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

Percentage Accurate: 99.8% → 99.8%

Time: 11.1s

Alternatives: 6

Speedup: 3.6×

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

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% 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, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 99.8%

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

   1. pow2 99.8%

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

7. Applied egg-rr 99.8%

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

Alternative 2: 99.0% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return Math.abs(x) * Math.abs((((Math.pow(x, 4.0) * (0.2 + (0.047619047619047616 * (x * x)))) + 2.0) / Math.sqrt(Math.PI)));
}

Python

def code(x):
	return math.fabs(x) * math.fabs((((math.pow(x, 4.0) * (0.2 + (0.047619047619047616 * (x * x)))) + 2.0) / math.sqrt(math.pi)))

Julia

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

MATLAB

function tmp = code(x)
	tmp = abs(x) * abs(((((x ^ 4.0) * (0.2 + (0.047619047619047616 * (x * x)))) + 2.0) / sqrt(pi)));
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 99.8%

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

   1. pow2 99.8%

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

7. Applied egg-rr 99.8%

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

8. Taylor expanded in x around 0 99.0%

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

Alternative 3: 98.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \left|x\right| \cdot \left|\frac{2 + 0.047619047619047616 \cdot {x}^{6}}{\sqrt{\pi}}\right| \end{array} \]

FPCore

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

C

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

Java

public static double code(double x) {
	return Math.abs(x) * Math.abs(((2.0 + (0.047619047619047616 * Math.pow(x, 6.0))) / Math.sqrt(Math.PI)));
}

Python

def code(x):
	return math.fabs(x) * math.fabs(((2.0 + (0.047619047619047616 * math.pow(x, 6.0))) / math.sqrt(math.pi)))

Julia

function code(x)
	return Float64(abs(x) * abs(Float64(Float64(2.0 + Float64(0.047619047619047616 * (x ^ 6.0))) / sqrt(pi))))
end

MATLAB

function tmp = code(x)
	tmp = abs(x) * abs(((2.0 + (0.047619047619047616 * (x ^ 6.0))) / sqrt(pi)));
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 99.8%

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

   1. pow2 99.8%

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

7. Applied egg-rr 99.8%

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

8. Taylor expanded in x around 0 99.0%

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

9. Taylor expanded in x around inf 99.0%

   \[\leadsto \left|x\right| \cdot \left|\frac{\color{blue}{0.047619047619047616 \cdot {x}^{6}} + 2}{\sqrt{\pi}}\right| \]

10. Final simplification 99.0%

    \[\leadsto \left|x\right| \cdot \left|\frac{2 + 0.047619047619047616 \cdot {x}^{6}}{\sqrt{\pi}}\right| \]
11. Add Preprocessing

Alternative 4: 34.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({x}^{6} \cdot \left(x \cdot {\pi}^{-0.5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.4)
   (* (pow PI -0.5) (* x 2.0))
   (* 0.047619047619047616 (* (pow x 6.0) (* x (pow PI -0.5))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.4) {
		tmp = pow(((double) M_PI), -0.5) * (x * 2.0);
	} else {
		tmp = 0.047619047619047616 * (pow(x, 6.0) * (x * pow(((double) M_PI), -0.5)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.4) {
		tmp = Math.pow(Math.PI, -0.5) * (x * 2.0);
	} else {
		tmp = 0.047619047619047616 * (Math.pow(x, 6.0) * (x * Math.pow(Math.PI, -0.5)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.4:
		tmp = math.pow(math.pi, -0.5) * (x * 2.0)
	else:
		tmp = 0.047619047619047616 * (math.pow(x, 6.0) * (x * math.pow(math.pi, -0.5)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.4)
		tmp = Float64((pi ^ -0.5) * Float64(x * 2.0));
	else
		tmp = Float64(0.047619047619047616 * Float64((x ^ 6.0) * Float64(x * (pi ^ -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.4)
		tmp = (pi ^ -0.5) * (x * 2.0);
	else
		tmp = 0.047619047619047616 * ((x ^ 6.0) * (x * (pi ^ -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.4], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[x, 6.0], $MachinePrecision] * N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.40000000000000002

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-*l* 98.9%

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

   7. Simplified 98.9%

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

      1. *-un-lft-identity 98.9%

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

      2. inv-pow 98.9%

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

      3. sqrt-pow1 98.9%

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

      4. metadata-eval 98.9%

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

   9. Applied egg-rr 98.9%

      \[\leadsto \left|\color{blue}{\left(1 \cdot {\pi}^{-0.5}\right)} \cdot \left(\left|x\right| \cdot 2\right)\right| \]
   10. Step-by-step derivation

       1. *-lft-identity 98.9%

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

   11. Simplified 98.9%

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

       1. *-commutative 98.9%

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

       2. add-sqr-sqrt 98.1%

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

       3. fabs-sqr 98.1%

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

       4. add-sqr-sqrt 98.9%

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

       5. *-commutative 98.9%

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

       6. add-sqr-sqrt 42.9%

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

       7. fabs-sqr 42.9%

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

       8. add-sqr-sqrt 45.2%

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

   13. Applied egg-rr 45.2%

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

   if 0.40000000000000002 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 99.2%

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

      1. add-sqr-sqrt 99.1%

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

      2. fabs-sqr 99.1%

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

      3. add-sqr-sqrt 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-*l* 99.2%

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

      6. add-sqr-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. add-sqr-sqrt 0.1%

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

      9. inv-pow 0.1%

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

      10. sqrt-pow1 0.1%

          \[\leadsto \left({x}^{6} \cdot \left(x \cdot \color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}\right)\right) \cdot 0.047619047619047616 \]

      11. metadata-eval 0.1%

          \[\leadsto \left({x}^{6} \cdot \left(x \cdot {\pi}^{\color{blue}{-0.5}}\right)\right) \cdot 0.047619047619047616 \]

   7. Applied egg-rr 0.1%

      \[\leadsto \color{blue}{\left({x}^{6} \cdot \left(x \cdot {\pi}^{-0.5}\right)\right) \cdot 0.047619047619047616} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({x}^{6} \cdot \left(x \cdot {\pi}^{-0.5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 34.1% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (* (pow PI -0.5) (* x 2.0))
   (* (pow PI -0.5) (* 0.047619047619047616 (* x (pow x 6.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = pow(((double) M_PI), -0.5) * (x * 2.0);
	} else {
		tmp = pow(((double) M_PI), -0.5) * (0.047619047619047616 * (x * pow(x, 6.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = Math.pow(Math.PI, -0.5) * (x * 2.0);
	} else {
		tmp = Math.pow(Math.PI, -0.5) * (0.047619047619047616 * (x * Math.pow(x, 6.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = math.pow(math.pi, -0.5) * (x * 2.0)
	else:
		tmp = math.pow(math.pi, -0.5) * (0.047619047619047616 * (x * math.pow(x, 6.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64((pi ^ -0.5) * Float64(x * 2.0));
	else
		tmp = Float64((pi ^ -0.5) * Float64(0.047619047619047616 * Float64(x * (x ^ 6.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = (pi ^ -0.5) * (x * 2.0);
	else
		tmp = (pi ^ -0.5) * (0.047619047619047616 * (x * (x ^ 6.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(0.047619047619047616 * N[(x * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 65.2%

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

      1. *-commutative 65.2%

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

      2. associate-*l* 65.2%

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

   7. Simplified 65.2%

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

      1. *-un-lft-identity 65.2%

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

      2. inv-pow 65.2%

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

      3. sqrt-pow1 65.2%

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

      4. metadata-eval 65.2%

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

   9. Applied egg-rr 65.2%

      \[\leadsto \left|\color{blue}{\left(1 \cdot {\pi}^{-0.5}\right)} \cdot \left(\left|x\right| \cdot 2\right)\right| \]
   10. Step-by-step derivation

       1. *-lft-identity 65.2%

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

   11. Simplified 65.2%

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

       1. *-commutative 65.2%

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

       2. add-sqr-sqrt 64.7%

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

       3. fabs-sqr 64.7%

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

       4. add-sqr-sqrt 65.2%

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

       5. *-commutative 65.2%

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

       6. add-sqr-sqrt 27.3%

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

       7. fabs-sqr 27.3%

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

       8. add-sqr-sqrt 28.9%

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

   13. Applied egg-rr 28.9%

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

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 39.7%

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

      1. add-sqr-sqrt 39.7%

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

      2. fabs-sqr 39.7%

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

      3. add-sqr-sqrt 39.7%

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

      4. associate-*r* 39.7%

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

      5. *-commutative 39.7%

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

      6. add-sqr-sqrt 1.6%

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

      7. fabs-sqr 1.6%

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

      8. add-sqr-sqrt 3.4%

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

      9. inv-pow 3.4%

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

      10. sqrt-pow1 3.4%

          \[\leadsto \left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right) \cdot \color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \]

      11. metadata-eval 3.4%

          \[\leadsto \left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right) \cdot {\pi}^{\color{blue}{-0.5}} \]

   7. Applied egg-rr 3.4%

      \[\leadsto \color{blue}{\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right) \cdot {\pi}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 34.1% accurate, 17.4× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (pow PI -0.5) (* x 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 65.2%

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

   1. *-commutative 65.2%

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

   2. associate-*l* 65.2%

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

7. Simplified 65.2%

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

   1. *-un-lft-identity 65.2%

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

   2. inv-pow 65.2%

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

   3. sqrt-pow1 65.2%

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

   4. metadata-eval 65.2%

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

9. Applied egg-rr 65.2%

   \[\leadsto \left|\color{blue}{\left(1 \cdot {\pi}^{-0.5}\right)} \cdot \left(\left|x\right| \cdot 2\right)\right| \]
10. Step-by-step derivation

    1. *-lft-identity 65.2%

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

11. Simplified 65.2%

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

    1. *-commutative 65.2%

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

    2. add-sqr-sqrt 64.7%

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

    3. fabs-sqr 64.7%

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

    4. add-sqr-sqrt 65.2%

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

    5. *-commutative 65.2%

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

    6. add-sqr-sqrt 27.3%

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

    7. fabs-sqr 27.3%

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

    8. add-sqr-sqrt 28.9%

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

13. Applied egg-rr 28.9%

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

Reproduce

herbie shell --seed 2024118 
(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)))))))