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

Percentage Accurate: 99.8% → 99.8%

Time: 13.8s

Alternatives: 12

Speedup: 1.5×

• 29.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.2 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 34.5% accurate, 3.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  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 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(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[(x * 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. add-sqr-sqrt 99.4%

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

   2. sqrt-prod 69.8%

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

   3. sqr-abs 69.8%

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

   4. pow2 69.8%

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

   5. sqrt-pow1 30.7%

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

   6. metadata-eval 30.7%

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

   7. pow1 30.7%

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

   8. *-un-lft-identity 30.7%

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

6. Applied egg-rr 30.7%

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

   1. *-lft-identity 30.7%

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

8. Simplified 30.7%

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

   1. metadata-eval 30.7%

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

   2. fma-undefine 30.7%

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

   3. metadata-eval 30.7%

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

10. Applied egg-rr 30.7%

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

Alternative 3: 99.2% accurate, 3.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	return fabs(x) * fabs((((0.047619047619047616 * 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(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[(0.047619047619047616 * N[Power[x, 6.0], $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 inf 99.1%

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

Alternative 4: 34.3% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(x * N[Abs[N[(N[(2.0 + N[(0.2 * N[Power[x, 4.0], $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $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. Step-by-step derivation

   1. add-sqr-sqrt 99.4%

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

   2. sqrt-prod 69.8%

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

   3. sqr-abs 69.8%

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

   4. pow2 69.8%

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

   5. sqrt-pow1 30.7%

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

   6. metadata-eval 30.7%

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

   7. pow1 30.7%

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

   8. *-un-lft-identity 30.7%

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

6. Applied egg-rr 30.7%

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

   1. *-lft-identity 30.7%

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

8. Simplified 30.7%

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

9. Taylor expanded in x around 0 30.1%

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

10. Final simplification 30.1%

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

Alternative 5: 34.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7:\\ \;\;\;\;x \cdot \left|\frac{0.2 \cdot {x}^{4} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({x}^{6} \cdot \frac{x}{\sqrt{\pi}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.7)
   (*
    x
    (fabs
     (/
      (+ (* 0.2 (pow x 4.0)) (fma 0.6666666666666666 (* x x) 2.0))
      (sqrt PI))))
   (* 0.047619047619047616 (* (pow x 6.0) (/ x (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.7) {
		tmp = x * fabs((((0.2 * pow(x, 4.0)) + fma(0.6666666666666666, (x * x), 2.0)) / sqrt(((double) M_PI))));
	} else {
		tmp = 0.047619047619047616 * (pow(x, 6.0) * (x / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.7)
		tmp = Float64(x * abs(Float64(Float64(Float64(0.2 * (x ^ 4.0)) + fma(0.6666666666666666, Float64(x * x), 2.0)) / sqrt(pi))));
	else
		tmp = Float64(0.047619047619047616 * Float64((x ^ 6.0) * Float64(x / sqrt(pi))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.7], N[(x * N[Abs[N[(N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[x, 6.0], $MachinePrecision] * N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7000000000000002

   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. add-sqr-sqrt 99.4%

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

      2. sqrt-prod 69.8%

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

      3. sqr-abs 69.8%

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

      4. pow2 69.8%

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

      5. sqrt-pow1 30.7%

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

      6. metadata-eval 30.7%

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

      7. pow1 30.7%

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

      8. *-un-lft-identity 30.7%

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

   6. Applied egg-rr 30.7%

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

      1. *-lft-identity 30.7%

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

   8. Simplified 30.7%

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

   9. Taylor expanded in x around 0 30.6%

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

   if 2.7000000000000002 < 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 38.9%

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

      1. add-sqr-sqrt 38.9%

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

      2. fabs-sqr 38.9%

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

      3. add-sqr-sqrt 38.9%

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

      4. *-commutative 38.9%

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

      5. add-sqr-sqrt 1.7%

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

      6. fabs-sqr 1.7%

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

      7. add-sqr-sqrt 3.7%

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

      8. associate-*r* 3.7%

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

      9. sqrt-div 3.7%

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

      10. metadata-eval 3.7%

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

      11. un-div-inv 3.7%

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

   7. Applied egg-rr 3.7%

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

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7:\\ \;\;\;\;x \cdot \left|\frac{0.2 \cdot {x}^{4} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({x}^{6} \cdot \frac{x}{\sqrt{\pi}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.3% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({x}^{6} \cdot \frac{x}{\sqrt{\pi}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (fabs (* x (/ (fma 0.6666666666666666 (pow x 2.0) 2.0) (sqrt PI))))
   (* 0.047619047619047616 (* (pow x 6.0) (/ x (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = fabs((x * (fma(0.6666666666666666, pow(x, 2.0), 2.0) / sqrt(((double) M_PI)))));
	} else {
		tmp = 0.047619047619047616 * (pow(x, 6.0) * (x / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = abs(Float64(x * Float64(fma(0.6666666666666666, (x ^ 2.0), 2.0) / sqrt(pi))));
	else
		tmp = Float64(0.047619047619047616 * Float64((x ^ 6.0) * Float64(x / sqrt(pi))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[Abs[N[(x * N[(N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[x, 6.0], $MachinePrecision] * N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

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

   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 91.3%

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

   7. Simplified 91.3%

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

   8. Taylor expanded in x around 0 87.1%

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

      1. associate-*r* 87.1%

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

      2. distribute-rgt-out 87.1%

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

      3. unpow-1 87.1%

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

      4. metadata-eval 87.1%

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

      5. pow-sqr 87.1%

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

      6. rem-sqrt-square 87.1%

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

      7. rem-square-sqrt 87.1%

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

      8. fabs-sqr 87.1%

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

      9. rem-square-sqrt 87.1%

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

      10. fma-define 87.1%

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

   10. Simplified 87.1%

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

       1. associate-*r* 87.1%

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

       2. metadata-eval 87.1%

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

       3. pow-flip 87.1%

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

       4. pow1/2 87.1%

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

       5. div-inv 86.7%

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

       6. clear-num 86.6%

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

       7. fma-undefine 86.6%

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

       8. pow2 86.6%

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

       9. distribute-lft-in 86.6%

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

       10. clear-num 86.6%

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

       11. pow2 86.6%

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

       12. clear-num 86.7%

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

   12. Applied egg-rr 86.7%

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

       1. distribute-lft-out 86.7%

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

       2. fma-define 86.7%

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

       3. associate-*l/ 86.7%

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

       4. associate-/l* 87.1%

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

   14. Simplified 87.1%

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

   if 2.2000000000000002 < x

   1. Initial program 99.8%

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

   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 38.9%

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

      1. add-sqr-sqrt 38.9%

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

      2. fabs-sqr 38.9%

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

      3. add-sqr-sqrt 38.9%

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

      4. *-commutative 38.9%

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

      5. add-sqr-sqrt 1.7%

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

      6. fabs-sqr 1.7%

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

      7. add-sqr-sqrt 3.7%

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

      8. associate-*r* 3.7%

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

      9. sqrt-div 3.7%

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

      10. metadata-eval 3.7%

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

      11. un-div-inv 3.7%

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

   7. Applied egg-rr 3.7%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({x}^{6} \cdot \frac{x}{\sqrt{\pi}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.2% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\sqrt{\pi}}\\ \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({x}^{6} \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (sqrt PI))))
   (if (<= x 2.2)
     (* t_0 (fma 0.6666666666666666 (pow x 2.0) 2.0))
     (* 0.047619047619047616 (* (pow x 6.0) t_0)))))

C

double code(double x) {
	double t_0 = x / sqrt(((double) M_PI));
	double tmp;
	if (x <= 2.2) {
		tmp = t_0 * fma(0.6666666666666666, pow(x, 2.0), 2.0);
	} else {
		tmp = 0.047619047619047616 * (pow(x, 6.0) * t_0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x / sqrt(pi))
	tmp = 0.0
	if (x <= 2.2)
		tmp = Float64(t_0 * fma(0.6666666666666666, (x ^ 2.0), 2.0));
	else
		tmp = Float64(0.047619047619047616 * Float64((x ^ 6.0) * t_0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.2], N[(t$95$0 * N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[x, 6.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

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

   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 91.3%

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

   7. Simplified 91.3%

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

   8. Taylor expanded in x around 0 87.1%

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

      1. associate-*r* 87.1%

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

      2. distribute-rgt-out 87.1%

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

      3. unpow-1 87.1%

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

      4. metadata-eval 87.1%

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

      5. pow-sqr 87.1%

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

      6. rem-sqrt-square 87.1%

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

      7. rem-square-sqrt 87.1%

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

      8. fabs-sqr 87.1%

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

      9. rem-square-sqrt 87.1%

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

      10. fma-define 87.1%

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

   10. Simplified 87.1%

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

       1. add-sqr-sqrt 28.8%

          \[\leadsto \left|\color{blue}{\sqrt{x \cdot \left({\pi}^{-0.5} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)} \cdot \sqrt{x \cdot \left({\pi}^{-0.5} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}}\right| \]

       2. fabs-sqr 28.8%

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

       3. add-sqr-sqrt 30.5%

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

       4. associate-*r* 30.5%

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

       5. metadata-eval 30.5%

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

       6. pow-flip 30.5%

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

       7. pow1/2 30.5%

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

       8. div-inv 30.3%

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

   12. Applied egg-rr 30.3%

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

   if 2.2000000000000002 < x

   1. Initial program 99.8%

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

   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 38.9%

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

      1. add-sqr-sqrt 38.9%

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

      2. fabs-sqr 38.9%

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

      3. add-sqr-sqrt 38.9%

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

      4. *-commutative 38.9%

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

      5. add-sqr-sqrt 1.7%

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

      6. fabs-sqr 1.7%

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

      7. add-sqr-sqrt 3.7%

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

      8. associate-*r* 3.7%

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

      9. sqrt-div 3.7%

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

      10. metadata-eval 3.7%

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

      11. un-div-inv 3.7%

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

   7. Applied egg-rr 3.7%

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

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\frac{x}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left({x}^{6} \cdot \frac{x}{\sqrt{\pi}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.4% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[x, 6.0], $MachinePrecision] * N[(x / N[Sqrt[Pi], $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 65.5%

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

      1. *-commutative 65.5%

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

      2. *-commutative 65.5%

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

      3. associate-*l* 65.5%

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

      4. rem-square-sqrt 28.4%

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

      5. fabs-sqr 28.4%

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

      6. rem-square-sqrt 65.5%

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

   7. Simplified 65.5%

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

      1. add-sqr-sqrt 28.5%

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

      2. fabs-sqr 28.5%

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

      3. add-sqr-sqrt 30.2%

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

      4. *-commutative 30.2%

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

      5. *-commutative 30.2%

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

      6. sqrt-div 30.2%

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

      7. metadata-eval 30.2%

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

      8. un-div-inv 30.2%

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

   9. Applied egg-rr 30.2%

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

   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 38.9%

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

      1. add-sqr-sqrt 38.9%

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

      2. fabs-sqr 38.9%

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

      3. add-sqr-sqrt 38.9%

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

      4. *-commutative 38.9%

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

      5. add-sqr-sqrt 1.7%

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

      6. fabs-sqr 1.7%

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

      7. add-sqr-sqrt 3.7%

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

      8. associate-*r* 3.7%

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

      9. sqrt-div 3.7%

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

      10. metadata-eval 3.7%

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

      11. un-div-inv 3.7%

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

   7. Applied egg-rr 3.7%

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

4. Final simplification 30.2%

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

Alternative 9: 34.4% accurate, 8.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (* x (/ 2.0 (sqrt PI)))
   (* 0.047619047619047616 (* (pow x 7.0) (pow PI -0.5)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[x, 7.0], $MachinePrecision] * N[Power[Pi, -0.5], $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 65.5%

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

      1. *-commutative 65.5%

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

      2. *-commutative 65.5%

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

      3. associate-*l* 65.5%

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

      4. rem-square-sqrt 28.4%

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

      5. fabs-sqr 28.4%

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

      6. rem-square-sqrt 65.5%

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

   7. Simplified 65.5%

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

      1. add-sqr-sqrt 28.5%

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

      2. fabs-sqr 28.5%

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

      3. add-sqr-sqrt 30.2%

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

      4. *-commutative 30.2%

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

      5. *-commutative 30.2%

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

      6. sqrt-div 30.2%

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

      7. metadata-eval 30.2%

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

      8. un-div-inv 30.2%

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

   9. Applied egg-rr 30.2%

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

   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 38.9%

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

      1. add-sqr-sqrt 38.9%

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

      2. fabs-sqr 38.9%

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

      3. add-sqr-sqrt 38.9%

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

      4. associate-*r* 38.9%

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

      5. *-commutative 38.9%

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

      6. add-sqr-sqrt 1.7%

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

      7. fabs-sqr 1.7%

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

      8. add-sqr-sqrt 3.7%

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

      9. inv-pow 3.7%

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

      10. sqrt-pow1 3.7%

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

      11. metadata-eval 3.7%

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

   7. Applied egg-rr 3.7%

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

      1. metadata-eval 3.7%

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

      2. rem-square-sqrt 1.7%

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

      3. metadata-eval 1.7%

         \[\leadsto \left(\left|0.047619047619047616\right| \cdot \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot {x}^{\color{blue}{\left(2 \cdot 3\right)}}\right)\right) \cdot {\pi}^{-0.5} \]

      4. pow-sqr 1.7%

         \[\leadsto \left(\left|0.047619047619047616\right| \cdot \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left({x}^{3} \cdot {x}^{3}\right)}\right)\right) \cdot {\pi}^{-0.5} \]

      5. fabs-sqr 1.7%

         \[\leadsto \left(\left|0.047619047619047616\right| \cdot \left(\color{blue}{\left|\sqrt{x} \cdot \sqrt{x}\right|} \cdot \left({x}^{3} \cdot {x}^{3}\right)\right)\right) \cdot {\pi}^{-0.5} \]

      6. rem-square-sqrt 38.9%

         \[\leadsto \left(\left|0.047619047619047616\right| \cdot \left(\left|\color{blue}{x}\right| \cdot \left({x}^{3} \cdot {x}^{3}\right)\right)\right) \cdot {\pi}^{-0.5} \]

      7. fabs-sqr 38.9%

         \[\leadsto \left(\left|0.047619047619047616\right| \cdot \left(\left|x\right| \cdot \color{blue}{\left|{x}^{3} \cdot {x}^{3}\right|}\right)\right) \cdot {\pi}^{-0.5} \]

      8. pow-sqr 38.9%

         \[\leadsto \left(\left|0.047619047619047616\right| \cdot \left(\left|x\right| \cdot \left|\color{blue}{{x}^{\left(2 \cdot 3\right)}}\right|\right)\right) \cdot {\pi}^{-0.5} \]

      9. metadata-eval 38.9%

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

      10. fabs-mul 38.9%

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

      11. fabs-mul 38.9%

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

      12. *-commutative 38.9%

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

      13. fabs-mul 38.9%

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

      14. metadata-eval 38.9%

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

      15. *-commutative 38.9%

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

      16. fabs-mul 38.9%

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

   9. Simplified 3.7%

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

       1. pow1 3.7%

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

       2. associate-*r* 3.7%

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

       3. *-commutative 3.7%

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

   11. Applied egg-rr 3.7%

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

       1. unpow1 3.7%

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

       2. associate-*r* 3.7%

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

       3. *-commutative 3.7%

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

   13. Simplified 3.7%

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

4. Final simplification 30.2%

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

Alternative 10: 34.4% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   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 65.5%

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

      1. *-commutative 65.5%

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

      2. *-commutative 65.5%

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

      3. associate-*l* 65.5%

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

      4. rem-square-sqrt 28.4%

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

      5. fabs-sqr 28.4%

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

      6. rem-square-sqrt 65.5%

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

   7. Simplified 65.5%

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

      1. add-sqr-sqrt 28.5%

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

      2. fabs-sqr 28.5%

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

      3. add-sqr-sqrt 30.2%

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

      4. *-commutative 30.2%

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

      5. *-commutative 30.2%

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

      6. sqrt-div 30.2%

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

      7. metadata-eval 30.2%

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

      8. un-div-inv 30.2%

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

   9. Applied egg-rr 30.2%

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

   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 38.9%

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

      1. add-sqr-sqrt 38.9%

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

      2. fabs-sqr 38.9%

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

      3. add-sqr-sqrt 38.9%

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

      4. associate-*r* 38.9%

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

      5. *-commutative 38.9%

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

      6. add-sqr-sqrt 1.7%

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

      7. fabs-sqr 1.7%

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

      8. add-sqr-sqrt 3.7%

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

      9. inv-pow 3.7%

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

      10. sqrt-pow1 3.7%

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

      11. metadata-eval 3.7%

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

   7. Applied egg-rr 3.7%

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

      1. add-exp-log 3.7%

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

      2. log-pow 1.7%

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

   9. Applied egg-rr 1.7%

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

       1. pow1 1.7%

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

       2. associate-*l* 1.7%

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

       3. *-commutative 1.7%

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

       4. associate-*l* 1.7%

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

       5. *-commutative 1.7%

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

       6. exp-to-pow 3.7%

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

       7. metadata-eval 3.7%

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

       8. pow-flip 3.7%

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

       9. pow1/2 3.7%

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

       10. div-inv 3.7%

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

   11. Applied egg-rr 3.7%

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

       1. unpow1 3.7%

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

       2. associate-*r/ 3.7%

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

       3. pow-plus 3.7%

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

       4. metadata-eval 3.7%

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

   13. Simplified 3.7%

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

4. Final simplification 30.2%

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

Alternative 11: 34.4% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.0022675736961451248 \cdot \frac{{x}^{14}}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (* x (/ 2.0 (sqrt PI)))
   (sqrt (* 0.0022675736961451248 (/ (pow x 14.0) PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = sqrt((0.0022675736961451248 * (pow(x, 14.0) / ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.sqrt((0.0022675736961451248 * (Math.pow(x, 14.0) / Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.sqrt((0.0022675736961451248 * (math.pow(x, 14.0) / math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = sqrt(Float64(0.0022675736961451248 * Float64((x ^ 14.0) / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = sqrt((0.0022675736961451248 * ((x ^ 14.0) / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.0022675736961451248 * N[(N[Power[x, 14.0], $MachinePrecision] / Pi), $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 65.5%

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

      1. *-commutative 65.5%

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

      2. *-commutative 65.5%

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

      3. associate-*l* 65.5%

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

      4. rem-square-sqrt 28.4%

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

      5. fabs-sqr 28.4%

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

      6. rem-square-sqrt 65.5%

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

   7. Simplified 65.5%

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

      1. add-sqr-sqrt 28.5%

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

      2. fabs-sqr 28.5%

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

      3. add-sqr-sqrt 30.2%

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

      4. *-commutative 30.2%

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

      5. *-commutative 30.2%

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

      6. sqrt-div 30.2%

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

      7. metadata-eval 30.2%

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

      8. un-div-inv 30.2%

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

   9. Applied egg-rr 30.2%

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

   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 38.9%

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

      1. add-sqr-sqrt 38.9%

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

      2. fabs-sqr 38.9%

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

      3. add-sqr-sqrt 38.9%

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

      4. associate-*r* 38.9%

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

      5. *-commutative 38.9%

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

      6. add-sqr-sqrt 1.7%

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

      7. fabs-sqr 1.7%

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

      8. add-sqr-sqrt 3.7%

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

      9. inv-pow 3.7%

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

      10. sqrt-pow1 3.7%

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

      11. metadata-eval 3.7%

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

   7. Applied egg-rr 3.7%

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

      1. metadata-eval 3.7%

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

      2. rem-square-sqrt 1.7%

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

      3. metadata-eval 1.7%

         \[\leadsto \left(\left|0.047619047619047616\right| \cdot \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot {x}^{\color{blue}{\left(2 \cdot 3\right)}}\right)\right) \cdot {\pi}^{-0.5} \]

      4. pow-sqr 1.7%

         \[\leadsto \left(\left|0.047619047619047616\right| \cdot \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left({x}^{3} \cdot {x}^{3}\right)}\right)\right) \cdot {\pi}^{-0.5} \]

      5. fabs-sqr 1.7%

         \[\leadsto \left(\left|0.047619047619047616\right| \cdot \left(\color{blue}{\left|\sqrt{x} \cdot \sqrt{x}\right|} \cdot \left({x}^{3} \cdot {x}^{3}\right)\right)\right) \cdot {\pi}^{-0.5} \]

      6. rem-square-sqrt 38.9%

         \[\leadsto \left(\left|0.047619047619047616\right| \cdot \left(\left|\color{blue}{x}\right| \cdot \left({x}^{3} \cdot {x}^{3}\right)\right)\right) \cdot {\pi}^{-0.5} \]

      7. fabs-sqr 38.9%

         \[\leadsto \left(\left|0.047619047619047616\right| \cdot \left(\left|x\right| \cdot \color{blue}{\left|{x}^{3} \cdot {x}^{3}\right|}\right)\right) \cdot {\pi}^{-0.5} \]

      8. pow-sqr 38.9%

         \[\leadsto \left(\left|0.047619047619047616\right| \cdot \left(\left|x\right| \cdot \left|\color{blue}{{x}^{\left(2 \cdot 3\right)}}\right|\right)\right) \cdot {\pi}^{-0.5} \]

      9. metadata-eval 38.9%

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

      10. fabs-mul 38.9%

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

      11. fabs-mul 38.9%

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

      12. *-commutative 38.9%

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

      13. fabs-mul 38.9%

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

      14. metadata-eval 38.9%

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

      15. *-commutative 38.9%

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

      16. fabs-mul 38.9%

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

   9. Simplified 3.7%

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

       1. add-sqr-sqrt 3.6%

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

       2. sqrt-unprod 33.9%

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

       3. swap-sqr 33.9%

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

       4. pow-prod-up 33.9%

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

       5. metadata-eval 33.9%

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

       6. *-commutative 33.9%

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

       7. *-commutative 33.9%

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

       8. swap-sqr 33.9%

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

       9. pow-prod-up 33.9%

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

       10. metadata-eval 33.9%

           \[\leadsto \sqrt{{\pi}^{-1} \cdot \left({x}^{\color{blue}{14}} \cdot \left(0.047619047619047616 \cdot 0.047619047619047616\right)\right)} \]

       11. metadata-eval 33.9%

           \[\leadsto \sqrt{{\pi}^{-1} \cdot \left({x}^{14} \cdot \color{blue}{0.0022675736961451248}\right)} \]

   11. Applied egg-rr 33.9%

       \[\leadsto \color{blue}{\sqrt{{\pi}^{-1} \cdot \left({x}^{14} \cdot 0.0022675736961451248\right)}} \]
   12. Step-by-step derivation

       1. associate-*r* 33.9%

          \[\leadsto \sqrt{\color{blue}{\left({\pi}^{-1} \cdot {x}^{14}\right) \cdot 0.0022675736961451248}} \]

       2. *-commutative 33.9%

          \[\leadsto \sqrt{\color{blue}{0.0022675736961451248 \cdot \left({\pi}^{-1} \cdot {x}^{14}\right)}} \]

       3. metadata-eval 33.9%

          \[\leadsto \sqrt{0.0022675736961451248 \cdot \left({\pi}^{-1} \cdot {x}^{\color{blue}{\left(2 \cdot 7\right)}}\right)} \]

       4. pow-sqr 33.9%

          \[\leadsto \sqrt{0.0022675736961451248 \cdot \left({\pi}^{-1} \cdot \color{blue}{\left({x}^{7} \cdot {x}^{7}\right)}\right)} \]

       5. unpow-1 33.9%

          \[\leadsto \sqrt{0.0022675736961451248 \cdot \left(\color{blue}{\frac{1}{\pi}} \cdot \left({x}^{7} \cdot {x}^{7}\right)\right)} \]

       6. associate-*l/ 33.9%

          \[\leadsto \sqrt{0.0022675736961451248 \cdot \color{blue}{\frac{1 \cdot \left({x}^{7} \cdot {x}^{7}\right)}{\pi}}} \]

       7. *-lft-identity 33.9%

          \[\leadsto \sqrt{0.0022675736961451248 \cdot \frac{\color{blue}{{x}^{7} \cdot {x}^{7}}}{\pi}} \]

       8. pow-sqr 33.9%

          \[\leadsto \sqrt{0.0022675736961451248 \cdot \frac{\color{blue}{{x}^{\left(2 \cdot 7\right)}}}{\pi}} \]

       9. metadata-eval 33.9%

          \[\leadsto \sqrt{0.0022675736961451248 \cdot \frac{{x}^{\color{blue}{14}}}{\pi}} \]

   13. Simplified 33.9%

       \[\leadsto \color{blue}{\sqrt{0.0022675736961451248 \cdot \frac{{x}^{14}}{\pi}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.0022675736961451248 \cdot \frac{{x}^{14}}{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 34.4% 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%

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

5. Taylor expanded in x around 0 65.5%

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

   1. *-commutative 65.5%

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

   2. *-commutative 65.5%

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

   3. associate-*l* 65.5%

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

   4. rem-square-sqrt 28.4%

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

   5. fabs-sqr 28.4%

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

   6. rem-square-sqrt 65.5%

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

7. Simplified 65.5%

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

   1. add-sqr-sqrt 28.5%

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

   2. fabs-sqr 28.5%

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

   3. add-sqr-sqrt 30.2%

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

   4. *-commutative 30.2%

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

   5. *-commutative 30.2%

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

   6. sqrt-div 30.2%

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

   7. metadata-eval 30.2%

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

   8. un-div-inv 30.2%

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

9. Applied egg-rr 30.2%

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

10. Final simplification 30.2%

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

Reproduce

herbie shell --seed 2024114 
(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)))))))