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

Percentage Accurate: 99.9% → 99.9%

Time: 9.6s

Alternatives: 11

Speedup: 3.1×

• 28.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

   1. div-inv 99.8%

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

   2. add-sqr-sqrt 99.4%

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

   3. sqrt-prod 68.5%

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

   4. sqr-abs 68.5%

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

   5. sqrt-prod 35.7%

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

   6. add-sqr-sqrt 99.8%

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

   7. inv-pow 99.8%

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

   8. sqrt-pow2 99.8%

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

   9. metadata-eval 99.8%

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

5. Applied egg-rr 99.8%

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

6. Taylor expanded in x around 0 99.8%

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

   1. fma-def 99.9%

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

8. Simplified 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

   1. div-inv 99.8%

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

   2. add-sqr-sqrt 99.4%

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

   3. sqrt-prod 68.5%

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

   4. sqr-abs 68.5%

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

   5. sqrt-prod 35.7%

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

   6. add-sqr-sqrt 99.8%

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

   7. inv-pow 99.8%

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

   8. sqrt-pow2 99.8%

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

   9. metadata-eval 99.8%

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

5. Applied egg-rr 99.8%

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

   1. metadata-eval 99.4%

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

   2. fma-udef 99.4%

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

   3. metadata-eval 99.4%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 3: 99.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[Abs[N[(N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] + N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / 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 99.4%

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

   1. expm1-log1p-u 99.2%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left|x\right|}{\sqrt{\pi}}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   2. expm1-udef 35.8%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\left|x\right|}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   3. add-sqr-sqrt 35.8%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left|x\right|} \cdot \sqrt{\left|x\right|}}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   4. sqrt-prod 35.8%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left|x\right| \cdot \left|x\right|}}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   5. sqr-abs 35.8%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{x \cdot x}}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   6. sqrt-prod 2.9%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   7. add-sqr-sqrt 6.4%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{x}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

5. Applied egg-rr 6.4%

   \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]
6. Step-by-step derivation

   1. expm1-def 69.7%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   2. expm1-log1p 99.4%

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

7. Simplified 99.4%

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

   1. metadata-eval 99.4%

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

   2. fma-udef 99.4%

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

   3. metadata-eval 99.4%

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

9. Applied egg-rr 99.4%

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

10. Final simplification 99.4%

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

Alternative 4: 33.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;x \cdot \frac{2 + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (*
    x
    (/
     (+ 2.0 (+ (* 0.2 (pow x 4.0)) (* 0.6666666666666666 (pow x 2.0))))
     (sqrt PI)))
   (fabs
    (*
     (sqrt (/ 1.0 PI))
     (+ (* 0.2 (pow x 5.0)) (* 0.047619047619047616 (pow x 7.0)))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = x * ((2.0 + ((0.2 * pow(x, 4.0)) + (0.6666666666666666 * pow(x, 2.0)))) / sqrt(((double) M_PI)));
	} else {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * ((0.2 * pow(x, 5.0)) + (0.047619047619047616 * pow(x, 7.0)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = x * ((2.0 + ((0.2 * Math.pow(x, 4.0)) + (0.6666666666666666 * Math.pow(x, 2.0)))) / Math.sqrt(Math.PI));
	} else {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * ((0.2 * Math.pow(x, 5.0)) + (0.047619047619047616 * Math.pow(x, 7.0)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.2:
		tmp = x * ((2.0 + ((0.2 * math.pow(x, 4.0)) + (0.6666666666666666 * math.pow(x, 2.0)))) / math.sqrt(math.pi))
	else:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * ((0.2 * math.pow(x, 5.0)) + (0.047619047619047616 * math.pow(x, 7.0)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = Float64(x * Float64(Float64(2.0 + Float64(Float64(0.2 * (x ^ 4.0)) + Float64(0.6666666666666666 * (x ^ 2.0)))) / sqrt(pi)));
	else
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(0.2 * (x ^ 5.0)) + Float64(0.047619047619047616 * (x ^ 7.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.2)
		tmp = x * ((2.0 + ((0.2 * (x ^ 4.0)) + (0.6666666666666666 * (x ^ 2.0)))) / sqrt(pi));
	else
		tmp = abs((sqrt((1.0 / pi)) * ((0.2 * (x ^ 5.0)) + (0.047619047619047616 * (x ^ 7.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[(x * N[(N[(2.0 + N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 93.5%

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

      1. div-inv 94.0%

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

      2. add-sqr-sqrt 35.7%

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

      3. fabs-sqr 35.7%

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

      4. add-sqr-sqrt 37.2%

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

      5. *-commutative 37.2%

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

      6. add-sqr-sqrt 37.2%

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

      7. fabs-sqr 37.2%

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

      8. add-sqr-sqrt 37.2%

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

   6. Applied egg-rr 37.2%

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

      1. fma-def 37.2%

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

      2. fma-udef 37.2%

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

      3. associate-+r+ 37.2%

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

   8. Applied egg-rr 37.2%

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

   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 99.4%

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

      1. div-inv 99.8%

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

      2. add-sqr-sqrt 99.4%

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

      3. sqrt-prod 68.5%

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

      4. sqr-abs 68.5%

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

      5. sqrt-prod 35.7%

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

      6. add-sqr-sqrt 99.8%

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

      7. inv-pow 99.8%

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

      8. sqrt-pow2 99.8%

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

      9. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 99.8%

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

      1. fma-def 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around inf 33.2%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.2 \cdot \left({x}^{5} \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   10. Step-by-step derivation

       1. +-commutative 33.2%

          \[\leadsto \left|\color{blue}{0.2 \cdot \left({x}^{5} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

       2. associate-*r* 33.2%

          \[\leadsto \left|\color{blue}{\left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}}} + 0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

       3. associate-*r* 33.2%

          \[\leadsto \left|\left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

       4. distribute-rgt-out 33.2%

          \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)}\right| \]

   11. Simplified 33.2%

       \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;x \cdot \frac{2 + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \end{array} \]

Alternative 5: 98.7% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

   1. expm1-log1p-u 99.2%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left|x\right|}{\sqrt{\pi}}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   2. expm1-udef 35.8%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\left|x\right|}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   3. add-sqr-sqrt 35.8%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left|x\right|} \cdot \sqrt{\left|x\right|}}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   4. sqrt-prod 35.8%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left|x\right| \cdot \left|x\right|}}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   5. sqr-abs 35.8%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{x \cdot x}}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   6. sqrt-prod 2.9%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   7. add-sqr-sqrt 6.4%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\frac{\color{blue}{x}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

5. Applied egg-rr 6.4%

   \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]
6. Step-by-step derivation

   1. expm1-def 69.7%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   2. expm1-log1p 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in x around inf 98.2%

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

9. Final simplification 98.2%

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

Alternative 6: 33.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;x \cdot \frac{2 + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.6)
   (*
    x
    (/
     (+ 2.0 (+ (* 0.2 (pow x 4.0)) (* 0.6666666666666666 (pow x 2.0))))
     (sqrt PI)))
   (fabs
    (* 0.047619047619047616 (* (sqrt (/ 1.0 PI)) (* (pow x 6.0) (fabs x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.6) {
		tmp = x * ((2.0 + ((0.2 * pow(x, 4.0)) + (0.6666666666666666 * pow(x, 2.0)))) / sqrt(((double) M_PI)));
	} else {
		tmp = fabs((0.047619047619047616 * (sqrt((1.0 / ((double) M_PI))) * (pow(x, 6.0) * fabs(x)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.6) {
		tmp = x * ((2.0 + ((0.2 * Math.pow(x, 4.0)) + (0.6666666666666666 * Math.pow(x, 2.0)))) / Math.sqrt(Math.PI));
	} else {
		tmp = Math.abs((0.047619047619047616 * (Math.sqrt((1.0 / Math.PI)) * (Math.pow(x, 6.0) * Math.abs(x)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.6:
		tmp = x * ((2.0 + ((0.2 * math.pow(x, 4.0)) + (0.6666666666666666 * math.pow(x, 2.0)))) / math.sqrt(math.pi))
	else:
		tmp = math.fabs((0.047619047619047616 * (math.sqrt((1.0 / math.pi)) * (math.pow(x, 6.0) * math.fabs(x)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.6)
		tmp = Float64(x * Float64(Float64(2.0 + Float64(Float64(0.2 * (x ^ 4.0)) + Float64(0.6666666666666666 * (x ^ 2.0)))) / sqrt(pi)));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64(sqrt(Float64(1.0 / pi)) * Float64((x ^ 6.0) * abs(x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.6)
		tmp = x * ((2.0 + ((0.2 * (x ^ 4.0)) + (0.6666666666666666 * (x ^ 2.0)))) / sqrt(pi));
	else
		tmp = abs((0.047619047619047616 * (sqrt((1.0 / pi)) * ((x ^ 6.0) * abs(x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.6], N[(x * N[(N[(2.0 + N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[Power[x, 6.0], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.60000000000000009

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 93.5%

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

      1. div-inv 94.0%

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

      2. add-sqr-sqrt 35.7%

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

      3. fabs-sqr 35.7%

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

      4. add-sqr-sqrt 37.2%

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

      5. *-commutative 37.2%

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

      6. add-sqr-sqrt 37.2%

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

      7. fabs-sqr 37.2%

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

      8. add-sqr-sqrt 37.2%

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

   6. Applied egg-rr 37.2%

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

      1. fma-def 37.2%

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

      2. fma-udef 37.2%

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

      3. associate-+r+ 37.2%

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

   8. Applied egg-rr 37.2%

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

   if 2.60000000000000009 < 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%

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

   4. Taylor expanded in x around inf 32.9%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;x \cdot \frac{2 + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right|\\ \end{array} \]

Alternative 7: 33.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;x \cdot \frac{2 + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{7}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.6)
   (*
    x
    (/
     (+ 2.0 (+ (* 0.2 (pow x 4.0)) (* 0.6666666666666666 (pow x 2.0))))
     (sqrt PI)))
   (fabs (* 0.047619047619047616 (* (sqrt (/ 1.0 PI)) (pow x 7.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.6) {
		tmp = x * ((2.0 + ((0.2 * pow(x, 4.0)) + (0.6666666666666666 * pow(x, 2.0)))) / sqrt(((double) M_PI)));
	} else {
		tmp = fabs((0.047619047619047616 * (sqrt((1.0 / ((double) M_PI))) * pow(x, 7.0))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.6) {
		tmp = x * ((2.0 + ((0.2 * Math.pow(x, 4.0)) + (0.6666666666666666 * Math.pow(x, 2.0)))) / Math.sqrt(Math.PI));
	} else {
		tmp = Math.abs((0.047619047619047616 * (Math.sqrt((1.0 / Math.PI)) * Math.pow(x, 7.0))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.6:
		tmp = x * ((2.0 + ((0.2 * math.pow(x, 4.0)) + (0.6666666666666666 * math.pow(x, 2.0)))) / math.sqrt(math.pi))
	else:
		tmp = math.fabs((0.047619047619047616 * (math.sqrt((1.0 / math.pi)) * math.pow(x, 7.0))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.6)
		tmp = Float64(x * Float64(Float64(2.0 + Float64(Float64(0.2 * (x ^ 4.0)) + Float64(0.6666666666666666 * (x ^ 2.0)))) / sqrt(pi)));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64(sqrt(Float64(1.0 / pi)) * (x ^ 7.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.6)
		tmp = x * ((2.0 + ((0.2 * (x ^ 4.0)) + (0.6666666666666666 * (x ^ 2.0)))) / sqrt(pi));
	else
		tmp = abs((0.047619047619047616 * (sqrt((1.0 / pi)) * (x ^ 7.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.6], N[(x * N[(N[(2.0 + N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.60000000000000009

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 93.5%

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

      1. div-inv 94.0%

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

      2. add-sqr-sqrt 35.7%

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

      3. fabs-sqr 35.7%

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

      4. add-sqr-sqrt 37.2%

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

      5. *-commutative 37.2%

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

      6. add-sqr-sqrt 37.2%

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

      7. fabs-sqr 37.2%

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

      8. add-sqr-sqrt 37.2%

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

   6. Applied egg-rr 37.2%

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

      1. fma-def 37.2%

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

      2. fma-udef 37.2%

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

      3. associate-+r+ 37.2%

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

   8. Applied egg-rr 37.2%

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

   if 2.60000000000000009 < x

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

      1. div-inv 99.8%

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

      2. add-sqr-sqrt 99.4%

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

      3. sqrt-prod 68.5%

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

      4. sqr-abs 68.5%

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

      5. sqrt-prod 35.7%

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

      6. add-sqr-sqrt 99.8%

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

      7. inv-pow 99.8%

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

      8. sqrt-pow2 99.8%

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

      9. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 99.8%

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

      1. fma-def 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around inf 32.9%

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

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;x \cdot \frac{2 + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{7}\right)\right|\\ \end{array} \]

Alternative 8: 33.9% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[(x * N[(N[(2.0 + N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[Power[x, 7.0], $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 99.4%

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

   4. Taylor expanded in x around 0 93.5%

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

      1. div-inv 94.0%

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

      2. add-sqr-sqrt 35.7%

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

      3. fabs-sqr 35.7%

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

      4. add-sqr-sqrt 37.2%

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

      5. *-commutative 37.2%

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

      6. add-sqr-sqrt 37.2%

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

      7. fabs-sqr 37.2%

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

      8. add-sqr-sqrt 37.2%

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

   6. Applied egg-rr 37.2%

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

   7. Taylor expanded in x around 0 37.1%

      \[\leadsto \frac{\color{blue}{2 + 0.6666666666666666 \cdot {x}^{2}}}{\sqrt{\pi}} \cdot x \]

   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 99.4%

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

      1. div-inv 99.8%

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

      2. add-sqr-sqrt 99.4%

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

      3. sqrt-prod 68.5%

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

      4. sqr-abs 68.5%

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

      5. sqrt-prod 35.7%

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

      6. add-sqr-sqrt 99.8%

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

      7. inv-pow 99.8%

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

      8. sqrt-pow2 99.8%

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

      9. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 99.8%

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

      1. fma-def 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around inf 32.9%

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

4. Final simplification 37.1%

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

Alternative 9: 33.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.3:\\ \;\;\;\;x \cdot \frac{2 + 0.6666666666666666 \cdot {x}^{2}}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{0.2 \cdot {x}^{4}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.3)
   (* x (/ (+ 2.0 (* 0.6666666666666666 (pow x 2.0))) (sqrt PI)))
   (* x (/ (* 0.2 (pow x 4.0)) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 2.3) {
		tmp = x * ((2.0 + (0.6666666666666666 * pow(x, 2.0))) / sqrt(((double) M_PI)));
	} else {
		tmp = x * ((0.2 * pow(x, 4.0)) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.3) {
		tmp = x * ((2.0 + (0.6666666666666666 * Math.pow(x, 2.0))) / Math.sqrt(Math.PI));
	} else {
		tmp = x * ((0.2 * Math.pow(x, 4.0)) / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.3:
		tmp = x * ((2.0 + (0.6666666666666666 * math.pow(x, 2.0))) / math.sqrt(math.pi))
	else:
		tmp = x * ((0.2 * math.pow(x, 4.0)) / math.sqrt(math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.3)
		tmp = Float64(x * Float64(Float64(2.0 + Float64(0.6666666666666666 * (x ^ 2.0))) / sqrt(pi)));
	else
		tmp = Float64(x * Float64(Float64(0.2 * (x ^ 4.0)) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.3)
		tmp = x * ((2.0 + (0.6666666666666666 * (x ^ 2.0))) / sqrt(pi));
	else
		tmp = x * ((0.2 * (x ^ 4.0)) / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.3], N[(x * N[(N[(2.0 + N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2999999999999998

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 93.5%

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

      1. div-inv 94.0%

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

      2. add-sqr-sqrt 35.7%

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

      3. fabs-sqr 35.7%

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

      4. add-sqr-sqrt 37.2%

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

      5. *-commutative 37.2%

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

      6. add-sqr-sqrt 37.2%

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

      7. fabs-sqr 37.2%

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

      8. add-sqr-sqrt 37.2%

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

   6. Applied egg-rr 37.2%

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

   7. Taylor expanded in x around 0 37.1%

      \[\leadsto \frac{\color{blue}{2 + 0.6666666666666666 \cdot {x}^{2}}}{\sqrt{\pi}} \cdot x \]

   if 2.2999999999999998 < x

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 93.5%

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

      1. div-inv 94.0%

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

      2. add-sqr-sqrt 35.7%

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

      3. fabs-sqr 35.7%

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

      4. add-sqr-sqrt 37.2%

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

      5. *-commutative 37.2%

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

      6. add-sqr-sqrt 37.2%

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

      7. fabs-sqr 37.2%

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

      8. add-sqr-sqrt 37.2%

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

   6. Applied egg-rr 37.2%

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

   7. Taylor expanded in x around inf 3.8%

      \[\leadsto \frac{\color{blue}{0.2 \cdot {x}^{4}}}{\sqrt{\pi}} \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.3:\\ \;\;\;\;x \cdot \frac{2 + 0.6666666666666666 \cdot {x}^{2}}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{0.2 \cdot {x}^{4}}{\sqrt{\pi}}\\ \end{array} \]

Alternative 10: 33.8% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{0.2 \cdot {x}^{4}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.75)
   (* x (/ 2.0 (sqrt PI)))
   (* x (/ (* 0.2 (pow x 4.0)) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.75) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = x * ((0.2 * pow(x, 4.0)) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.75) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = x * ((0.2 * Math.pow(x, 4.0)) / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.75:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = x * ((0.2 * math.pow(x, 4.0)) / math.sqrt(math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.75)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(x * Float64(Float64(0.2 * (x ^ 4.0)) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.75)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = x * ((0.2 * (x ^ 4.0)) / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.75], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.75

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 93.5%

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

      1. div-inv 94.0%

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

      2. add-sqr-sqrt 35.7%

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

      3. fabs-sqr 35.7%

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

      4. add-sqr-sqrt 37.2%

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

      5. *-commutative 37.2%

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

      6. add-sqr-sqrt 37.2%

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

      7. fabs-sqr 37.2%

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

      8. add-sqr-sqrt 37.2%

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

   6. Applied egg-rr 37.2%

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

   7. Taylor expanded in x around 0 36.8%

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

   if 1.75 < x

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 93.5%

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

      1. div-inv 94.0%

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

      2. add-sqr-sqrt 35.7%

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

      3. fabs-sqr 35.7%

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

      4. add-sqr-sqrt 37.2%

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

      5. *-commutative 37.2%

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

      6. add-sqr-sqrt 37.2%

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

      7. fabs-sqr 37.2%

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

      8. add-sqr-sqrt 37.2%

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

   6. Applied egg-rr 37.2%

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

   7. Taylor expanded in x around inf 3.8%

      \[\leadsto \frac{\color{blue}{0.2 \cdot {x}^{4}}}{\sqrt{\pi}} \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{0.2 \cdot {x}^{4}}{\sqrt{\pi}}\\ \end{array} \]

Alternative 11: 33.8% accurate, 9.5× 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 99.4%

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

4. Taylor expanded in x around 0 93.5%

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

   1. div-inv 94.0%

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

   2. add-sqr-sqrt 35.7%

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

   3. fabs-sqr 35.7%

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

   4. add-sqr-sqrt 37.2%

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

   5. *-commutative 37.2%

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

   6. add-sqr-sqrt 37.2%

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

   7. fabs-sqr 37.2%

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

   8. add-sqr-sqrt 37.2%

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

6. Applied egg-rr 37.2%

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

7. Taylor expanded in x around 0 36.8%

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

8. Final simplification 36.8%

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

Reproduce

herbie shell --seed 2023319 
(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)))))))