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

Percentage Accurate: 99.8% → 99.8%

Time: 13.4s

Alternatives: 12

Speedup: 1.5×

• 28.8% 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: 99.8% 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: 99.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot \left(x \cdot x\right)\\ t_1 := \left|x\right| \cdot \left(\left|x\right| \cdot t\_0\right)\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + 0.6666666666666666 \cdot t\_0\right) + 0.2 \cdot t\_1\right) + 0.047619047619047616 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot t\_1\right)\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (* x x))) (t_1 (* (fabs x) (* (fabs x) t_0))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* 0.6666666666666666 t_0)) (* 0.2 t_1))
      (* 0.047619047619047616 (* (fabs x) (* (fabs x) t_1))))))))

C

double code(double x) {
	double t_0 = fabs(x) * (x * x);
	double t_1 = fabs(x) * (fabs(x) * t_0);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (fabs(x) * (fabs(x) * t_1))))));
}

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * (x * x);
	double t_1 = Math.abs(x) * (Math.abs(x) * t_0);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (Math.abs(x) * (Math.abs(x) * t_1))))));
}

Python

def code(x):
	t_0 = math.fabs(x) * (x * x)
	t_1 = math.fabs(x) * (math.fabs(x) * t_0)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (math.fabs(x) * (math.fabs(x) * t_1))))))

Julia

function code(x)
	t_0 = Float64(abs(x) * Float64(x * x))
	t_1 = Float64(abs(x) * Float64(abs(x) * t_0))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(0.6666666666666666 * t_0)) + Float64(0.2 * t_1)) + Float64(0.047619047619047616 * Float64(abs(x) * Float64(abs(x) * t_1))))))
end

MATLAB

function tmp = code(x)
	t_0 = abs(x) * (x * x);
	t_1 = abs(x) * (abs(x) * t_0);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (abs(x) * (abs(x) * t_1))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$0), $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[(0.6666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.2 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 76.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right) + 0.6666666666666666 \cdot {x}^{3}\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (sqrt (/ 1.0 PI))
   (+
    (*
     (fabs x)
     (+ 2.0 (fma 0.2 (pow x 4.0) (* 0.047619047619047616 (pow x 6.0)))))
    (* 0.6666666666666666 (pow x 3.0))))))

C

double code(double x) {
	return fabs((sqrt((1.0 / ((double) M_PI))) * ((fabs(x) * (2.0 + fma(0.2, pow(x, 4.0), (0.047619047619047616 * pow(x, 6.0))))) + (0.6666666666666666 * pow(x, 3.0)))));
}

Julia

function code(x)
	return abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(abs(x) * Float64(2.0 + fma(0.2, (x ^ 4.0), Float64(0.047619047619047616 * (x ^ 6.0))))) + Float64(0.6666666666666666 * (x ^ 3.0)))))
end

Wolfram

code[x_] := N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Abs[x], $MachinePrecision] * N[(2.0 + N[(0.2 * N[Power[x, 4.0], $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.3%

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

5. Taylor expanded in x around 0 99.9%

   \[\leadsto \left|\color{blue}{0.6666666666666666 \cdot \left(\left({x}^{2} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + \left(0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6} + 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right)\right)}\right| \]

7. Simplified 79.4%

   \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right) + 0.6666666666666666 \cdot {x}^{3}\right)}\right| \]

8. Final simplification 79.4%

   \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \left(2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right) + 0.6666666666666666 \cdot {x}^{3}\right)\right| \]
9. Add Preprocessing

Alternative 3: 34.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\right) \cdot {\pi}^{-0.5} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (*
   x
   (+
    (fma 0.2 (pow x 4.0) (* 0.047619047619047616 (pow x 6.0)))
    (fma 0.6666666666666666 (pow x 2.0) 2.0)))
  (pow PI -0.5)))

C

double code(double x) {
	return (x * (fma(0.2, pow(x, 4.0), (0.047619047619047616 * pow(x, 6.0))) + fma(0.6666666666666666, pow(x, 2.0), 2.0))) * pow(((double) M_PI), -0.5);
}

Julia

function code(x)
	return Float64(Float64(x * Float64(fma(0.2, (x ^ 4.0), Float64(0.047619047619047616 * (x ^ 6.0))) + fma(0.6666666666666666, (x ^ 2.0), 2.0))) * (pi ^ -0.5))
end

Wolfram

code[x_] := N[(N[(x * N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.3%

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

6. Applied egg-rr 42.1%

   \[\leadsto \color{blue}{\left(x \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\right) \cdot {\pi}^{-0.5}} \]

7. Final simplification 42.1%

   \[\leadsto \left(x \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\right) \cdot {\pi}^{-0.5} \]
8. Add Preprocessing

Alternative 4: 34.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (/
   (+
    (fma 0.2 (pow x 4.0) (* 0.047619047619047616 (pow x 6.0)))
    (fma 0.6666666666666666 (pow x 2.0) 2.0))
   (sqrt PI))))

C

double code(double x) {
	return x * ((fma(0.2, pow(x, 4.0), (0.047619047619047616 * pow(x, 6.0))) + fma(0.6666666666666666, pow(x, 2.0), 2.0)) / sqrt(((double) M_PI)));
}

Julia

function code(x)
	return Float64(x * Float64(Float64(fma(0.2, (x ^ 4.0), Float64(0.047619047619047616 * (x ^ 6.0))) + fma(0.6666666666666666, (x ^ 2.0), 2.0)) / sqrt(pi)))
end

Wolfram

code[x_] := N[(x * N[(N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.3%

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

5. Taylor expanded in x around 0 99.3%

   \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \frac{1}{2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)}\right|}} \]
6. Step-by-step derivation

   1. associate-*r/ 99.3%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\frac{\sqrt{\pi} \cdot 1}{2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)}}\right|} \]

   2. +-commutative 99.3%

      \[\leadsto \frac{\left|x\right|}{\left|\frac{\sqrt{\pi} \cdot 1}{\color{blue}{\left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right) + 2}}\right|} \]

   3. associate-+r+ 99.3%

      \[\leadsto \frac{\left|x\right|}{\left|\frac{\sqrt{\pi} \cdot 1}{\color{blue}{\left(\left(0.047619047619047616 \cdot {x}^{6} + 0.2 \cdot {x}^{4}\right) + 0.6666666666666666 \cdot {x}^{2}\right)} + 2}\right|} \]

   4. +-commutative 99.3%

      \[\leadsto \frac{\left|x\right|}{\left|\frac{\sqrt{\pi} \cdot 1}{\left(\color{blue}{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)} + 0.6666666666666666 \cdot {x}^{2}\right) + 2}\right|} \]

   5. fma-define 99.3%

      \[\leadsto \frac{\left|x\right|}{\left|\frac{\sqrt{\pi} \cdot 1}{\left(\color{blue}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)} + 0.6666666666666666 \cdot {x}^{2}\right) + 2}\right|} \]

   6. associate-+r+ 99.3%

      \[\leadsto \frac{\left|x\right|}{\left|\frac{\sqrt{\pi} \cdot 1}{\color{blue}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \left(0.6666666666666666 \cdot {x}^{2} + 2\right)}}\right|} \]

   7. fma-undefine 99.3%

      \[\leadsto \frac{\left|x\right|}{\left|\frac{\sqrt{\pi} \cdot 1}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \color{blue}{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)}}\right|} \]

   8. *-rgt-identity 99.3%

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

7. Simplified 42.1%

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

8. Final simplification 42.1%

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

Alternative 5: 99.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left|\frac{x}{\sqrt{\pi}} \cdot \left(\left(0.047619047619047616 \cdot {x}^{6} + 0.2 \cdot {x}^{4}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ x (sqrt PI))
   (+
    (+ (* 0.047619047619047616 (pow x 6.0)) (* 0.2 (pow x 4.0)))
    (fma 0.6666666666666666 (* x x) 2.0)))))

C

double code(double x) {
	return fabs(((x / sqrt(((double) M_PI))) * (((0.047619047619047616 * pow(x, 6.0)) + (0.2 * pow(x, 4.0))) + fma(0.6666666666666666, (x * x), 2.0))));
}

Julia

function code(x)
	return abs(Float64(Float64(x / sqrt(pi)) * Float64(Float64(Float64(0.047619047619047616 * (x ^ 6.0)) + Float64(0.2 * (x ^ 4.0))) + fma(0.6666666666666666, Float64(x * x), 2.0))))
end

Wolfram

code[x_] := N[Abs[N[(N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.3%

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

   1. *-un-lft-identity 99.3%

      \[\leadsto \left|\color{blue}{\left(1 \cdot \frac{\left|x\right|}{\sqrt{\pi}}\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   2. add-sqr-sqrt 40.5%

      \[\leadsto \left|\left(1 \cdot \frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\sqrt{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   3. fabs-sqr 40.5%

      \[\leadsto \left|\left(1 \cdot \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   4. add-sqr-sqrt 99.3%

      \[\leadsto \left|\left(1 \cdot \frac{\color{blue}{x}}{\sqrt{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

6. Applied egg-rr 99.3%

   \[\leadsto \left|\color{blue}{\left(1 \cdot \frac{x}{\sqrt{\pi}}\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]
7. Step-by-step derivation

   1. *-lft-identity 99.3%

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

8. Simplified 99.3%

   \[\leadsto \left|\color{blue}{\frac{x}{\sqrt{\pi}}} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]
9. Step-by-step derivation

   1. metadata-eval 99.3%

      \[\leadsto \left|\frac{x}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{1}{5}}, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   2. fma-undefine 99.3%

      \[\leadsto \left|\frac{x}{\sqrt{\pi}} \cdot \left(\color{blue}{\left(\frac{1}{5} \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   3. metadata-eval 99.3%

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

10. Applied egg-rr 99.3%

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

11. Final simplification 99.3%

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

Alternative 6: 34.3% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;{\pi}^{-0.5} \cdot \frac{x}{0.5 + \left({x}^{2} \cdot -0.16666666666666666 + {x}^{4} \cdot 0.005555555555555556\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-{x}^{7}}{\sqrt{\pi} \cdot -21}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.1)
   (*
    (pow PI -0.5)
    (/
     x
     (+
      0.5
      (+
       (* (pow x 2.0) -0.16666666666666666)
       (* (pow x 4.0) 0.005555555555555556)))))
   (/ (- (pow x 7.0)) (* (sqrt PI) -21.0))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.1) {
		tmp = pow(((double) M_PI), -0.5) * (x / (0.5 + ((pow(x, 2.0) * -0.16666666666666666) + (pow(x, 4.0) * 0.005555555555555556))));
	} else {
		tmp = -pow(x, 7.0) / (sqrt(((double) M_PI)) * -21.0);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.1) {
		tmp = Math.pow(Math.PI, -0.5) * (x / (0.5 + ((Math.pow(x, 2.0) * -0.16666666666666666) + (Math.pow(x, 4.0) * 0.005555555555555556))));
	} else {
		tmp = -Math.pow(x, 7.0) / (Math.sqrt(Math.PI) * -21.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.1:
		tmp = math.pow(math.pi, -0.5) * (x / (0.5 + ((math.pow(x, 2.0) * -0.16666666666666666) + (math.pow(x, 4.0) * 0.005555555555555556))))
	else:
		tmp = -math.pow(x, 7.0) / (math.sqrt(math.pi) * -21.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.1)
		tmp = Float64((pi ^ -0.5) * Float64(x / Float64(0.5 + Float64(Float64((x ^ 2.0) * -0.16666666666666666) + Float64((x ^ 4.0) * 0.005555555555555556)))));
	else
		tmp = Float64(Float64(-(x ^ 7.0)) / Float64(sqrt(pi) * -21.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.1)
		tmp = (pi ^ -0.5) * (x / (0.5 + (((x ^ 2.0) * -0.16666666666666666) + ((x ^ 4.0) * 0.005555555555555556))));
	else
		tmp = -(x ^ 7.0) / (sqrt(pi) * -21.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.1], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x / N[(0.5 + N[(N[(N[Power[x, 2.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Power[x, 7.0], $MachinePrecision]) / N[(N[Sqrt[Pi], $MachinePrecision] * -21.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.10000000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in x around 0 98.7%

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

      1. add-sqr-sqrt 55.0%

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

      2. fabs-sqr 55.0%

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

      3. add-sqr-sqrt 56.7%

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

      4. *-un-lft-identity 56.7%

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

      5. add-sqr-sqrt 57.0%

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

      6. fabs-sqr 57.0%

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

      7. add-sqr-sqrt 56.7%

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

      8. div-inv 56.7%

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

      9. times-frac 57.2%

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

      10. pow1/2 57.2%

          \[\leadsto \frac{1}{\color{blue}{{\pi}^{0.5}}} \cdot \frac{x}{\frac{1}{0.2 \cdot {x}^{4} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}} \]

      11. pow-flip 57.2%

          \[\leadsto \color{blue}{{\pi}^{\left(-0.5\right)}} \cdot \frac{x}{\frac{1}{0.2 \cdot {x}^{4} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}} \]

      12. metadata-eval 57.2%

          \[\leadsto {\pi}^{\color{blue}{-0.5}} \cdot \frac{x}{\frac{1}{0.2 \cdot {x}^{4} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}} \]

   7. Applied egg-rr 57.2%

      \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \frac{x}{\frac{1}{\mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}}} \]

   8. Taylor expanded in x around 0 57.2%

      \[\leadsto {\pi}^{-0.5} \cdot \frac{x}{\color{blue}{0.5 + \left(-0.16666666666666666 \cdot {x}^{2} + 0.005555555555555556 \cdot {x}^{4}\right)}} \]

   if 0.10000000000000001 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 99.9%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r/ 99.9%

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

      4. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

      1. associate-*r/ 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. frac-2neg 99.9%

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

       2. div-inv 100.0%

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

       3. add-sqr-sqrt 0.0%

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

       4. fabs-sqr 0.0%

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

       5. add-sqr-sqrt 0.1%

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

       6. add-sqr-sqrt 0.1%

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

       7. fabs-sqr 0.1%

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

       8. add-sqr-sqrt 0.1%

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

       9. distribute-neg-frac 0.1%

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

       10. distribute-rgt-neg-in 0.1%

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

       11. metadata-eval 0.1%

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

   11. Applied egg-rr 0.1%

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

       1. associate-*r/ 0.1%

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

       2. *-rgt-identity 0.1%

          \[\leadsto \frac{\color{blue}{-x}}{\frac{\sqrt{\pi} \cdot -21}{{x}^{6}}} \]

       3. distribute-neg-frac 0.1%

          \[\leadsto \color{blue}{-\frac{x}{\frac{\sqrt{\pi} \cdot -21}{{x}^{6}}}} \]

       4. associate-/r/ 0.1%

          \[\leadsto -\color{blue}{\frac{x}{\sqrt{\pi} \cdot -21} \cdot {x}^{6}} \]

       5. distribute-rgt-neg-out 0.1%

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

       6. associate-*l/ 0.1%

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

       7. distribute-rgt-neg-out 0.1%

          \[\leadsto \frac{\color{blue}{-x \cdot {x}^{6}}}{\sqrt{\pi} \cdot -21} \]

       8. *-commutative 0.1%

          \[\leadsto \frac{-\color{blue}{{x}^{6} \cdot x}}{\sqrt{\pi} \cdot -21} \]

       9. pow-plus 0.1%

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

       10. metadata-eval 0.1%

           \[\leadsto \frac{-{x}^{\color{blue}{7}}}{\sqrt{\pi} \cdot -21} \]

   13. Simplified 0.1%

       \[\leadsto \color{blue}{\frac{-{x}^{7}}{\sqrt{\pi} \cdot -21}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;{\pi}^{-0.5} \cdot \frac{x}{0.5 + \left({x}^{2} \cdot -0.16666666666666666 + {x}^{4} \cdot 0.005555555555555556\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-{x}^{7}}{\sqrt{\pi} \cdot -21}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.3% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;\frac{x \cdot {\pi}^{-0.5}}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-{x}^{7}}{\sqrt{\pi} \cdot -21}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.1)
   (/ (* x (pow PI -0.5)) (fma (pow x 2.0) -0.16666666666666666 0.5))
   (/ (- (pow x 7.0)) (* (sqrt PI) -21.0))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.1) {
		tmp = (x * pow(((double) M_PI), -0.5)) / fma(pow(x, 2.0), -0.16666666666666666, 0.5);
	} else {
		tmp = -pow(x, 7.0) / (sqrt(((double) M_PI)) * -21.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.1)
		tmp = Float64(Float64(x * (pi ^ -0.5)) / fma((x ^ 2.0), -0.16666666666666666, 0.5));
	else
		tmp = Float64(Float64(-(x ^ 7.0)) / Float64(sqrt(pi) * -21.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.1], N[(N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, 2.0], $MachinePrecision] * -0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision], N[((-N[Power[x, 7.0], $MachinePrecision]) / N[(N[Sqrt[Pi], $MachinePrecision] * -21.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.10000000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in x around 0 98.7%

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

   6. Taylor expanded in x around 0 98.4%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{-0.16666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\pi}\right) + 0.5 \cdot \sqrt{\pi}}\right|} \]
   7. Step-by-step derivation

      1. associate-*r* 98.4%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot \sqrt{\pi}} + 0.5 \cdot \sqrt{\pi}\right|} \]

      2. distribute-rgt-out 98.4%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot \left(-0.16666666666666666 \cdot {x}^{2} + 0.5\right)}\right|} \]

      3. *-commutative 98.4%

         \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \left(\color{blue}{{x}^{2} \cdot -0.16666666666666666} + 0.5\right)\right|} \]

   8. Simplified 98.4%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)}\right|} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 54.7%

         \[\leadsto \frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\left|\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)\right|} \]

      2. fabs-sqr 54.7%

         \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\left|\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)\right|} \]

      3. add-sqr-sqrt 56.4%

         \[\leadsto \frac{\color{blue}{x}}{\left|\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)\right|} \]

      4. *-un-lft-identity 56.4%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{\left|\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)\right|} \]

      5. add-sqr-sqrt 56.7%

         \[\leadsto \frac{1 \cdot x}{\left|\color{blue}{\sqrt{\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)} \cdot \sqrt{\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)}}\right|} \]

      6. fabs-sqr 56.7%

         \[\leadsto \frac{1 \cdot x}{\color{blue}{\sqrt{\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)} \cdot \sqrt{\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)}}} \]

      7. add-sqr-sqrt 56.4%

         \[\leadsto \frac{1 \cdot x}{\color{blue}{\sqrt{\pi} \cdot \left({x}^{2} \cdot -0.16666666666666666 + 0.5\right)}} \]

      8. times-frac 56.8%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{\pi}} \cdot \frac{x}{{x}^{2} \cdot -0.16666666666666666 + 0.5}} \]

      9. pow1/2 56.8%

         \[\leadsto \frac{1}{\color{blue}{{\pi}^{0.5}}} \cdot \frac{x}{{x}^{2} \cdot -0.16666666666666666 + 0.5} \]

      10. pow-flip 56.8%

          \[\leadsto \color{blue}{{\pi}^{\left(-0.5\right)}} \cdot \frac{x}{{x}^{2} \cdot -0.16666666666666666 + 0.5} \]

      11. metadata-eval 56.8%

          \[\leadsto {\pi}^{\color{blue}{-0.5}} \cdot \frac{x}{{x}^{2} \cdot -0.16666666666666666 + 0.5} \]

      12. fma-define 56.8%

          \[\leadsto {\pi}^{-0.5} \cdot \frac{x}{\color{blue}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]

   10. Applied egg-rr 56.8%

       \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \frac{x}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]
   11. Step-by-step derivation

       1. associate-*r/ 56.8%

          \[\leadsto \color{blue}{\frac{{\pi}^{-0.5} \cdot x}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]

       2. *-commutative 56.8%

          \[\leadsto \frac{\color{blue}{x \cdot {\pi}^{-0.5}}}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)} \]

   12. Simplified 56.8%

       \[\leadsto \color{blue}{\frac{x \cdot {\pi}^{-0.5}}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}} \]

   if 0.10000000000000001 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 99.9%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r/ 99.9%

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

      4. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

      1. associate-*r/ 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. frac-2neg 99.9%

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

       2. div-inv 100.0%

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

       3. add-sqr-sqrt 0.0%

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

       4. fabs-sqr 0.0%

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

       5. add-sqr-sqrt 0.1%

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

       6. add-sqr-sqrt 0.1%

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

       7. fabs-sqr 0.1%

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

       8. add-sqr-sqrt 0.1%

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

       9. distribute-neg-frac 0.1%

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

       10. distribute-rgt-neg-in 0.1%

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

       11. metadata-eval 0.1%

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

   11. Applied egg-rr 0.1%

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

       1. associate-*r/ 0.1%

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

       2. *-rgt-identity 0.1%

          \[\leadsto \frac{\color{blue}{-x}}{\frac{\sqrt{\pi} \cdot -21}{{x}^{6}}} \]

       3. distribute-neg-frac 0.1%

          \[\leadsto \color{blue}{-\frac{x}{\frac{\sqrt{\pi} \cdot -21}{{x}^{6}}}} \]

       4. associate-/r/ 0.1%

          \[\leadsto -\color{blue}{\frac{x}{\sqrt{\pi} \cdot -21} \cdot {x}^{6}} \]

       5. distribute-rgt-neg-out 0.1%

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

       6. associate-*l/ 0.1%

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

       7. distribute-rgt-neg-out 0.1%

          \[\leadsto \frac{\color{blue}{-x \cdot {x}^{6}}}{\sqrt{\pi} \cdot -21} \]

       8. *-commutative 0.1%

          \[\leadsto \frac{-\color{blue}{{x}^{6} \cdot x}}{\sqrt{\pi} \cdot -21} \]

       9. pow-plus 0.1%

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

       10. metadata-eval 0.1%

           \[\leadsto \frac{-{x}^{\color{blue}{7}}}{\sqrt{\pi} \cdot -21} \]

   13. Simplified 0.1%

       \[\leadsto \color{blue}{\frac{-{x}^{7}}{\sqrt{\pi} \cdot -21}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;\frac{x \cdot {\pi}^{-0.5}}{\mathsf{fma}\left({x}^{2}, -0.16666666666666666, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-{x}^{7}}{\sqrt{\pi} \cdot -21}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;{\pi}^{-0.5} \cdot \frac{x}{0.5 + {x}^{2} \cdot -0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{-{x}^{7}}{\sqrt{\pi} \cdot -21}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.1)
   (* (pow PI -0.5) (/ x (+ 0.5 (* (pow x 2.0) -0.16666666666666666))))
   (/ (- (pow x 7.0)) (* (sqrt PI) -21.0))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.1) {
		tmp = pow(((double) M_PI), -0.5) * (x / (0.5 + (pow(x, 2.0) * -0.16666666666666666)));
	} else {
		tmp = -pow(x, 7.0) / (sqrt(((double) M_PI)) * -21.0);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.1) {
		tmp = Math.pow(Math.PI, -0.5) * (x / (0.5 + (Math.pow(x, 2.0) * -0.16666666666666666)));
	} else {
		tmp = -Math.pow(x, 7.0) / (Math.sqrt(Math.PI) * -21.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.1:
		tmp = math.pow(math.pi, -0.5) * (x / (0.5 + (math.pow(x, 2.0) * -0.16666666666666666)))
	else:
		tmp = -math.pow(x, 7.0) / (math.sqrt(math.pi) * -21.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.1)
		tmp = Float64((pi ^ -0.5) * Float64(x / Float64(0.5 + Float64((x ^ 2.0) * -0.16666666666666666))));
	else
		tmp = Float64(Float64(-(x ^ 7.0)) / Float64(sqrt(pi) * -21.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.1)
		tmp = (pi ^ -0.5) * (x / (0.5 + ((x ^ 2.0) * -0.16666666666666666)));
	else
		tmp = -(x ^ 7.0) / (sqrt(pi) * -21.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.1], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x / N[(0.5 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Power[x, 7.0], $MachinePrecision]) / N[(N[Sqrt[Pi], $MachinePrecision] * -21.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.10000000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in x around 0 98.7%

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

      1. add-sqr-sqrt 55.0%

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

      2. fabs-sqr 55.0%

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

      3. add-sqr-sqrt 56.7%

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

      4. *-un-lft-identity 56.7%

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

      5. add-sqr-sqrt 57.0%

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

      6. fabs-sqr 57.0%

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

      7. add-sqr-sqrt 56.7%

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

      8. div-inv 56.7%

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

      9. times-frac 57.2%

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

      10. pow1/2 57.2%

          \[\leadsto \frac{1}{\color{blue}{{\pi}^{0.5}}} \cdot \frac{x}{\frac{1}{0.2 \cdot {x}^{4} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}} \]

      11. pow-flip 57.2%

          \[\leadsto \color{blue}{{\pi}^{\left(-0.5\right)}} \cdot \frac{x}{\frac{1}{0.2 \cdot {x}^{4} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}} \]

      12. metadata-eval 57.2%

          \[\leadsto {\pi}^{\color{blue}{-0.5}} \cdot \frac{x}{\frac{1}{0.2 \cdot {x}^{4} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}} \]

   7. Applied egg-rr 57.2%

      \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \frac{x}{\frac{1}{\mathsf{fma}\left(0.2, {x}^{4}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}}} \]

   8. Taylor expanded in x around 0 56.8%

      \[\leadsto {\pi}^{-0.5} \cdot \frac{x}{\color{blue}{0.5 + -0.16666666666666666 \cdot {x}^{2}}} \]

   if 0.10000000000000001 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 99.9%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r/ 99.9%

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

      4. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

      1. associate-*r/ 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. frac-2neg 99.9%

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

       2. div-inv 100.0%

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

       3. add-sqr-sqrt 0.0%

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

       4. fabs-sqr 0.0%

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

       5. add-sqr-sqrt 0.1%

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

       6. add-sqr-sqrt 0.1%

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

       7. fabs-sqr 0.1%

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

       8. add-sqr-sqrt 0.1%

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

       9. distribute-neg-frac 0.1%

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

       10. distribute-rgt-neg-in 0.1%

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

       11. metadata-eval 0.1%

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

   11. Applied egg-rr 0.1%

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

       1. associate-*r/ 0.1%

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

       2. *-rgt-identity 0.1%

          \[\leadsto \frac{\color{blue}{-x}}{\frac{\sqrt{\pi} \cdot -21}{{x}^{6}}} \]

       3. distribute-neg-frac 0.1%

          \[\leadsto \color{blue}{-\frac{x}{\frac{\sqrt{\pi} \cdot -21}{{x}^{6}}}} \]

       4. associate-/r/ 0.1%

          \[\leadsto -\color{blue}{\frac{x}{\sqrt{\pi} \cdot -21} \cdot {x}^{6}} \]

       5. distribute-rgt-neg-out 0.1%

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

       6. associate-*l/ 0.1%

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

       7. distribute-rgt-neg-out 0.1%

          \[\leadsto \frac{\color{blue}{-x \cdot {x}^{6}}}{\sqrt{\pi} \cdot -21} \]

       8. *-commutative 0.1%

          \[\leadsto \frac{-\color{blue}{{x}^{6} \cdot x}}{\sqrt{\pi} \cdot -21} \]

       9. pow-plus 0.1%

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

       10. metadata-eval 0.1%

           \[\leadsto \frac{-{x}^{\color{blue}{7}}}{\sqrt{\pi} \cdot -21} \]

   13. Simplified 0.1%

       \[\leadsto \color{blue}{\frac{-{x}^{7}}{\sqrt{\pi} \cdot -21}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;{\pi}^{-0.5} \cdot \frac{x}{0.5 + {x}^{2} \cdot -0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{-{x}^{7}}{\sqrt{\pi} \cdot -21}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.2% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-{x}^{7}}{\sqrt{\pi} \cdot -21}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (* x (/ 2.0 (sqrt PI)))
   (/ (- (pow x 7.0)) (* (sqrt PI) -21.0))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = -pow(x, 7.0) / (sqrt(((double) M_PI)) * -21.0);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = -Math.pow(x, 7.0) / (Math.sqrt(Math.PI) * -21.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = -math.pow(x, 7.0) / (math.sqrt(math.pi) * -21.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(Float64(-(x ^ 7.0)) / Float64(sqrt(pi) * -21.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = -(x ^ 7.0) / (sqrt(pi) * -21.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Power[x, 7.0], $MachinePrecision]) / N[(N[Sqrt[Pi], $MachinePrecision] * -21.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 72.7%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{0.5 \cdot \sqrt{\pi}}\right|} \]
   6. Step-by-step derivation

      1. *-commutative 72.7%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

   7. Simplified 72.7%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

   8. Taylor expanded in x around 0 72.7%

      \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|0.5 \cdot \sqrt{\pi}\right|}} \]
   9. Step-by-step derivation

      1. fabs-neg 72.7%

         \[\leadsto \frac{\color{blue}{\left|-x\right|}}{\left|0.5 \cdot \sqrt{\pi}\right|} \]

      2. *-commutative 72.7%

         \[\leadsto \frac{\left|-x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

      3. fabs-div 72.7%

         \[\leadsto \color{blue}{\left|\frac{-x}{\sqrt{\pi} \cdot 0.5}\right|} \]

      4. neg-mul-1 72.7%

         \[\leadsto \left|\frac{\color{blue}{-1 \cdot x}}{\sqrt{\pi} \cdot 0.5}\right| \]

      5. *-commutative 72.7%

         \[\leadsto \left|\frac{-1 \cdot x}{\color{blue}{0.5 \cdot \sqrt{\pi}}}\right| \]

      6. times-frac 72.7%

         \[\leadsto \left|\color{blue}{\frac{-1}{0.5} \cdot \frac{x}{\sqrt{\pi}}}\right| \]

      7. metadata-eval 72.7%

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

      8. metadata-eval 72.7%

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

      9. distribute-lft-neg-in 72.7%

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

      10. fabs-neg 72.7%

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

      11. rem-square-sqrt 39.4%

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

      12. fabs-sqr 39.4%

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

      13. rem-square-sqrt 40.7%

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

      14. *-commutative 40.7%

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

      15. metadata-eval 40.7%

          \[\leadsto \frac{x}{\sqrt{\pi}} \cdot \color{blue}{\frac{1}{0.5}} \]

      16. times-frac 40.7%

          \[\leadsto \color{blue}{\frac{x \cdot 1}{\sqrt{\pi} \cdot 0.5}} \]

      17. associate-*r/ 41.0%

          \[\leadsto \color{blue}{x \cdot \frac{1}{\sqrt{\pi} \cdot 0.5}} \]

      18. *-commutative 41.0%

          \[\leadsto x \cdot \frac{1}{\color{blue}{0.5 \cdot \sqrt{\pi}}} \]

      19. associate-/r* 41.0%

          \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{0.5}}{\sqrt{\pi}}} \]

      20. metadata-eval 41.0%

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

   10. Simplified 41.0%

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

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around inf 31.0%

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

      1. associate-*r* 31.0%

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

      2. *-commutative 31.0%

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

      3. associate-*r/ 31.0%

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

      4. metadata-eval 31.0%

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

   7. Simplified 31.0%

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

      1. associate-*r/ 31.0%

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

   9. Applied egg-rr 31.0%

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

       1. frac-2neg 31.0%

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

       2. div-inv 31.0%

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

       3. add-sqr-sqrt 2.3%

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

       4. fabs-sqr 2.3%

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

       5. add-sqr-sqrt 3.9%

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

       6. add-sqr-sqrt 3.9%

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

       7. fabs-sqr 3.9%

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

       8. add-sqr-sqrt 3.9%

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

       9. distribute-neg-frac 3.9%

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

       10. distribute-rgt-neg-in 3.9%

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

       11. metadata-eval 3.9%

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

   11. Applied egg-rr 3.9%

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

       1. associate-*r/ 3.9%

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

       2. *-rgt-identity 3.9%

          \[\leadsto \frac{\color{blue}{-x}}{\frac{\sqrt{\pi} \cdot -21}{{x}^{6}}} \]

       3. distribute-neg-frac 3.9%

          \[\leadsto \color{blue}{-\frac{x}{\frac{\sqrt{\pi} \cdot -21}{{x}^{6}}}} \]

       4. associate-/r/ 3.9%

          \[\leadsto -\color{blue}{\frac{x}{\sqrt{\pi} \cdot -21} \cdot {x}^{6}} \]

       5. distribute-rgt-neg-out 3.9%

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

       6. associate-*l/ 3.9%

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

       7. distribute-rgt-neg-out 3.9%

          \[\leadsto \frac{\color{blue}{-x \cdot {x}^{6}}}{\sqrt{\pi} \cdot -21} \]

       8. *-commutative 3.9%

          \[\leadsto \frac{-\color{blue}{{x}^{6} \cdot x}}{\sqrt{\pi} \cdot -21} \]

       9. pow-plus 3.9%

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

       10. metadata-eval 3.9%

           \[\leadsto \frac{-{x}^{\color{blue}{7}}}{\sqrt{\pi} \cdot -21} \]

   13. Simplified 3.9%

       \[\leadsto \color{blue}{\frac{-{x}^{7}}{\sqrt{\pi} \cdot -21}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-{x}^{7}}{\sqrt{\pi} \cdot -21}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 34.2% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (* x (/ 2.0 (sqrt PI)))
   (* 0.047619047619047616 (/ (pow x 7.0) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = 0.047619047619047616 * (pow(x, 7.0) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

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

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = 0.047619047619047616 * (math.pow(x, 7.0) / math.sqrt(math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(0.047619047619047616 * Float64((x ^ 7.0) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = 0.047619047619047616 * ((x ^ 7.0) / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[x, 7.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 72.7%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{0.5 \cdot \sqrt{\pi}}\right|} \]
   6. Step-by-step derivation

      1. *-commutative 72.7%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

   7. Simplified 72.7%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

   8. Taylor expanded in x around 0 72.7%

      \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|0.5 \cdot \sqrt{\pi}\right|}} \]
   9. Step-by-step derivation

      1. fabs-neg 72.7%

         \[\leadsto \frac{\color{blue}{\left|-x\right|}}{\left|0.5 \cdot \sqrt{\pi}\right|} \]

      2. *-commutative 72.7%

         \[\leadsto \frac{\left|-x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

      3. fabs-div 72.7%

         \[\leadsto \color{blue}{\left|\frac{-x}{\sqrt{\pi} \cdot 0.5}\right|} \]

      4. neg-mul-1 72.7%

         \[\leadsto \left|\frac{\color{blue}{-1 \cdot x}}{\sqrt{\pi} \cdot 0.5}\right| \]

      5. *-commutative 72.7%

         \[\leadsto \left|\frac{-1 \cdot x}{\color{blue}{0.5 \cdot \sqrt{\pi}}}\right| \]

      6. times-frac 72.7%

         \[\leadsto \left|\color{blue}{\frac{-1}{0.5} \cdot \frac{x}{\sqrt{\pi}}}\right| \]

      7. metadata-eval 72.7%

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

      8. metadata-eval 72.7%

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

      9. distribute-lft-neg-in 72.7%

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

      10. fabs-neg 72.7%

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

      11. rem-square-sqrt 39.4%

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

      12. fabs-sqr 39.4%

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

      13. rem-square-sqrt 40.7%

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

      14. *-commutative 40.7%

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

      15. metadata-eval 40.7%

          \[\leadsto \frac{x}{\sqrt{\pi}} \cdot \color{blue}{\frac{1}{0.5}} \]

      16. times-frac 40.7%

          \[\leadsto \color{blue}{\frac{x \cdot 1}{\sqrt{\pi} \cdot 0.5}} \]

      17. associate-*r/ 41.0%

          \[\leadsto \color{blue}{x \cdot \frac{1}{\sqrt{\pi} \cdot 0.5}} \]

      18. *-commutative 41.0%

          \[\leadsto x \cdot \frac{1}{\color{blue}{0.5 \cdot \sqrt{\pi}}} \]

      19. associate-/r* 41.0%

          \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{0.5}}{\sqrt{\pi}}} \]

      20. metadata-eval 41.0%

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

   10. Simplified 41.0%

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

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around inf 31.0%

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

      1. associate-*r* 31.0%

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

      2. *-commutative 31.0%

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

      3. associate-*r/ 31.0%

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

      4. metadata-eval 31.0%

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

   7. Simplified 31.0%

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

      1. associate-*r/ 31.0%

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

   9. Applied egg-rr 31.0%

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

   10. Taylor expanded in x around 0 31.0%

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

       1. fabs-mul 31.0%

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

       2. metadata-eval 31.0%

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

       3. associate-/r* 31.0%

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

       4. associate-*l/ 31.0%

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

       5. *-lft-identity 31.0%

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

       6. associate-/r* 31.0%

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

       7. fabs-neg 31.0%

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

       8. metadata-eval 31.0%

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

       9. fabs-mul 31.0%

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

       10. fabs-div 31.0%

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

       11. neg-mul-1 31.0%

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

       12. times-frac 31.0%

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

       13. metadata-eval 31.0%

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

       14. metadata-eval 31.0%

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

       15. associate-/l* 30.9%

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

   12. Simplified 3.9%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 34.0% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \frac{x}{\sqrt{\pi}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.3%

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

5. Taylor expanded in x around 0 72.7%

   \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{0.5 \cdot \sqrt{\pi}}\right|} \]
6. Step-by-step derivation

   1. *-commutative 72.7%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

7. Simplified 72.7%

   \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]
8. Step-by-step derivation

   1. add-sqr-sqrt 39.4%

      \[\leadsto \frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\left|\sqrt{\pi} \cdot 0.5\right|} \]

   2. fabs-sqr 39.4%

      \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\left|\sqrt{\pi} \cdot 0.5\right|} \]

   3. add-sqr-sqrt 39.5%

      \[\leadsto \frac{\sqrt{x} \cdot \sqrt{x}}{\left|\color{blue}{\sqrt{\sqrt{\pi} \cdot 0.5} \cdot \sqrt{\sqrt{\pi} \cdot 0.5}}\right|} \]

   4. fabs-sqr 39.5%

      \[\leadsto \frac{\sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{\sqrt{\pi} \cdot 0.5} \cdot \sqrt{\sqrt{\pi} \cdot 0.5}}} \]

   5. add-sqr-sqrt 40.9%

      \[\leadsto \frac{\color{blue}{x}}{\sqrt{\sqrt{\pi} \cdot 0.5} \cdot \sqrt{\sqrt{\pi} \cdot 0.5}} \]

   6. *-un-lft-identity 40.9%

      \[\leadsto \frac{\color{blue}{1 \cdot x}}{\sqrt{\sqrt{\pi} \cdot 0.5} \cdot \sqrt{\sqrt{\pi} \cdot 0.5}} \]

   7. add-sqr-sqrt 40.7%

      \[\leadsto \frac{1 \cdot x}{\color{blue}{\sqrt{\pi} \cdot 0.5}} \]

   8. *-commutative 40.7%

      \[\leadsto \frac{1 \cdot x}{\color{blue}{0.5 \cdot \sqrt{\pi}}} \]

   9. times-frac 40.7%

      \[\leadsto \color{blue}{\frac{1}{0.5} \cdot \frac{x}{\sqrt{\pi}}} \]

   10. metadata-eval 40.7%

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

9. Applied egg-rr 40.7%

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

10. Final simplification 40.7%

    \[\leadsto 2 \cdot \frac{x}{\sqrt{\pi}} \]
11. Add Preprocessing

Alternative 12: 34.2% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{2}{\sqrt{\pi}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.3%

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

5. Taylor expanded in x around 0 72.7%

   \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{0.5 \cdot \sqrt{\pi}}\right|} \]
6. Step-by-step derivation

   1. *-commutative 72.7%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

7. Simplified 72.7%

   \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

8. Taylor expanded in x around 0 72.7%

   \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|0.5 \cdot \sqrt{\pi}\right|}} \]
9. Step-by-step derivation

   1. fabs-neg 72.7%

      \[\leadsto \frac{\color{blue}{\left|-x\right|}}{\left|0.5 \cdot \sqrt{\pi}\right|} \]

   2. *-commutative 72.7%

      \[\leadsto \frac{\left|-x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

   3. fabs-div 72.7%

      \[\leadsto \color{blue}{\left|\frac{-x}{\sqrt{\pi} \cdot 0.5}\right|} \]

   4. neg-mul-1 72.7%

      \[\leadsto \left|\frac{\color{blue}{-1 \cdot x}}{\sqrt{\pi} \cdot 0.5}\right| \]

   5. *-commutative 72.7%

      \[\leadsto \left|\frac{-1 \cdot x}{\color{blue}{0.5 \cdot \sqrt{\pi}}}\right| \]

   6. times-frac 72.7%

      \[\leadsto \left|\color{blue}{\frac{-1}{0.5} \cdot \frac{x}{\sqrt{\pi}}}\right| \]

   7. metadata-eval 72.7%

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

   8. metadata-eval 72.7%

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

   9. distribute-lft-neg-in 72.7%

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

   10. fabs-neg 72.7%

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

   11. rem-square-sqrt 39.4%

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

   12. fabs-sqr 39.4%

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

   13. rem-square-sqrt 40.7%

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

   14. *-commutative 40.7%

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

   15. metadata-eval 40.7%

       \[\leadsto \frac{x}{\sqrt{\pi}} \cdot \color{blue}{\frac{1}{0.5}} \]

   16. times-frac 40.7%

       \[\leadsto \color{blue}{\frac{x \cdot 1}{\sqrt{\pi} \cdot 0.5}} \]

   17. associate-*r/ 41.0%

       \[\leadsto \color{blue}{x \cdot \frac{1}{\sqrt{\pi} \cdot 0.5}} \]

   18. *-commutative 41.0%

       \[\leadsto x \cdot \frac{1}{\color{blue}{0.5 \cdot \sqrt{\pi}}} \]

   19. associate-/r* 41.0%

       \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{0.5}}{\sqrt{\pi}}} \]

   20. metadata-eval 41.0%

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

10. Simplified 41.0%

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

11. Final simplification 41.0%

    \[\leadsto x \cdot \frac{2}{\sqrt{\pi}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024032 
(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)))))))