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

Percentage Accurate: 99.8% → 99.9%

Time: 10.8s

Alternatives: 11

Speedup: 3.5×

• 28.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 3.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  x_m
  (fabs
   (/
    (+
     (fma 0.2 (pow x_m 4.0) (* 0.047619047619047616 (pow x_m 6.0)))
     (fma 0.6666666666666666 (* x_m x_m) 2.0))
    (sqrt PI)))))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * N[Abs[N[(N[(N[(0.2 * N[Power[x$95$m, 4.0], $MachinePrecision] + N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. add-sqr-sqrt 99.4%

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

   2. sqrt-prod 64.0%

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

   3. sqr-abs 64.0%

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

   4. pow2 64.0%

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

   5. sqrt-pow1 39.2%

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

   6. metadata-eval 39.2%

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

   7. pow1 39.2%

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

   8. *-un-lft-identity 39.2%

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

6. Applied egg-rr 39.2%

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

   1. *-lft-identity 39.2%

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

8. Simplified 39.2%

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

Alternative 2: 99.9% accurate, 3.5× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  x_m
  (fabs
   (*
    (pow PI -0.5)
    (+
     2.0
     (+
      (* 0.047619047619047616 (pow x_m 6.0))
      (+ (* 0.2 (pow x_m 4.0)) (* 0.6666666666666666 (pow x_m 2.0)))))))))

C

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return x_m * Math.abs((Math.pow(Math.PI, -0.5) * (2.0 + ((0.047619047619047616 * Math.pow(x_m, 6.0)) + ((0.2 * Math.pow(x_m, 4.0)) + (0.6666666666666666 * Math.pow(x_m, 2.0)))))));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return x_m * math.fabs((math.pow(math.pi, -0.5) * (2.0 + ((0.047619047619047616 * math.pow(x_m, 6.0)) + ((0.2 * math.pow(x_m, 4.0)) + (0.6666666666666666 * math.pow(x_m, 2.0)))))))

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = x_m * abs(((pi ^ -0.5) * (2.0 + ((0.047619047619047616 * (x_m ^ 6.0)) + ((0.2 * (x_m ^ 4.0)) + (0.6666666666666666 * (x_m ^ 2.0)))))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(2.0 + N[(N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.2 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $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 99.8%

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

   1. add-sqr-sqrt 99.4%

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

   2. sqrt-prod 64.0%

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

   3. sqr-abs 64.0%

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

   4. pow2 64.0%

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

   5. sqrt-pow1 39.2%

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

   6. metadata-eval 39.2%

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

   7. pow1 39.2%

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

   8. *-un-lft-identity 39.2%

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

6. Applied egg-rr 39.2%

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

   1. *-lft-identity 39.2%

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

8. Simplified 39.2%

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

9. Taylor expanded in x around 0 39.2%

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

    1. *-un-lft-identity 39.2%

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

    2. pow1/2 39.2%

       \[\leadsto x \cdot \left|\left(1 \cdot \color{blue}{{\left(\frac{1}{\pi}\right)}^{0.5}}\right) \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

    3. inv-pow 39.2%

       \[\leadsto x \cdot \left|\left(1 \cdot {\color{blue}{\left({\pi}^{-1}\right)}}^{0.5}\right) \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

    4. pow-pow 39.2%

       \[\leadsto x \cdot \left|\left(1 \cdot \color{blue}{{\pi}^{\left(-1 \cdot 0.5\right)}}\right) \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

    5. metadata-eval 39.2%

       \[\leadsto x \cdot \left|\left(1 \cdot {\pi}^{\color{blue}{-0.5}}\right) \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]

11. Applied egg-rr 39.2%

    \[\leadsto x \cdot \left|\color{blue}{\left(1 \cdot {\pi}^{-0.5}\right)} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]
12. Step-by-step derivation

    1. *-lft-identity 39.2%

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

13. Simplified 39.2%

    \[\leadsto x \cdot \left|\color{blue}{{\pi}^{-0.5}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]
14. Add Preprocessing

Alternative 3: 99.2% accurate, 4.5× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  x_m
  (fabs
   (/
    (+
     (* 0.047619047619047616 (pow x_m 6.0))
     (fma 0.6666666666666666 (* x_m x_m) 2.0))
    (sqrt PI)))))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * N[Abs[N[(N[(N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. add-sqr-sqrt 99.4%

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

   2. sqrt-prod 64.0%

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

   3. sqr-abs 64.0%

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

   4. pow2 64.0%

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

   5. sqrt-pow1 39.2%

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

   6. metadata-eval 39.2%

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

   7. pow1 39.2%

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

   8. *-un-lft-identity 39.2%

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

6. Applied egg-rr 39.2%

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

   1. *-lft-identity 39.2%

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

8. Simplified 39.2%

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

9. Taylor expanded in x around inf 39.1%

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

Alternative 4: 98.8% accurate, 5.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(x\_m \cdot 2 + 0.047619047619047616 \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left|x\_m\right|\right)\right)\right)\right)\right| \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt PI))
   (+
    (* x_m 2.0)
    (*
     0.047619047619047616
     (* (* x_m x_m) (* (* x_m x_m) (* (* x_m x_m) (fabs x_m)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((x_m * 2.0) + (0.047619047619047616 * ((x_m * x_m) * ((x_m * x_m) * ((x_m * x_m) * fabs(x_m))))))));
}

Java

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

Python

x_m = math.fabs(x)
def code(x_m):
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((x_m * 2.0) + (0.047619047619047616 * ((x_m * x_m) * ((x_m * x_m) * ((x_m * x_m) * math.fabs(x_m))))))))

Julia

x_m = abs(x)
function code(x_m)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(x_m * 2.0) + Float64(0.047619047619047616 * Float64(Float64(x_m * x_m) * Float64(Float64(x_m * x_m) * Float64(Float64(x_m * x_m) * abs(x_m))))))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = abs(((1.0 / sqrt(pi)) * ((x_m * 2.0) + (0.047619047619047616 * ((x_m * x_m) * ((x_m * x_m) * ((x_m * x_m) * abs(x_m))))))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(x$95$m * 2.0), $MachinePrecision] + N[(0.047619047619047616 * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[Abs[x$95$m], $MachinePrecision]), $MachinePrecision]), $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 99.8%

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

5. Taylor expanded in x around 0 97.7%

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

   1. rem-square-sqrt 37.3%

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

   2. fabs-sqr 37.3%

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

   3. rem-square-sqrt 97.7%

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

7. Simplified 97.7%

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

8. Final simplification 97.7%

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

Alternative 5: 99.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.2:\\ \;\;\;\;\left(x\_m \cdot {\pi}^{-0.5}\right) \cdot \mathsf{fma}\left(0.6666666666666666, {x\_m}^{2}, 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x\_m}^{7}\right)\right|\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.2)
   (* (* x_m (pow PI -0.5)) (fma 0.6666666666666666 (pow x_m 2.0) 2.0))
   (fabs (* 0.047619047619047616 (* (sqrt (/ 1.0 PI)) (pow x_m 7.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.2) {
		tmp = (x_m * pow(((double) M_PI), -0.5)) * fma(0.6666666666666666, pow(x_m, 2.0), 2.0);
	} else {
		tmp = fabs((0.047619047619047616 * (sqrt((1.0 / ((double) M_PI))) * pow(x_m, 7.0))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.2)
		tmp = Float64(Float64(x_m * (pi ^ -0.5)) * fma(0.6666666666666666, (x_m ^ 2.0), 2.0));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64(sqrt(Float64(1.0 / pi)) * (x_m ^ 7.0))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.2], N[(N[(x$95$m * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[Power[x$95$m, 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 99.8%

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

   5. Taylor expanded in x around 0 90.4%

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

      1. associate-*r* 90.4%

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

      2. rem-square-sqrt 37.8%

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

      3. fabs-sqr 37.8%

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

      4. rem-square-sqrt 89.8%

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

      5. pow-plus 89.8%

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

      6. metadata-eval 89.8%

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

      7. rem-cube-cbrt 89.8%

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

      8. cube-prod 89.8%

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

      9. rem-square-sqrt 37.8%

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

      10. fabs-sqr 37.8%

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

      11. rem-square-sqrt 90.4%

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

      12. cube-prod 90.4%

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

      13. rem-cube-cbrt 90.4%

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

      14. *-commutative 90.4%

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

      15. associate-*r* 90.4%

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

   7. Simplified 90.4%

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

   8. Taylor expanded in x around 0 90.4%

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

      1. associate-*l* 90.4%

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

      2. +-commutative 90.4%

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

      3. fma-define 90.4%

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

   10. Simplified 90.4%

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

       1. add-sqr-sqrt 37.5%

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

       2. fabs-sqr 37.5%

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

       3. add-sqr-sqrt 39.2%

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

       4. *-commutative 39.2%

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

       5. *-commutative 39.2%

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

       6. inv-pow 39.2%

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

       7. sqrt-pow1 39.2%

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

       8. metadata-eval 39.2%

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

   12. Applied egg-rr 39.2%

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

       1. pow1 39.2%

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

       2. *-commutative 39.2%

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

   14. Applied egg-rr 39.2%

       \[\leadsto \color{blue}{{\left(x \cdot \left({\pi}^{-0.5} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\right)}^{1}} \]
   15. Step-by-step derivation

       1. unpow1 39.2%

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

       2. associate-*r* 39.2%

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

   16. Simplified 39.2%

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

   if 2.2000000000000002 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 32.4%

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

      1. *-commutative 32.4%

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

      2. rem-square-sqrt 2.1%

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

      3. fabs-sqr 2.1%

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

      4. rem-square-sqrt 32.4%

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

      5. pow-plus 32.4%

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

      6. rem-square-sqrt 2.1%

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

      7. fabs-sqr 2.1%

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

      8. rem-square-sqrt 32.4%

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

      9. metadata-eval 32.4%

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

   7. Simplified 32.4%

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

Alternative 6: 99.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.2:\\ \;\;\;\;x\_m \cdot \left({\pi}^{-0.5} \cdot \left(2 + 0.6666666666666666 \cdot {x\_m}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x\_m}^{7}\right)\right|\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.2)
   (* x_m (* (pow PI -0.5) (+ 2.0 (* 0.6666666666666666 (pow x_m 2.0)))))
   (fabs (* 0.047619047619047616 (* (sqrt (/ 1.0 PI)) (pow x_m 7.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.2) {
		tmp = x_m * (pow(((double) M_PI), -0.5) * (2.0 + (0.6666666666666666 * pow(x_m, 2.0))));
	} else {
		tmp = fabs((0.047619047619047616 * (sqrt((1.0 / ((double) M_PI))) * pow(x_m, 7.0))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.2) {
		tmp = x_m * (Math.pow(Math.PI, -0.5) * (2.0 + (0.6666666666666666 * Math.pow(x_m, 2.0))));
	} else {
		tmp = Math.abs((0.047619047619047616 * (Math.sqrt((1.0 / Math.PI)) * Math.pow(x_m, 7.0))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.2:
		tmp = x_m * (math.pow(math.pi, -0.5) * (2.0 + (0.6666666666666666 * math.pow(x_m, 2.0))))
	else:
		tmp = math.fabs((0.047619047619047616 * (math.sqrt((1.0 / math.pi)) * math.pow(x_m, 7.0))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.2)
		tmp = Float64(x_m * Float64((pi ^ -0.5) * Float64(2.0 + Float64(0.6666666666666666 * (x_m ^ 2.0)))));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64(sqrt(Float64(1.0 / pi)) * (x_m ^ 7.0))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.2)
		tmp = x_m * ((pi ^ -0.5) * (2.0 + (0.6666666666666666 * (x_m ^ 2.0))));
	else
		tmp = abs((0.047619047619047616 * (sqrt((1.0 / pi)) * (x_m ^ 7.0))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.2], N[(x$95$m * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(2.0 + N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[Power[x$95$m, 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 99.8%

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

   5. Taylor expanded in x around 0 90.4%

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

      1. associate-*r* 90.4%

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

      2. rem-square-sqrt 37.8%

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

      3. fabs-sqr 37.8%

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

      4. rem-square-sqrt 89.8%

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

      5. pow-plus 89.8%

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

      6. metadata-eval 89.8%

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

      7. rem-cube-cbrt 89.8%

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

      8. cube-prod 89.8%

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

      9. rem-square-sqrt 37.8%

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

      10. fabs-sqr 37.8%

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

      11. rem-square-sqrt 90.4%

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

      12. cube-prod 90.4%

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

      13. rem-cube-cbrt 90.4%

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

      14. *-commutative 90.4%

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

      15. associate-*r* 90.4%

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

   7. Simplified 90.4%

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

   8. Taylor expanded in x around 0 90.4%

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

      1. associate-*l* 90.4%

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

      2. +-commutative 90.4%

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

      3. fma-define 90.4%

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

   10. Simplified 90.4%

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

       1. add-sqr-sqrt 37.5%

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

       2. fabs-sqr 37.5%

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

       3. add-sqr-sqrt 39.2%

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

       4. *-commutative 39.2%

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

       5. *-commutative 39.2%

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

       6. inv-pow 39.2%

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

       7. sqrt-pow1 39.2%

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

       8. metadata-eval 39.2%

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

   12. Applied egg-rr 39.2%

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

       1. fma-undefine 39.2%

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

   14. Applied egg-rr 39.2%

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

   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 99.8%

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

   5. Taylor expanded in x around inf 32.4%

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

      1. *-commutative 32.4%

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

      2. rem-square-sqrt 2.1%

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

      3. fabs-sqr 2.1%

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

      4. rem-square-sqrt 32.4%

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

      5. pow-plus 32.4%

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

      6. rem-square-sqrt 2.1%

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

      7. fabs-sqr 2.1%

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

      8. rem-square-sqrt 32.4%

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

      9. metadata-eval 32.4%

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

   7. Simplified 32.4%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;x \cdot \left({\pi}^{-0.5} \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{7}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.8% accurate, 6.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\_m \cdot {\pi}^{-0.5}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (* 2.0 (log1p (expm1 (* x_m (pow PI -0.5))))))

C

x_m = fabs(x);
double code(double x_m) {
	return 2.0 * log1p(expm1((x_m * pow(((double) M_PI), -0.5))));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 2.0 * Math.log1p(Math.expm1((x_m * Math.pow(Math.PI, -0.5))));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 2.0 * math.log1p(math.expm1((x_m * math.pow(math.pi, -0.5))))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(2.0 * log1p(expm1(Float64(x_m * (pi ^ -0.5)))))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(2.0 * N[Log[1 + N[(Exp[N[(x$95$m * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 71.3%

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

   1. *-commutative 71.3%

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

   2. associate-*l* 71.3%

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

   3. *-commutative 71.3%

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

   4. rem-square-sqrt 37.3%

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

   5. fabs-sqr 37.3%

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

   6. rem-square-sqrt 71.3%

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

7. Simplified 71.3%

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

8. Taylor expanded in x around 0 71.3%

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

   1. *-commutative 71.3%

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

   2. associate-*r* 71.3%

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

   3. unpow-1 71.3%

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

   4. metadata-eval 71.3%

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

   5. pow-sqr 71.3%

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

   6. rem-sqrt-square 71.3%

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

   7. rem-square-sqrt 71.3%

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

   8. fabs-sqr 71.3%

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

   9. rem-square-sqrt 71.3%

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

   10. *-commutative 71.3%

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

   11. rem-square-sqrt 37.2%

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

   12. fabs-sqr 37.2%

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

   13. rem-square-sqrt 39.0%

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

10. Simplified 39.0%

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

    1. log1p-expm1-u 38.9%

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

12. Applied egg-rr 38.9%

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

Alternative 8: 88.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.75:\\ \;\;\;\;2 \cdot \left(x\_m \cdot {\pi}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x\_m}^{3}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.75)
   (* 2.0 (* x_m (pow PI -0.5)))
   (* 0.6666666666666666 (* (sqrt (/ 1.0 PI)) (pow x_m 3.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.75) {
		tmp = 2.0 * (x_m * pow(((double) M_PI), -0.5));
	} else {
		tmp = 0.6666666666666666 * (sqrt((1.0 / ((double) M_PI))) * pow(x_m, 3.0));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.75) {
		tmp = 2.0 * (x_m * Math.pow(Math.PI, -0.5));
	} else {
		tmp = 0.6666666666666666 * (Math.sqrt((1.0 / Math.PI)) * Math.pow(x_m, 3.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.75:
		tmp = 2.0 * (x_m * math.pow(math.pi, -0.5))
	else:
		tmp = 0.6666666666666666 * (math.sqrt((1.0 / math.pi)) * math.pow(x_m, 3.0))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.75)
		tmp = Float64(2.0 * Float64(x_m * (pi ^ -0.5)));
	else
		tmp = Float64(0.6666666666666666 * Float64(sqrt(Float64(1.0 / pi)) * (x_m ^ 3.0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.75)
		tmp = 2.0 * (x_m * (pi ^ -0.5));
	else
		tmp = 0.6666666666666666 * (sqrt((1.0 / pi)) * (x_m ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.75], N[(2.0 * N[(x$95$m * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.6666666666666666 * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[Power[x$95$m, 3.0], $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 99.8%

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

   5. Taylor expanded in x around 0 71.3%

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

      1. *-commutative 71.3%

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

      2. associate-*l* 71.3%

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

      3. *-commutative 71.3%

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

      4. rem-square-sqrt 37.3%

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

      5. fabs-sqr 37.3%

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

      6. rem-square-sqrt 71.3%

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

   7. Simplified 71.3%

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

   8. Taylor expanded in x around 0 71.3%

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

      1. *-commutative 71.3%

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

      2. associate-*r* 71.3%

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

      3. unpow-1 71.3%

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

      4. metadata-eval 71.3%

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

      5. pow-sqr 71.3%

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

      6. rem-sqrt-square 71.3%

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

      7. rem-square-sqrt 71.3%

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

      8. fabs-sqr 71.3%

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

      9. rem-square-sqrt 71.3%

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

      10. *-commutative 71.3%

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

      11. rem-square-sqrt 37.2%

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

      12. fabs-sqr 37.2%

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

      13. rem-square-sqrt 39.0%

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

   10. Simplified 39.0%

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

   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 99.8%

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

   5. Taylor expanded in x around 0 90.4%

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

      1. associate-*r* 90.4%

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

      2. rem-square-sqrt 37.8%

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

      3. fabs-sqr 37.8%

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

      4. rem-square-sqrt 89.8%

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

      5. pow-plus 89.8%

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

      6. metadata-eval 89.8%

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

      7. rem-cube-cbrt 89.8%

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

      8. cube-prod 89.8%

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

      9. rem-square-sqrt 37.8%

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

      10. fabs-sqr 37.8%

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

      11. rem-square-sqrt 90.4%

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

      12. cube-prod 90.4%

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

      13. rem-cube-cbrt 90.4%

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

      14. *-commutative 90.4%

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

      15. associate-*r* 90.4%

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

   7. Simplified 90.4%

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

   8. Taylor expanded in x around 0 90.4%

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

      1. associate-*l* 90.4%

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

      2. +-commutative 90.4%

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

      3. fma-define 90.4%

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

   10. Simplified 90.4%

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

       1. add-sqr-sqrt 37.5%

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

       2. fabs-sqr 37.5%

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

       3. add-sqr-sqrt 39.2%

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

       4. *-commutative 39.2%

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

       5. *-commutative 39.2%

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

       6. inv-pow 39.2%

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

       7. sqrt-pow1 39.2%

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

       8. metadata-eval 39.2%

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

   12. Applied egg-rr 39.2%

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

   13. Taylor expanded in x around inf 4.0%

       \[\leadsto \color{blue}{0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.2% accurate, 8.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x\_m \cdot \left({\pi}^{-0.5} \cdot \left(2 + 0.6666666666666666 \cdot {x\_m}^{2}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (* x_m (* (pow PI -0.5) (+ 2.0 (* 0.6666666666666666 (pow x_m 2.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	return x_m * (pow(((double) M_PI), -0.5) * (2.0 + (0.6666666666666666 * pow(x_m, 2.0))));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return x_m * (Math.pow(Math.PI, -0.5) * (2.0 + (0.6666666666666666 * Math.pow(x_m, 2.0))));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return x_m * (math.pow(math.pi, -0.5) * (2.0 + (0.6666666666666666 * math.pow(x_m, 2.0))))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(x_m * Float64((pi ^ -0.5) * Float64(2.0 + Float64(0.6666666666666666 * (x_m ^ 2.0)))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = x_m * ((pi ^ -0.5) * (2.0 + (0.6666666666666666 * (x_m ^ 2.0))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(2.0 + N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $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 99.8%

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

5. Taylor expanded in x around 0 90.4%

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

   1. associate-*r* 90.4%

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

   2. rem-square-sqrt 37.8%

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

   3. fabs-sqr 37.8%

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

   4. rem-square-sqrt 89.8%

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

   5. pow-plus 89.8%

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

   6. metadata-eval 89.8%

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

   7. rem-cube-cbrt 89.8%

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

   8. cube-prod 89.8%

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

   9. rem-square-sqrt 37.8%

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

   10. fabs-sqr 37.8%

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

   11. rem-square-sqrt 90.4%

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

   12. cube-prod 90.4%

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

   13. rem-cube-cbrt 90.4%

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

   14. *-commutative 90.4%

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

   15. associate-*r* 90.4%

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

7. Simplified 90.4%

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

8. Taylor expanded in x around 0 90.4%

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

   1. associate-*l* 90.4%

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

   2. +-commutative 90.4%

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

   3. fma-define 90.4%

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

10. Simplified 90.4%

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

    1. add-sqr-sqrt 37.5%

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

    2. fabs-sqr 37.5%

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

    3. add-sqr-sqrt 39.2%

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

    4. *-commutative 39.2%

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

    5. *-commutative 39.2%

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

    6. inv-pow 39.2%

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

    7. sqrt-pow1 39.2%

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

    8. metadata-eval 39.2%

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

12. Applied egg-rr 39.2%

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

    1. fma-undefine 39.2%

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

14. Applied egg-rr 39.2%

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

15. Final simplification 39.2%

    \[\leadsto x \cdot \left({\pi}^{-0.5} \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right) \]
16. Add Preprocessing

Alternative 10: 67.7% accurate, 17.4× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* 2.0 (* x_m (pow PI -0.5))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(2.0 * N[(x$95$m * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 71.3%

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

   1. *-commutative 71.3%

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

   2. associate-*l* 71.3%

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

   3. *-commutative 71.3%

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

   4. rem-square-sqrt 37.3%

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

   5. fabs-sqr 37.3%

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

   6. rem-square-sqrt 71.3%

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

7. Simplified 71.3%

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

8. Taylor expanded in x around 0 71.3%

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

   1. *-commutative 71.3%

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

   2. associate-*r* 71.3%

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

   3. unpow-1 71.3%

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

   4. metadata-eval 71.3%

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

   5. pow-sqr 71.3%

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

   6. rem-sqrt-square 71.3%

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

   7. rem-square-sqrt 71.3%

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

   8. fabs-sqr 71.3%

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

   9. rem-square-sqrt 71.3%

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

   10. *-commutative 71.3%

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

   11. rem-square-sqrt 37.2%

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

   12. fabs-sqr 37.2%

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

   13. rem-square-sqrt 39.0%

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

10. Simplified 39.0%

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

Alternative 11: 4.1% accurate, 1849.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.0;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 0.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.0

Julia

x_m = abs(x)
function code(x_m)
	return 0.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 0.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.0

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 71.3%

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

   1. *-commutative 71.3%

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

   2. associate-*l* 71.3%

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

   3. *-commutative 71.3%

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

   4. rem-square-sqrt 37.3%

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

   5. fabs-sqr 37.3%

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

   6. rem-square-sqrt 71.3%

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

7. Simplified 71.3%

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

8. Taylor expanded in x around 0 71.3%

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

   1. *-commutative 71.3%

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

   2. associate-*r* 71.3%

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

   3. unpow-1 71.3%

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

   4. metadata-eval 71.3%

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

   5. pow-sqr 71.3%

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

   6. rem-sqrt-square 71.3%

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

   7. rem-square-sqrt 71.3%

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

   8. fabs-sqr 71.3%

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

   9. rem-square-sqrt 71.3%

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

   10. *-commutative 71.3%

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

   11. rem-square-sqrt 37.2%

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

   12. fabs-sqr 37.2%

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

   13. rem-square-sqrt 39.0%

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

10. Simplified 39.0%

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

    1. expm1-log1p-u 38.9%

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

    2. expm1-undefine 4.6%

       \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)} - 1\right)} \]

12. Applied egg-rr 4.6%

    \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)} - 1\right)} \]
13. Step-by-step derivation

    1. sub-neg 4.6%

       \[\leadsto 2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)} + \left(-1\right)\right)} \]

    2. metadata-eval 4.6%

       \[\leadsto 2 \cdot \left(e^{\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)} + \color{blue}{-1}\right) \]

    3. +-commutative 4.6%

       \[\leadsto 2 \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)}\right)} \]

    4. log1p-undefine 4.6%

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

    5. rem-exp-log 4.7%

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

    6. +-commutative 4.7%

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

    7. fma-define 4.7%

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

14. Simplified 4.7%

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

15. Taylor expanded in x around 0 4.3%

    \[\leadsto 2 \cdot \left(-1 + \color{blue}{1}\right) \]

16. Final simplification 4.3%

    \[\leadsto 0 \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024137 
(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)))))))