.0) * (0.6666666666666666 + (pow(x_m, 2.0) * (0.2 + (0.047619047619047616 * pow(x_m, 2.0))))))));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(Math.PI, -0.5) * (x_m * (2.0 + (Math.pow(x_m, 2.0) * (0.6666666666666666 + (Math.pow(x_m, 2.0) * (0.2 + (0.047619047619047616 * Math.pow(x_m, 2.0))))))));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(math.pi, -0.5) * (x_m * (2.0 + (math.pow(x_m, 2.0) * (0.6666666666666666 + (math.pow(x_m, 2.0) * (0.2 + (0.047619047619047616 * math.pow(x_m, 2.0))))))))

Julia

x_m = abs(x)
function code(x_m)
	return Float64((pi ^ -0.5) * Float64(x_m * Float64(2.0 + Float64((x_m ^ 2.0) * Float64(0.6666666666666666 + Float64((x_m ^ 2.0) * Float64(0.2 + Float64(0.047619047619047616 * (x_m ^ 2.0)))))))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (pi ^ -0.5) * (x_m * (2.0 + ((x_m ^ 2.0) * (0.6666666666666666 + ((x_m ^ 2.0) * (0.2 + (0.047619047619047616 * (x_m ^ 2.0))))))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x$95$m * N[(2.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.6666666666666666 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.2 + N[(0.047619047619047616 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 34.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.047619047619047616, x \cdot {x}^{6}, \mathsf{fma}\left(x, 2, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right) \cdot {\pi}^{-0.5}} \]

5. Taylor expanded in x around 0 34.3%

   \[\leadsto \color{blue}{\left(x \cdot \left(2 + {x}^{2} \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.2 + 0.047619047619047616 \cdot {x}^{2}\right)\right)\right)\right)} \cdot {\pi}^{-0.5} \]
6. Step-by-step derivation

   1. *-commutative 34.3%

      \[\leadsto \left(x \cdot \left(2 + {x}^{2} \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.2 + \color{blue}{{x}^{2} \cdot 0.047619047619047616}\right)\right)\right)\right) \cdot {\pi}^{-0.5} \]

7. Simplified 34.3%

   \[\leadsto \color{blue}{\left(x \cdot \left(2 + {x}^{2} \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.2 + {x}^{2} \cdot 0.047619047619047616\right)\right)\right)\right)} \cdot {\pi}^{-0.5} \]

8. Final simplification 34.3%

   \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(2 + {x}^{2} \cdot \left(0.6666666666666666 + {x}^{2} \cdot \left(0.2 + 0.047619047619047616 \cdot {x}^{2}\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 4: 99.6% accurate, 5.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.2:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x\_m \cdot \left(2 + {x\_m}^{2} \cdot \left(0.6666666666666666 + 0.2 \cdot {x\_m}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left({x\_m}^{7} \cdot \left(0.047619047619047616 + \frac{0.2}{{x\_m}^{2}}\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.2)
   (*
    (pow PI -0.5)
    (*
     x_m
     (+ 2.0 (* (pow x_m 2.0) (+ 0.6666666666666666 (* 0.2 (pow x_m 2.0)))))))
   (*
    (pow PI -0.5)
    (* (pow x_m 7.0) (+ 0.047619047619047616 (/ 0.2 (pow x_m 2.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.2) {
		tmp = pow(((double) M_PI), -0.5) * (x_m * (2.0 + (pow(x_m, 2.0) * (0.6666666666666666 + (0.2 * pow(x_m, 2.0))))));
	} else {
		tmp = pow(((double) M_PI), -0.5) * (pow(x_m, 7.0) * (0.047619047619047616 + (0.2 / pow(x_m, 2.0))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.2) {
		tmp = Math.pow(Math.PI, -0.5) * (x_m * (2.0 + (Math.pow(x_m, 2.0) * (0.6666666666666666 + (0.2 * Math.pow(x_m, 2.0))))));
	} else {
		tmp = Math.pow(Math.PI, -0.5) * (Math.pow(x_m, 7.0) * (0.047619047619047616 + (0.2 / Math.pow(x_m, 2.0))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.2:
		tmp = math.pow(math.pi, -0.5) * (x_m * (2.0 + (math.pow(x_m, 2.0) * (0.6666666666666666 + (0.2 * math.pow(x_m, 2.0))))))
	else:
		tmp = math.pow(math.pi, -0.5) * (math.pow(x_m, 7.0) * (0.047619047619047616 + (0.2 / math.pow(x_m, 2.0))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.2)
		tmp = Float64((pi ^ -0.5) * Float64(x_m * Float64(2.0 + Float64((x_m ^ 2.0) * Float64(0.6666666666666666 + Float64(0.2 * (x_m ^ 2.0)))))));
	else
		tmp = Float64((pi ^ -0.5) * Float64((x_m ^ 7.0) * Float64(0.047619047619047616 + Float64(0.2 / (x_m ^ 2.0)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.2)
		tmp = (pi ^ -0.5) * (x_m * (2.0 + ((x_m ^ 2.0) * (0.6666666666666666 + (0.2 * (x_m ^ 2.0))))));
	else
		tmp = (pi ^ -0.5) * ((x_m ^ 7.0) * (0.047619047619047616 + (0.2 / (x_m ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.2], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x$95$m * N[(2.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.6666666666666666 + N[(0.2 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(N[Power[x$95$m, 7.0], $MachinePrecision] * N[(0.047619047619047616 + N[(0.2 / N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

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

   4. Applied egg-rr 34.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.047619047619047616, x \cdot {x}^{6}, \mathsf{fma}\left(x, 2, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right) \cdot {\pi}^{-0.5}} \]

   5. Taylor expanded in x around 0 34.3%

      \[\leadsto \color{blue}{\left(x \cdot \left(2 + {x}^{2} \cdot \left(0.6666666666666666 + 0.2 \cdot {x}^{2}\right)\right)\right)} \cdot {\pi}^{-0.5} \]

   if 2.2000000000000002 < x

   1. Initial program 99.8%

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

   4. Applied egg-rr 34.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.047619047619047616, x \cdot {x}^{6}, \mathsf{fma}\left(x, 2, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right) \cdot {\pi}^{-0.5}} \]

   5. Taylor expanded in x around inf 1.5%

      \[\leadsto \color{blue}{\left({x}^{7} \cdot \left(0.047619047619047616 + 0.2 \cdot \frac{1}{{x}^{2}}\right)\right)} \cdot {\pi}^{-0.5} \]
   6. Step-by-step derivation

      1. associate-*r/ 1.5%

         \[\leadsto \left({x}^{7} \cdot \left(0.047619047619047616 + \color{blue}{\frac{0.2 \cdot 1}{{x}^{2}}}\right)\right) \cdot {\pi}^{-0.5} \]

      2. metadata-eval 1.5%

         \[\leadsto \left({x}^{7} \cdot \left(0.047619047619047616 + \frac{\color{blue}{0.2}}{{x}^{2}}\right)\right) \cdot {\pi}^{-0.5} \]

   7. Simplified 1.5%

      \[\leadsto \color{blue}{\left({x}^{7} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)} \cdot {\pi}^{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot \left(2 + {x}^{2} \cdot \left(0.6666666666666666 + 0.2 \cdot {x}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left({x}^{7} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 5.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.85:\\ \;\;\;\;x\_m \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + 0.6666666666666666 \cdot {x\_m}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left({x\_m}^{7} \cdot \left(0.047619047619047616 + \frac{0.2}{{x\_m}^{2}}\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.85)
   (* x_m (* (sqrt (/ 1.0 PI)) (+ 2.0 (* 0.6666666666666666 (pow x_m 2.0)))))
   (*
    (pow PI -0.5)
    (* (pow x_m 7.0) (+ 0.047619047619047616 (/ 0.2 (pow x_m 2.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * (sqrt((1.0 / ((double) M_PI))) * (2.0 + (0.6666666666666666 * pow(x_m, 2.0))));
	} else {
		tmp = pow(((double) M_PI), -0.5) * (pow(x_m, 7.0) * (0.047619047619047616 + (0.2 / pow(x_m, 2.0))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * (Math.sqrt((1.0 / Math.PI)) * (2.0 + (0.6666666666666666 * Math.pow(x_m, 2.0))));
	} else {
		tmp = Math.pow(Math.PI, -0.5) * (Math.pow(x_m, 7.0) * (0.047619047619047616 + (0.2 / Math.pow(x_m, 2.0))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.85:
		tmp = x_m * (math.sqrt((1.0 / math.pi)) * (2.0 + (0.6666666666666666 * math.pow(x_m, 2.0))))
	else:
		tmp = math.pow(math.pi, -0.5) * (math.pow(x_m, 7.0) * (0.047619047619047616 + (0.2 / math.pow(x_m, 2.0))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.85)
		tmp = Float64(x_m * Float64(sqrt(Float64(1.0 / pi)) * Float64(2.0 + Float64(0.6666666666666666 * (x_m ^ 2.0)))));
	else
		tmp = Float64((pi ^ -0.5) * Float64((x_m ^ 7.0) * Float64(0.047619047619047616 + Float64(0.2 / (x_m ^ 2.0)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.85)
		tmp = x_m * (sqrt((1.0 / pi)) * (2.0 + (0.6666666666666666 * (x_m ^ 2.0))));
	else
		tmp = (pi ^ -0.5) * ((x_m ^ 7.0) * (0.047619047619047616 + (0.2 / (x_m ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.85], N[(x$95$m * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(2.0 + N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(N[Power[x$95$m, 7.0], $MachinePrecision] * N[(0.047619047619047616 + N[(0.2 / N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   4. Applied egg-rr 34.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.047619047619047616, x \cdot {x}^{6}, \mathsf{fma}\left(x, 2, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right) \cdot {\pi}^{-0.5}} \]

   5. Taylor expanded in x around 0 34.3%

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{\pi}} + {x}^{2} \cdot \left(0.2 \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.6666666666666666 \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]

   6. Taylor expanded in x around 0 34.3%

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

      1. associate-*r* 34.3%

         \[\leadsto x \cdot \left(\color{blue}{\left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \sqrt{\frac{1}{\pi}}} + 2 \cdot \sqrt{\frac{1}{\pi}}\right) \]

      2. distribute-rgt-out 34.3%

         \[\leadsto x \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{2} + 2\right)\right)} \]

   8. Simplified 34.3%

      \[\leadsto \color{blue}{x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{2} + 2\right)\right)} \]

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   4. Applied egg-rr 34.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.047619047619047616, x \cdot {x}^{6}, \mathsf{fma}\left(x, 2, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right) \cdot {\pi}^{-0.5}} \]

   5. Taylor expanded in x around inf 1.5%

      \[\leadsto \color{blue}{\left({x}^{7} \cdot \left(0.047619047619047616 + 0.2 \cdot \frac{1}{{x}^{2}}\right)\right)} \cdot {\pi}^{-0.5} \]
   6. Step-by-step derivation

      1. associate-*r/ 1.5%

         \[\leadsto \left({x}^{7} \cdot \left(0.047619047619047616 + \color{blue}{\frac{0.2 \cdot 1}{{x}^{2}}}\right)\right) \cdot {\pi}^{-0.5} \]

      2. metadata-eval 1.5%

         \[\leadsto \left({x}^{7} \cdot \left(0.047619047619047616 + \frac{\color{blue}{0.2}}{{x}^{2}}\right)\right) \cdot {\pi}^{-0.5} \]

   7. Simplified 1.5%

      \[\leadsto \color{blue}{\left({x}^{7} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)} \cdot {\pi}^{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left({x}^{7} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 5.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {\pi}^{-0.5} \cdot \left(x\_m \cdot \left(2 + \left(0.2 + 0.047619047619047616 \cdot {x\_m}^{2}\right) \cdot {x\_m}^{4}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (pow PI -0.5)
  (*
   x_m
   (+ 2.0 (* (+ 0.2 (* 0.047619047619047616 (pow x_m 2.0))) (pow x_m 4.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(((double) M_PI), -0.5) * (x_m * (2.0 + ((0.2 + (0.047619047619047616 * pow(x_m, 2.0))) * pow(x_m, 4.0))));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(Math.PI, -0.5) * (x_m * (2.0 + ((0.2 + (0.047619047619047616 * Math.pow(x_m, 2.0))) * Math.pow(x_m, 4.0))));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(math.pi, -0.5) * (x_m * (2.0 + ((0.2 + (0.047619047619047616 * math.pow(x_m, 2.0))) * math.pow(x_m, 4.0))))

Julia

x_m = abs(x)
function code(x_m)
	return Float64((pi ^ -0.5) * Float64(x_m * Float64(2.0 + Float64(Float64(0.2 + Float64(0.047619047619047616 * (x_m ^ 2.0))) * (x_m ^ 4.0)))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (pi ^ -0.5) * (x_m * (2.0 + ((0.2 + (0.047619047619047616 * (x_m ^ 2.0))) * (x_m ^ 4.0))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x$95$m * N[(2.0 + N[(N[(0.2 + N[(0.047619047619047616 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 34.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.047619047619047616, x \cdot {x}^{6}, \mathsf{fma}\left(x, 2, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right) \cdot {\pi}^{-0.5}} \]

5. Taylor expanded in x around inf 34.2%

   \[\leadsto \mathsf{fma}\left(0.047619047619047616, x \cdot {x}^{6}, \mathsf{fma}\left(x, 2, \color{blue}{0.2 \cdot {x}^{5}}\right)\right) \cdot {\pi}^{-0.5} \]

6. Taylor expanded in x around 0 34.2%

   \[\leadsto \color{blue}{\left(x \cdot \left(2 + {x}^{4} \cdot \left(0.2 + 0.047619047619047616 \cdot {x}^{2}\right)\right)\right)} \cdot {\pi}^{-0.5} \]
7. Step-by-step derivation

   1. *-commutative 34.2%

      \[\leadsto \left(x \cdot \left(2 + {x}^{4} \cdot \left(0.2 + \color{blue}{{x}^{2} \cdot 0.047619047619047616}\right)\right)\right) \cdot {\pi}^{-0.5} \]

8. Simplified 34.2%

   \[\leadsto \color{blue}{\left(x \cdot \left(2 + {x}^{4} \cdot \left(0.2 + {x}^{2} \cdot 0.047619047619047616\right)\right)\right)} \cdot {\pi}^{-0.5} \]

9. Final simplification 34.2%

   \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(2 + \left(0.2 + 0.047619047619047616 \cdot {x}^{2}\right) \cdot {x}^{4}\right)\right) \]
10. Add Preprocessing

Alternative 7: 99.3% accurate, 8.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.2:\\ \;\;\;\;x\_m \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + 0.6666666666666666 \cdot {x\_m}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x\_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.2)
   (* x_m (* (sqrt (/ 1.0 PI)) (+ 2.0 (* 0.6666666666666666 (pow x_m 2.0)))))
   (* (pow PI -0.5) (* 0.047619047619047616 (pow x_m 7.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.2) {
		tmp = x_m * (sqrt((1.0 / ((double) M_PI))) * (2.0 + (0.6666666666666666 * pow(x_m, 2.0))));
	} else {
		tmp = pow(((double) M_PI), -0.5) * (0.047619047619047616 * pow(x_m, 7.0));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.2) {
		tmp = x_m * (Math.sqrt((1.0 / Math.PI)) * (2.0 + (0.6666666666666666 * Math.pow(x_m, 2.0))));
	} else {
		tmp = Math.pow(Math.PI, -0.5) * (0.047619047619047616 * Math.pow(x_m, 7.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.2:
		tmp = x_m * (math.sqrt((1.0 / math.pi)) * (2.0 + (0.6666666666666666 * math.pow(x_m, 2.0))))
	else:
		tmp = math.pow(math.pi, -0.5) * (0.047619047619047616 * math.pow(x_m, 7.0))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.2)
		tmp = Float64(x_m * Float64(sqrt(Float64(1.0 / pi)) * Float64(2.0 + Float64(0.6666666666666666 * (x_m ^ 2.0)))));
	else
		tmp = Float64((pi ^ -0.5) * Float64(0.047619047619047616 * (x_m ^ 7.0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.2)
		tmp = x_m * (sqrt((1.0 / pi)) * (2.0 + (0.6666666666666666 * (x_m ^ 2.0))));
	else
		tmp = (pi ^ -0.5) * (0.047619047619047616 * (x_m ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.2], N[(x$95$m * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(2.0 + N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(0.047619047619047616 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

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

   4. Applied egg-rr 34.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.047619047619047616, x \cdot {x}^{6}, \mathsf{fma}\left(x, 2, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right) \cdot {\pi}^{-0.5}} \]

   5. Taylor expanded in x around 0 34.3%

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{\pi}} + {x}^{2} \cdot \left(0.2 \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.6666666666666666 \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]

   6. Taylor expanded in x around 0 34.3%

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

      1. associate-*r* 34.3%

         \[\leadsto x \cdot \left(\color{blue}{\left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \sqrt{\frac{1}{\pi}}} + 2 \cdot \sqrt{\frac{1}{\pi}}\right) \]

      2. distribute-rgt-out 34.3%

         \[\leadsto x \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{2} + 2\right)\right)} \]

   8. Simplified 34.3%

      \[\leadsto \color{blue}{x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{2} + 2\right)\right)} \]

   if 2.2000000000000002 < x

   1. Initial program 99.8%

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

   4. Applied egg-rr 34.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.047619047619047616, x \cdot {x}^{6}, \mathsf{fma}\left(x, 2, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right) \cdot {\pi}^{-0.5}} \]

   5. Taylor expanded in x around inf 3.9%

      \[\leadsto \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right)} \cdot {\pi}^{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.3% accurate, 8.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.2:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x\_m \cdot \left(2 + 0.6666666666666666 \cdot {x\_m}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x\_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.2)
   (* (pow PI -0.5) (* x_m (+ 2.0 (* 0.6666666666666666 (pow x_m 2.0)))))
   (* (pow PI -0.5) (* 0.047619047619047616 (pow x_m 7.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.2) {
		tmp = pow(((double) M_PI), -0.5) * (x_m * (2.0 + (0.6666666666666666 * pow(x_m, 2.0))));
	} else {
		tmp = pow(((double) M_PI), -0.5) * (0.047619047619047616 * pow(x_m, 7.0));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.2) {
		tmp = Math.pow(Math.PI, -0.5) * (x_m * (2.0 + (0.6666666666666666 * Math.pow(x_m, 2.0))));
	} else {
		tmp = Math.pow(Math.PI, -0.5) * (0.047619047619047616 * Math.pow(x_m, 7.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.2:
		tmp = math.pow(math.pi, -0.5) * (x_m * (2.0 + (0.6666666666666666 * math.pow(x_m, 2.0))))
	else:
		tmp = math.pow(math.pi, -0.5) * (0.047619047619047616 * math.pow(x_m, 7.0))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.2)
		tmp = Float64((pi ^ -0.5) * Float64(x_m * Float64(2.0 + Float64(0.6666666666666666 * (x_m ^ 2.0)))));
	else
		tmp = Float64((pi ^ -0.5) * Float64(0.047619047619047616 * (x_m ^ 7.0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.2)
		tmp = (pi ^ -0.5) * (x_m * (2.0 + (0.6666666666666666 * (x_m ^ 2.0))));
	else
		tmp = (pi ^ -0.5) * (0.047619047619047616 * (x_m ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.2], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x$95$m * N[(2.0 + N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(0.047619047619047616 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

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

   4. Applied egg-rr 34.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.047619047619047616, x \cdot {x}^{6}, \mathsf{fma}\left(x, 2, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right) \cdot {\pi}^{-0.5}} \]

   5. Taylor expanded in x around 0 34.3%

      \[\leadsto \color{blue}{\left(x \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right)} \cdot {\pi}^{-0.5} \]

   if 2.2000000000000002 < x

   1. Initial program 99.8%

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

   4. Applied egg-rr 34.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.047619047619047616, x \cdot {x}^{6}, \mathsf{fma}\left(x, 2, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right) \cdot {\pi}^{-0.5}} \]

   5. Taylor expanded in x around inf 3.9%

      \[\leadsto \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right)} \cdot {\pi}^{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.0% accurate, 8.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.85:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x\_m \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x\_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.85)
   (* (pow PI -0.5) (* x_m 2.0))
   (* (pow PI -0.5) (* 0.047619047619047616 (pow x_m 7.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = pow(((double) M_PI), -0.5) * (x_m * 2.0);
	} else {
		tmp = pow(((double) M_PI), -0.5) * (0.047619047619047616 * pow(x_m, 7.0));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = Math.pow(Math.PI, -0.5) * (x_m * 2.0);
	} else {
		tmp = Math.pow(Math.PI, -0.5) * (0.047619047619047616 * Math.pow(x_m, 7.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.85:
		tmp = math.pow(math.pi, -0.5) * (x_m * 2.0)
	else:
		tmp = math.pow(math.pi, -0.5) * (0.047619047619047616 * math.pow(x_m, 7.0))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.85)
		tmp = Float64((pi ^ -0.5) * Float64(x_m * 2.0));
	else
		tmp = Float64((pi ^ -0.5) * Float64(0.047619047619047616 * (x_m ^ 7.0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.85)
		tmp = (pi ^ -0.5) * (x_m * 2.0);
	else
		tmp = (pi ^ -0.5) * (0.047619047619047616 * (x_m ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.85], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(0.047619047619047616 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   4. Applied egg-rr 34.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.047619047619047616, x \cdot {x}^{6}, \mathsf{fma}\left(x, 2, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right) \cdot {\pi}^{-0.5}} \]

   5. Taylor expanded in x around 0 34.2%

      \[\leadsto \color{blue}{\left(2 \cdot x\right)} \cdot {\pi}^{-0.5} \]
   6. Step-by-step derivation

      1. *-commutative 34.2%

         \[\leadsto \color{blue}{\left(x \cdot 2\right)} \cdot {\pi}^{-0.5} \]

   7. Simplified 34.2%

      \[\leadsto \color{blue}{\left(x \cdot 2\right)} \cdot {\pi}^{-0.5} \]

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   4. Applied egg-rr 34.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.047619047619047616, x \cdot {x}^{6}, \mathsf{fma}\left(x, 2, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right) \cdot {\pi}^{-0.5}} \]

   5. Taylor expanded in x around inf 3.9%

      \[\leadsto \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right)} \cdot {\pi}^{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.0% accurate, 8.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.75:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x\_m \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.6666666666666666 \cdot \frac{{x\_m}^{3}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.75)
   (* (pow PI -0.5) (* x_m 2.0))
   (* 0.6666666666666666 (/ (pow x_m 3.0) (sqrt PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.75) {
		tmp = pow(((double) M_PI), -0.5) * (x_m * 2.0);
	} else {
		tmp = 0.6666666666666666 * (pow(x_m, 3.0) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.75) {
		tmp = Math.pow(Math.PI, -0.5) * (x_m * 2.0);
	} else {
		tmp = 0.6666666666666666 * (Math.pow(x_m, 3.0) / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.75:
		tmp = math.pow(math.pi, -0.5) * (x_m * 2.0)
	else:
		tmp = 0.6666666666666666 * (math.pow(x_m, 3.0) / math.sqrt(math.pi))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.75)
		tmp = Float64((pi ^ -0.5) * Float64(x_m * 2.0));
	else
		tmp = Float64(0.6666666666666666 * Float64((x_m ^ 3.0) / sqrt(pi)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.75)
		tmp = (pi ^ -0.5) * (x_m * 2.0);
	else
		tmp = 0.6666666666666666 * ((x_m ^ 3.0) / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.75], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.6666666666666666 * N[(N[Power[x$95$m, 3.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.75

   1. Initial program 99.8%

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

   4. Applied egg-rr 34.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.047619047619047616, x \cdot {x}^{6}, \mathsf{fma}\left(x, 2, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right) \cdot {\pi}^{-0.5}} \]

   5. Taylor expanded in x around 0 34.2%

      \[\leadsto \color{blue}{\left(2 \cdot x\right)} \cdot {\pi}^{-0.5} \]
   6. Step-by-step derivation

      1. *-commutative 34.2%

         \[\leadsto \color{blue}{\left(x \cdot 2\right)} \cdot {\pi}^{-0.5} \]

   7. Simplified 34.2%

      \[\leadsto \color{blue}{\left(x \cdot 2\right)} \cdot {\pi}^{-0.5} \]

   if 1.75 < 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. Simplified 99.4%

      \[\leadsto \color{blue}{\left|\frac{\left|x\right| \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{4}, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6}\right)\right)}{\sqrt{\pi}}\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 22.9%

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

      1. add-sqr-sqrt 22.9%

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

      2. fabs-sqr 22.9%

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

      3. add-sqr-sqrt 22.9%

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

      4. associate-*r* 22.9%

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

      5. add-sqr-sqrt 2.0%

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

      6. fabs-sqr 2.0%

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

      7. add-sqr-sqrt 4.1%

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

      8. pow-plus 4.1%

         \[\leadsto \left(0.6666666666666666 \cdot \color{blue}{{x}^{\left(2 + 1\right)}}\right) \cdot \sqrt{\frac{1}{\pi}} \]

      9. metadata-eval 4.1%

         \[\leadsto \left(0.6666666666666666 \cdot {x}^{\color{blue}{3}}\right) \cdot \sqrt{\frac{1}{\pi}} \]

      10. associate-*r* 4.1%

          \[\leadsto \color{blue}{0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right)} \]

      11. *-commutative 4.1%

          \[\leadsto \color{blue}{\left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right) \cdot 0.6666666666666666} \]

      12. sqrt-div 4.1%

          \[\leadsto \left({x}^{3} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right) \cdot 0.6666666666666666 \]

      13. metadata-eval 4.1%

          \[\leadsto \left({x}^{3} \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right) \cdot 0.6666666666666666 \]

      14. un-div-inv 4.1%

          \[\leadsto \color{blue}{\frac{{x}^{3}}{\sqrt{\pi}}} \cdot 0.6666666666666666 \]

   7. Applied egg-rr 4.1%

      \[\leadsto \color{blue}{\frac{{x}^{3}}{\sqrt{\pi}} \cdot 0.6666666666666666} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.6666666666666666 \cdot \frac{{x}^{3}}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.6% accurate, 17.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {\pi}^{-0.5} \cdot \left(x\_m \cdot 2\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* (pow PI -0.5) (* x_m 2.0)))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(((double) M_PI), -0.5) * (x_m * 2.0);
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(Math.PI, -0.5) * (x_m * 2.0);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(math.pi, -0.5) * (x_m * 2.0)

Julia

x_m = abs(x)
function code(x_m)
	return Float64((pi ^ -0.5) * Float64(x_m * 2.0))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (pi ^ -0.5) * (x_m * 2.0);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x$95$m * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 34.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.047619047619047616, x \cdot {x}^{6}, \mathsf{fma}\left(x, 2, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right)\right) \cdot {\pi}^{-0.5}} \]

5. Taylor expanded in x around 0 34.2%

   \[\leadsto \color{blue}{\left(2 \cdot x\right)} \cdot {\pi}^{-0.5} \]
6. Step-by-step derivation

   1. *-commutative 34.2%

      \[\leadsto \color{blue}{\left(x \cdot 2\right)} \cdot {\pi}^{-0.5} \]

7. Simplified 34.2%

   \[\leadsto \color{blue}{\left(x \cdot 2\right)} \cdot {\pi}^{-0.5} \]

8. Final simplification 34.2%

   \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot 2\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024111 
(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)))))))