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

Percentage Accurate: 99.8% → 99.9%

Time: 11.8s

Alternatives: 9

Speedup: 3.0×

• 28.2% 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: 99.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(abs(x) * abs(Float64(Float64(Float64(Float64(0.2 * (x ^ 4.0)) + Float64(0.047619047619047616 * (x ^ 6.0))) + 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[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $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. Step-by-step derivation

   1. fma-undefine 99.8%

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

6. Applied egg-rr 99.8%

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

Alternative 2: 34.6% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return x * ((((0.2 * Math.pow(x, 4.0)) + (0.047619047619047616 * Math.pow(x, 6.0))) + 2.0) / Math.sqrt(Math.PI));
}

Python

def code(x):
	return x * ((((0.2 * math.pow(x, 4.0)) + (0.047619047619047616 * math.pow(x, 6.0))) + 2.0) / math.sqrt(math.pi))

Julia

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

MATLAB

function tmp = code(x)
	tmp = x * ((((0.2 * (x ^ 4.0)) + (0.047619047619047616 * (x ^ 6.0))) + 2.0) / sqrt(pi));
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. 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. Step-by-step derivation

   1. fma-undefine 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in x around 0 99.3%

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

   1. pow1 99.3%

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

9. Applied egg-rr 32.9%

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

    1. unpow1 32.9%

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

11. Simplified 32.9%

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

    1. fma-undefine 32.9%

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

    2. fma-undefine 32.9%

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

    3. associate-+r+ 32.9%

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

13. Applied egg-rr 32.9%

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

14. Final simplification 32.9%

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

Alternative 3: 34.6% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. 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. Step-by-step derivation

   1. fma-undefine 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in x around 0 99.3%

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

   1. pow1 99.3%

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

9. Applied egg-rr 32.9%

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

    1. unpow1 32.9%

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

11. Simplified 32.9%

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

12. Taylor expanded in x around 0 32.9%

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

Alternative 4: 34.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-10}:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 2e-10)
   (* x (/ 2.0 (sqrt PI)))
   (* (sqrt (/ 1.0 PI)) (* 0.047619047619047616 (pow x 7.0)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 2e-10) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = sqrt((1.0 / ((double) M_PI))) * (0.047619047619047616 * pow(x, 7.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 2e-10) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.sqrt((1.0 / Math.PI)) * (0.047619047619047616 * Math.pow(x, 7.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 2e-10:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.sqrt((1.0 / math.pi)) * (0.047619047619047616 * math.pow(x, 7.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 2e-10)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(sqrt(Float64(1.0 / pi)) * Float64(0.047619047619047616 * (x ^ 7.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 2e-10)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = sqrt((1.0 / pi)) * (0.047619047619047616 * (x ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 2e-10], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2.00000000000000007e-10

   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. Step-by-step derivation

      1. fma-undefine 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 99.8%

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

      1. pow1 99.8%

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

   9. Applied egg-rr 47.7%

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

       1. unpow1 47.7%

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

   11. Simplified 47.7%

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

   12. Taylor expanded in x around 0 47.7%

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

   if 2.00000000000000007e-10 < (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%

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

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 98.3%

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

      1. pow1 98.3%

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

   9. Applied egg-rr 0.1%

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

       1. unpow1 0.1%

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

   11. Simplified 0.1%

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

   12. Taylor expanded in x around inf 0.1%

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

       1. *-commutative 0.1%

          \[\leadsto \color{blue}{\left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right) \cdot 0.047619047619047616} \]

       2. *-commutative 0.1%

          \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot {x}^{7}\right)} \cdot 0.047619047619047616 \]

       3. associate-*l* 0.1%

          \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left({x}^{7} \cdot 0.047619047619047616\right)} \]

       4. *-commutative 0.1%

          \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right)} \]

   14. Simplified 0.1%

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

Alternative 5: 34.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-10}:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{0.047619047619047616 \cdot {x}^{6}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 2e-10)
   (* x (/ 2.0 (sqrt PI)))
   (* x (/ (* 0.047619047619047616 (pow x 6.0)) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 2e-10) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = x * ((0.047619047619047616 * pow(x, 6.0)) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 2e-10) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = x * ((0.047619047619047616 * Math.pow(x, 6.0)) / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 2e-10:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = x * ((0.047619047619047616 * math.pow(x, 6.0)) / math.sqrt(math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 2e-10)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(x * Float64(Float64(0.047619047619047616 * (x ^ 6.0)) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 2e-10)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = x * ((0.047619047619047616 * (x ^ 6.0)) / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 2e-10], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2.00000000000000007e-10

   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. Step-by-step derivation

      1. fma-undefine 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 99.8%

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

      1. pow1 99.8%

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

   9. Applied egg-rr 47.7%

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

       1. unpow1 47.7%

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

   11. Simplified 47.7%

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

   12. Taylor expanded in x around 0 47.7%

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

   if 2.00000000000000007e-10 < (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%

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

      1. fma-undefine 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 98.3%

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

      1. pow1 98.3%

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

   9. Applied egg-rr 0.1%

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

       1. unpow1 0.1%

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

   11. Simplified 0.1%

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

   12. Taylor expanded in x around inf 0.1%

       \[\leadsto x \cdot \frac{\color{blue}{0.047619047619047616 \cdot {x}^{6}}}{\sqrt{\pi}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 34.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-10}:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 2e-10)
   (* x (/ 2.0 (sqrt PI)))
   (* 0.047619047619047616 (* (pow x 7.0) (pow PI -0.5)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 2e-10) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = 0.047619047619047616 * (pow(x, 7.0) * pow(((double) M_PI), -0.5));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 2e-10) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = 0.047619047619047616 * (Math.pow(x, 7.0) * Math.pow(Math.PI, -0.5));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 2e-10:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = 0.047619047619047616 * (math.pow(x, 7.0) * math.pow(math.pi, -0.5))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 2e-10)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(0.047619047619047616 * Float64((x ^ 7.0) * (pi ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 2e-10)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = 0.047619047619047616 * ((x ^ 7.0) * (pi ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 2e-10], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[x, 7.0], $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2.00000000000000007e-10

   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. Step-by-step derivation

      1. fma-undefine 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 99.8%

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

      1. pow1 99.8%

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

   9. Applied egg-rr 47.7%

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

       1. unpow1 47.7%

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

   11. Simplified 47.7%

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

   12. Taylor expanded in x around 0 47.7%

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

   if 2.00000000000000007e-10 < (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%

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

   5. Taylor expanded in x around inf 98.0%

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

      1. expm1-log1p-u 96.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|0.047619047619047616 \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right|\right)\right)} \]

      2. expm1-undefine 96.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|0.047619047619047616 \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right|\right)} - 1} \]

      3. fabs-mul 96.8%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left|0.047619047619047616\right| \cdot \left|\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right|}\right)} - 1 \]

      4. metadata-eval 96.8%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{0.047619047619047616} \cdot \left|\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right|\right)} - 1 \]

      5. *-commutative 96.8%

         \[\leadsto e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left({x}^{6} \cdot \left|x\right|\right)}\right|\right)} - 1 \]

      6. inv-pow 96.8%

         \[\leadsto e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left|\sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right|\right)} - 1 \]

      7. sqrt-pow1 96.8%

         \[\leadsto e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left|\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right|\right)} - 1 \]

      8. metadata-eval 96.8%

         \[\leadsto e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left|{\pi}^{\color{blue}{-0.5}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right|\right)} - 1 \]

      9. *-commutative 96.8%

         \[\leadsto e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left|{\pi}^{-0.5} \cdot \color{blue}{\left(\left|x\right| \cdot {x}^{6}\right)}\right|\right)} - 1 \]

   7. Applied egg-rr 96.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left|{\pi}^{-0.5} \cdot \left(\left|x\right| \cdot {x}^{6}\right)\right|\right)} - 1} \]
   8. Step-by-step derivation

      1. log1p-undefine 96.8%

         \[\leadsto e^{\color{blue}{\log \left(1 + 0.047619047619047616 \cdot \left|{\pi}^{-0.5} \cdot \left(\left|x\right| \cdot {x}^{6}\right)\right|\right)}} - 1 \]

      2. rem-exp-log 98.0%

         \[\leadsto \color{blue}{\left(1 + 0.047619047619047616 \cdot \left|{\pi}^{-0.5} \cdot \left(\left|x\right| \cdot {x}^{6}\right)\right|\right)} - 1 \]

      3. +-commutative 98.0%

         \[\leadsto \color{blue}{\left(0.047619047619047616 \cdot \left|{\pi}^{-0.5} \cdot \left(\left|x\right| \cdot {x}^{6}\right)\right| + 1\right)} - 1 \]

      4. associate--l+ 98.0%

         \[\leadsto \color{blue}{0.047619047619047616 \cdot \left|{\pi}^{-0.5} \cdot \left(\left|x\right| \cdot {x}^{6}\right)\right| + \left(1 - 1\right)} \]

   9. Simplified 98.0%

      \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{6} \cdot \left(\left|x\right| \cdot {\pi}^{-0.5}\right)\right) + 0} \]
   10. Step-by-step derivation

       1. metadata-eval 98.0%

          \[\leadsto 0.047619047619047616 \cdot \left({x}^{6} \cdot \left(\left|x\right| \cdot {\pi}^{-0.5}\right)\right) + \color{blue}{\left(-0\right)} \]

       2. sub-neg 98.0%

          \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{6} \cdot \left(\left|x\right| \cdot {\pi}^{-0.5}\right)\right) - 0} \]

       3. *-commutative 98.0%

          \[\leadsto \color{blue}{\left({x}^{6} \cdot \left(\left|x\right| \cdot {\pi}^{-0.5}\right)\right) \cdot 0.047619047619047616} - 0 \]

       4. associate-*l* 98.0%

          \[\leadsto \color{blue}{{x}^{6} \cdot \left(\left(\left|x\right| \cdot {\pi}^{-0.5}\right) \cdot 0.047619047619047616\right)} - 0 \]

       5. add-sqr-sqrt 0.0%

          \[\leadsto {x}^{6} \cdot \left(\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot {\pi}^{-0.5}\right) \cdot 0.047619047619047616\right) - 0 \]

       6. fabs-sqr 0.0%

          \[\leadsto {x}^{6} \cdot \left(\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {\pi}^{-0.5}\right) \cdot 0.047619047619047616\right) - 0 \]

       7. add-sqr-sqrt 0.1%

          \[\leadsto {x}^{6} \cdot \left(\left(\color{blue}{x} \cdot {\pi}^{-0.5}\right) \cdot 0.047619047619047616\right) - 0 \]

   11. Applied egg-rr 0.1%

       \[\leadsto \color{blue}{{x}^{6} \cdot \left(\left(x \cdot {\pi}^{-0.5}\right) \cdot 0.047619047619047616\right) - 0} \]
   12. Step-by-step derivation

       1. --rgt-identity 0.1%

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

       2. associate-*r* 0.1%

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

       3. associate-*r* 0.1%

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

       4. *-commutative 0.1%

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

       5. *-commutative 0.1%

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

       6. *-commutative 0.1%

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

       7. pow-plus 0.1%

          \[\leadsto 0.047619047619047616 \cdot \left(\color{blue}{{x}^{\left(6 + 1\right)}} \cdot {\pi}^{-0.5}\right) \]

       8. metadata-eval 0.1%

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

   13. Simplified 0.1%

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

Alternative 7: 51.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 20:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{2} \cdot \frac{4}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 20.0)
   (* x (/ 2.0 (sqrt PI)))
   (sqrt (* (pow x 2.0) (/ 4.0 PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 20.0) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = sqrt((pow(x, 2.0) * (4.0 / ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 20.0) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.sqrt((Math.pow(x, 2.0) * (4.0 / Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 20.0:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.sqrt((math.pow(x, 2.0) * (4.0 / math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 20.0)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = sqrt(Float64((x ^ 2.0) * Float64(4.0 / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 20.0)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = sqrt(((x ^ 2.0) * (4.0 / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 20.0], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[x, 2.0], $MachinePrecision] * N[(4.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 20

   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. Step-by-step derivation

      1. fma-undefine 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 99.1%

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

      1. pow1 99.1%

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

   9. Applied egg-rr 47.2%

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

       1. unpow1 47.2%

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

   11. Simplified 47.2%

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

   12. Taylor expanded in x around 0 47.2%

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

   if 20 < (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 99.8%

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

   5. Taylor expanded in x around 0 5.6%

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

      1. fabs-mul 5.6%

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

      2. metadata-eval 5.6%

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

      3. inv-pow 5.6%

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

      4. sqrt-pow1 5.6%

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

      5. metadata-eval 5.6%

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

   7. Applied egg-rr 5.6%

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

      1. rem-square-sqrt 5.6%

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

      2. fabs-sqr 5.6%

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

      3. rem-square-sqrt 5.6%

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

      4. associate-*r* 5.6%

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

      5. *-commutative 5.6%

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

   9. Simplified 5.6%

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

       1. pow1 5.6%

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

       2. add-sqr-sqrt 0.0%

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

       3. fabs-sqr 0.0%

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

       4. add-sqr-sqrt 0.3%

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

   11. Applied egg-rr 0.3%

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

       1. unpow1 0.3%

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

   13. Simplified 0.3%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 55.8%

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

       3. log1p-expm1-u 86.7%

          \[\leadsto \sqrt{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \left(2 \cdot {\pi}^{-0.5}\right)\right)\right)} \cdot \left(x \cdot \left(2 \cdot {\pi}^{-0.5}\right)\right)} \]

       4. log1p-expm1-u 86.7%

          \[\leadsto \sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \left(2 \cdot {\pi}^{-0.5}\right)\right)\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \left(2 \cdot {\pi}^{-0.5}\right)\right)\right)}} \]

       5. log1p-expm1-u 86.7%

          \[\leadsto \sqrt{\color{blue}{\left(x \cdot \left(2 \cdot {\pi}^{-0.5}\right)\right)} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \left(2 \cdot {\pi}^{-0.5}\right)\right)\right)} \]

       6. *-commutative 86.7%

          \[\leadsto \sqrt{\color{blue}{\left(\left(2 \cdot {\pi}^{-0.5}\right) \cdot x\right)} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \left(2 \cdot {\pi}^{-0.5}\right)\right)\right)} \]

       7. log1p-expm1-u 55.8%

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

       8. *-commutative 55.8%

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

       9. swap-sqr 55.8%

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

       10. swap-sqr 55.8%

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

       11. metadata-eval 55.8%

           \[\leadsto \sqrt{\left(\color{blue}{4} \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)\right) \cdot \left(x \cdot x\right)} \]

       12. pow-prod-up 55.8%

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

       13. metadata-eval 55.8%

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

       14. pow2 55.8%

           \[\leadsto \sqrt{\left(4 \cdot {\pi}^{-1}\right) \cdot \color{blue}{{x}^{2}}} \]

   15. Applied egg-rr 55.8%

       \[\leadsto \color{blue}{\sqrt{\left(4 \cdot {\pi}^{-1}\right) \cdot {x}^{2}}} \]
   16. Step-by-step derivation

       1. *-commutative 55.8%

          \[\leadsto \sqrt{\color{blue}{{x}^{2} \cdot \left(4 \cdot {\pi}^{-1}\right)}} \]

       2. unpow-1 55.8%

          \[\leadsto \sqrt{{x}^{2} \cdot \left(4 \cdot \color{blue}{\frac{1}{\pi}}\right)} \]

       3. associate-*r/ 55.8%

          \[\leadsto \sqrt{{x}^{2} \cdot \color{blue}{\frac{4 \cdot 1}{\pi}}} \]

       4. metadata-eval 55.8%

          \[\leadsto \sqrt{{x}^{2} \cdot \frac{\color{blue}{4}}{\pi}} \]

   17. Simplified 55.8%

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

Alternative 8: 34.4% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around inf 98.9%

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

6. Taylor expanded in x around 0 98.9%

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

   1. associate-*l/ 98.9%

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

   2. add-sqr-sqrt 31.0%

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

   3. fabs-sqr 31.0%

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

   4. add-sqr-sqrt 98.9%

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

   5. *-commutative 98.9%

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

   6. fma-define 98.9%

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

8. Applied egg-rr 98.9%

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

   1. add-sqr-sqrt 30.9%

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

   2. fabs-sqr 30.9%

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

   3. add-sqr-sqrt 32.7%

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

10. Applied egg-rr 32.7%

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

Alternative 9: 34.7% 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 99.8%

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

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. Step-by-step derivation

   1. fma-undefine 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in x around 0 99.3%

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

   1. pow1 99.3%

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

9. Applied egg-rr 32.9%

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

    1. unpow1 32.9%

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

11. Simplified 32.9%

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

12. Taylor expanded in x around 0 32.9%

    \[\leadsto x \cdot \frac{\color{blue}{2}}{\sqrt{\pi}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024146 
(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)))))))