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

Percentage Accurate: 99.8% → 99.9%

Time: 11.0s

Alternatives: 9

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. inv-pow 39.2%

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

    3. sqrt-pow1 39.2%

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

    4. 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: 99.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(x\_m \cdot 2 + 0.6666666666666666 \cdot {x\_m}^{3}\right) + 0.2 \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left|x\_m\right|\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left(\left(x\_m \cdot x\_m\right) \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right)\right)\right)\right| \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt PI))
   (+
    (+
     (+ (* x_m 2.0) (* 0.6666666666666666 (pow x_m 3.0)))
     (* 0.2 (* (* x_m x_m) (* (* x_m x_m) (fabs x_m)))))
    (*
     0.047619047619047616
     (* (* x_m x_m) (* (* x_m x_m) (* x_m (* x_m x_m)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((x_m * 2.0) + (0.6666666666666666 * pow(x_m, 3.0))) + (0.2 * ((x_m * x_m) * ((x_m * x_m) * fabs(x_m))))) + (0.047619047619047616 * ((x_m * x_m) * ((x_m * x_m) * (x_m * (x_m * 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) + (0.6666666666666666 * Math.pow(x_m, 3.0))) + (0.2 * ((x_m * x_m) * ((x_m * x_m) * Math.abs(x_m))))) + (0.047619047619047616 * ((x_m * x_m) * ((x_m * x_m) * (x_m * (x_m * 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) + (0.6666666666666666 * math.pow(x_m, 3.0))) + (0.2 * ((x_m * x_m) * ((x_m * x_m) * math.fabs(x_m))))) + (0.047619047619047616 * ((x_m * x_m) * ((x_m * x_m) * (x_m * (x_m * x_m))))))))

Julia

x_m = abs(x)
function code(x_m)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(x_m * 2.0) + Float64(0.6666666666666666 * (x_m ^ 3.0))) + Float64(0.2 * Float64(Float64(x_m * x_m) * Float64(Float64(x_m * x_m) * abs(x_m))))) + Float64(0.047619047619047616 * Float64(Float64(x_m * x_m) * Float64(Float64(x_m * x_m) * Float64(x_m * Float64(x_m * x_m))))))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = abs(((1.0 / sqrt(pi)) * ((((x_m * 2.0) + (0.6666666666666666 * (x_m ^ 3.0))) + (0.2 * ((x_m * x_m) * ((x_m * x_m) * abs(x_m))))) + (0.047619047619047616 * ((x_m * x_m) * ((x_m * x_m) * (x_m * (x_m * 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[(N[(N[(x$95$m * 2.0), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2 * 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] + N[(0.047619047619047616 * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(x$95$m * N[(x$95$m * 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. Step-by-step derivation

   1. metadata-eval 99.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, \color{blue}{\frac{2}{3}} \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   2. *-commutative 99.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, \frac{2}{3} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left|x\right|\right)}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   3. sqr-abs 99.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, \frac{2}{3} \cdot \left(\color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)} \cdot \left|x\right|\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   4. fma-define 99.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\color{blue}{\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)} + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   5. *-commutative 99.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(\color{blue}{\left|x\right| \cdot 2} + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   6. add-sqr-sqrt 99.3%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(\color{blue}{\left(\sqrt{\left|x\right|} \cdot \sqrt{\left|x\right|}\right)} \cdot 2 + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   7. sqrt-prod 63.9%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(\color{blue}{\sqrt{\left|x\right| \cdot \left|x\right|}} \cdot 2 + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   8. sqr-abs 63.9%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(\sqrt{\color{blue}{x \cdot x}} \cdot 2 + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   9. pow2 63.9%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(\sqrt{\color{blue}{{x}^{2}}} \cdot 2 + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   10. sqrt-pow1 98.0%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(\color{blue}{{x}^{\left(\frac{2}{2}\right)}} \cdot 2 + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   11. metadata-eval 98.0%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left({x}^{\color{blue}{1}} \cdot 2 + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   12. pow1 98.0%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(\color{blue}{x} \cdot 2 + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   13. metadata-eval 98.0%

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

   14. pow3 98.0%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(x \cdot 2 + 0.6666666666666666 \cdot \color{blue}{{\left(\left|x\right|\right)}^{3}}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   15. add-sqr-sqrt 98.0%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(x \cdot 2 + 0.6666666666666666 \cdot {\color{blue}{\left(\sqrt{\left|x\right|} \cdot \sqrt{\left|x\right|}\right)}}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   16. sqrt-prod 98.0%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(x \cdot 2 + 0.6666666666666666 \cdot {\color{blue}{\left(\sqrt{\left|x\right| \cdot \left|x\right|}\right)}}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   17. sqr-abs 98.0%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(x \cdot 2 + 0.6666666666666666 \cdot {\left(\sqrt{\color{blue}{x \cdot x}}\right)}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   18. pow2 98.0%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(x \cdot 2 + 0.6666666666666666 \cdot {\left(\sqrt{\color{blue}{{x}^{2}}}\right)}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   19. sqrt-pow1 78.9%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(x \cdot 2 + 0.6666666666666666 \cdot {\color{blue}{\left({x}^{\left(\frac{2}{2}\right)}\right)}}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   20. metadata-eval 78.9%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(x \cdot 2 + 0.6666666666666666 \cdot {\left({x}^{\color{blue}{1}}\right)}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

   21. pow1 78.9%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(x \cdot 2 + 0.6666666666666666 \cdot {\color{blue}{x}}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

6. Applied egg-rr 78.9%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\color{blue}{\left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)} + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]
7. 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%

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

8. Applied egg-rr 74.9%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{\left(1 \cdot x\right)} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]
9. Step-by-step derivation

   1. *-lft-identity 39.2%

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

10. Simplified 74.9%

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\color{blue}{x} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]

11. Final simplification 74.9%

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right) + 0.2 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left|x\right|\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right| \]
12. Add Preprocessing

Alternative 4: 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 5: 98.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \left|\left(x\_m \cdot {\pi}^{-0.5}\right) \cdot \left(2 + 0.047619047619047616 \cdot \left(\left|x\_m\right| \cdot {x\_m}^{5}\right)\right)\right| \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (fabs
  (*
   (* x_m (pow PI -0.5))
   (+ 2.0 (* 0.047619047619047616 (* (fabs x_m) (pow x_m 5.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	return fabs(((x_m * pow(((double) M_PI), -0.5)) * (2.0 + (0.047619047619047616 * (fabs(x_m) * pow(x_m, 5.0))))));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.abs(((x_m * Math.pow(Math.PI, -0.5)) * (2.0 + (0.047619047619047616 * (Math.abs(x_m) * Math.pow(x_m, 5.0))))));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.fabs(((x_m * math.pow(math.pi, -0.5)) * (2.0 + (0.047619047619047616 * (math.fabs(x_m) * math.pow(x_m, 5.0))))))

Julia

x_m = abs(x)
function code(x_m)
	return abs(Float64(Float64(x_m * (pi ^ -0.5)) * Float64(2.0 + Float64(0.047619047619047616 * Float64(abs(x_m) * (x_m ^ 5.0))))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = abs(((x_m * (pi ^ -0.5)) * (2.0 + (0.047619047619047616 * (abs(x_m) * (x_m ^ 5.0))))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Abs[N[(N[(x$95$m * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(0.047619047619047616 * N[(N[Abs[x$95$m], $MachinePrecision] * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

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

   4. *-commutative 97.7%

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

7. Simplified 97.7%

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

8. Taylor expanded in x around 0 97.7%

   \[\leadsto \left|\color{blue}{x \cdot \left(0.047619047619047616 \cdot \left(\left({x}^{5} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
9. Step-by-step derivation

   1. distribute-rgt-in 97.7%

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

   2. associate-*r* 97.7%

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

   3. associate-*l* 97.7%

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

   4. *-commutative 97.7%

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

   5. associate-*r* 97.7%

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

   6. *-commutative 97.7%

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

   7. distribute-rgt-out 97.7%

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

   8. +-commutative 97.7%

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

10. Simplified 97.7%

    \[\leadsto \left|\color{blue}{\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(2 + 0.047619047619047616 \cdot \left({x}^{5} \cdot \left|x\right|\right)\right)}\right| \]

11. Final simplification 97.7%

    \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(2 + 0.047619047619047616 \cdot \left(\left|x\right| \cdot {x}^{5}\right)\right)\right| \]
12. Add Preprocessing

Alternative 6: 98.8% accurate, 8.3× speedup

Math

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

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 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 * 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 * 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 * 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(x_m * Float64(x_m * 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 * 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[(x$95$m * N[(x$95$m * 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%

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

   4. *-commutative 97.7%

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

7. Simplified 97.7%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{x \cdot 2} + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]
8. 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%

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

9. Applied egg-rr 97.7%

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

    1. *-lft-identity 39.2%

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

11. Simplified 97.7%

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

12. Final simplification 97.7%

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

Alternative 7: 98.8% accurate, 8.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.85)
   (* x_m (/ 2.0 (sqrt PI)))
   (* (* 0.047619047619047616 (pow x_m 6.0)) (* x_m (pow PI -0.5)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.85], N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision] * N[(x$95$m * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 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. *-commutative 71.3%

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

   7. Simplified 71.3%

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

      1. add-cube-cbrt 69.7%

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

      2. pow3 69.7%

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

      3. sqrt-div 69.7%

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

      4. metadata-eval 69.7%

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

      5. associate-*r* 69.7%

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

      6. pow1/2 69.7%

         \[\leadsto {\left(\sqrt[3]{\left|\left(\frac{1}{\color{blue}{{\pi}^{0.5}}} \cdot x\right) \cdot 2\right|}\right)}^{3} \]

      7. pow-flip 69.7%

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

      8. metadata-eval 69.7%

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

   9. Applied egg-rr 69.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\left|\left({\pi}^{-0.5} \cdot x\right) \cdot 2\right|}\right)}^{3}} \]
   10. Step-by-step derivation

       1. rem-cube-cbrt 71.3%

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

       2. add-sqr-sqrt 37.2%

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

       3. fabs-sqr 37.2%

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

       4. add-sqr-sqrt 39.0%

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

       5. *-commutative 39.0%

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

       6. *-commutative 39.0%

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

       7. add-sqr-sqrt 37.3%

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

       8. fabs-sqr 37.3%

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

       9. add-sqr-sqrt 71.3%

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

       10. metadata-eval 71.3%

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

       11. pow-flip 71.3%

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

       12. pow1/2 71.3%

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

       13. div-inv 70.8%

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

       14. add-sqr-sqrt 37.2%

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

       15. fabs-sqr 37.2%

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

       16. add-sqr-sqrt 38.7%

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

   11. Applied egg-rr 38.7%

       \[\leadsto \color{blue}{2 \cdot \frac{x}{\sqrt{\pi}}} \]
   12. Step-by-step derivation

       1. *-commutative 38.7%

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

       2. associate-*l/ 38.7%

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

       3. associate-/l* 39.0%

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

   13. Simplified 39.0%

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

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 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%

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

      4. *-commutative 97.7%

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

   7. Simplified 97.7%

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

   8. Taylor expanded in x around inf 32.4%

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

      1. associate-*l* 32.4%

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

      2. unpow-1 32.4%

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

      3. metadata-eval 32.4%

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

      4. pow-sqr 32.4%

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

      5. rem-sqrt-square 32.4%

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

      6. fabs-mul 32.4%

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

   10. Simplified 32.4%

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

       1. add-sqr-sqrt 32.3%

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

       2. fabs-sqr 32.3%

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

       3. add-sqr-sqrt 32.4%

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

       4. associate-*r* 32.4%

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

       5. *-commutative 32.4%

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

       6. add-sqr-sqrt 2.1%

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

       7. fabs-sqr 2.1%

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

       8. add-sqr-sqrt 3.9%

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

       9. *-commutative 3.9%

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

   12. Applied egg-rr 3.9%

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

Alternative 8: 98.8% accurate, 8.7× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.85)
   (* x_m (/ 2.0 (sqrt PI)))
   (* (pow PI -0.5) (* 0.047619047619047616 (pow x_m 7.0)))))

C

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

Python

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

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.85)
		tmp = Float64(x_m * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64((pi ^ -0.5) * Float64(0.047619047619047616 * (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 <= 1.85)
		tmp = x_m * (2.0 / sqrt(pi));
	else
		tmp = (pi ^ -0.5) * (0.047619047619047616 * (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, 1.85], N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(0.047619047619047616 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 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. *-commutative 71.3%

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

   7. Simplified 71.3%

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

      1. add-cube-cbrt 69.7%

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

      2. pow3 69.7%

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

      3. sqrt-div 69.7%

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

      4. metadata-eval 69.7%

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

      5. associate-*r* 69.7%

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

      6. pow1/2 69.7%

         \[\leadsto {\left(\sqrt[3]{\left|\left(\frac{1}{\color{blue}{{\pi}^{0.5}}} \cdot x\right) \cdot 2\right|}\right)}^{3} \]

      7. pow-flip 69.7%

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

      8. metadata-eval 69.7%

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

   9. Applied egg-rr 69.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\left|\left({\pi}^{-0.5} \cdot x\right) \cdot 2\right|}\right)}^{3}} \]
   10. Step-by-step derivation

       1. rem-cube-cbrt 71.3%

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

       2. add-sqr-sqrt 37.2%

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

       3. fabs-sqr 37.2%

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

       4. add-sqr-sqrt 39.0%

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

       5. *-commutative 39.0%

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

       6. *-commutative 39.0%

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

       7. add-sqr-sqrt 37.3%

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

       8. fabs-sqr 37.3%

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

       9. add-sqr-sqrt 71.3%

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

       10. metadata-eval 71.3%

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

       11. pow-flip 71.3%

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

       12. pow1/2 71.3%

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

       13. div-inv 70.8%

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

       14. add-sqr-sqrt 37.2%

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

       15. fabs-sqr 37.2%

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

       16. add-sqr-sqrt 38.7%

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

   11. Applied egg-rr 38.7%

       \[\leadsto \color{blue}{2 \cdot \frac{x}{\sqrt{\pi}}} \]
   12. Step-by-step derivation

       1. *-commutative 38.7%

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

       2. associate-*l/ 38.7%

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

       3. associate-/l* 39.0%

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

   13. Simplified 39.0%

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

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 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. associate-*r* 32.4%

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

      2. *-commutative 32.4%

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

      3. *-commutative 32.4%

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

      4. *-commutative 32.4%

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

      5. rem-square-sqrt 2.1%

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

      6. fabs-sqr 2.1%

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

      7. rem-square-sqrt 32.4%

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

      8. pow-plus 32.4%

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

      9. rem-square-sqrt 2.1%

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

      10. fabs-sqr 2.1%

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

      11. rem-square-sqrt 32.4%

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

      12. metadata-eval 32.4%

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

   7. Simplified 32.4%

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

      1. add-sqr-sqrt 3.8%

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

      2. fabs-sqr 3.8%

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

      3. add-sqr-sqrt 3.9%

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

      4. *-commutative 3.9%

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

      5. inv-pow 3.9%

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

      6. sqrt-pow1 3.9%

         \[\leadsto \left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \]

      7. metadata-eval 3.9%

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

   9. Applied egg-rr 3.9%

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

4. Final simplification 39.0%

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

Alternative 9: 67.7% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 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. *-commutative 71.3%

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

7. Simplified 71.3%

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

   1. add-cube-cbrt 69.7%

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

   2. pow3 69.7%

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

   3. sqrt-div 69.7%

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

   4. metadata-eval 69.7%

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

   5. associate-*r* 69.7%

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

   6. pow1/2 69.7%

      \[\leadsto {\left(\sqrt[3]{\left|\left(\frac{1}{\color{blue}{{\pi}^{0.5}}} \cdot x\right) \cdot 2\right|}\right)}^{3} \]

   7. pow-flip 69.7%

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

   8. metadata-eval 69.7%

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

9. Applied egg-rr 69.7%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\left|\left({\pi}^{-0.5} \cdot x\right) \cdot 2\right|}\right)}^{3}} \]
10. Step-by-step derivation

    1. rem-cube-cbrt 71.3%

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

    2. add-sqr-sqrt 37.2%

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

    3. fabs-sqr 37.2%

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

    4. add-sqr-sqrt 39.0%

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

    5. *-commutative 39.0%

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

    6. *-commutative 39.0%

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

    7. add-sqr-sqrt 37.3%

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

    8. fabs-sqr 37.3%

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

    9. add-sqr-sqrt 71.3%

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

    10. metadata-eval 71.3%

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

    11. pow-flip 71.3%

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

    12. pow1/2 71.3%

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

    13. div-inv 70.8%

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

    14. add-sqr-sqrt 37.2%

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

    15. fabs-sqr 37.2%

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

    16. add-sqr-sqrt 38.7%

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

11. Applied egg-rr 38.7%

    \[\leadsto \color{blue}{2 \cdot \frac{x}{\sqrt{\pi}}} \]
12. Step-by-step derivation

    1. *-commutative 38.7%

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

    2. associate-*l/ 38.7%

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

    3. associate-/l* 39.0%

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

13. Simplified 39.0%

    \[\leadsto \color{blue}{x \cdot \frac{2}{\sqrt{\pi}}} \]
14. 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)))))))