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

Percentage Accurate: 99.9% → 99.9%

Time: 7.3s

Alternatives: 7

Speedup: 3.7×

• 28.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

   1. div-inv 99.9%

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

   2. add-sqr-sqrt 37.6%

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

   3. fabs-sqr 37.6%

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

   4. add-sqr-sqrt 99.9%

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

   5. *-commutative 99.9%

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

   6. pow1/2 99.9%

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

   7. pow-flip 99.9%

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

   8. metadata-eval 99.9%

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

5. Applied egg-rr 99.9%

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

6. Taylor expanded in x around 0 99.9%

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

   1. fma-udef 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 99.4% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x):
	return math.fabs(((((0.6666666666666666 * (x * x)) + 2.0) + ((0.2 * math.pow(x, 4.0)) + (0.047619047619047616 * math.pow(x, 6.0)))) * (x / math.sqrt(math.pi))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

   1. div-inv 99.9%

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

   2. add-sqr-sqrt 37.6%

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

   3. fabs-sqr 37.6%

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

   4. add-sqr-sqrt 99.9%

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

   5. *-commutative 99.9%

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

   6. pow1/2 99.9%

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

   7. pow-flip 99.9%

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

   8. metadata-eval 99.9%

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

5. Applied egg-rr 99.9%

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

6. Taylor expanded in x around 0 99.9%

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

   1. fma-udef 99.9%

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

8. Applied egg-rr 99.9%

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

   1. *-commutative 99.9%

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

   2. metadata-eval 99.9%

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

   3. pow-flip 99.9%

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

   4. pow1/2 99.9%

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

   5. div-inv 99.4%

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

   6. expm1-log1p-u 68.1%

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

   7. expm1-udef 6.8%

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

10. Applied egg-rr 6.8%

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

    1. expm1-def 68.1%

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

    2. expm1-log1p 99.4%

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

12. Simplified 99.4%

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

13. Final simplification 99.4%

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

Alternative 3: 93.5% accurate, 3.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 1.0)
   (fabs (* x (/ 2.0 (sqrt PI))))
   (fabs (* 0.2 (* (pow x 5.0) (sqrt (/ 1.0 PI)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1

   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%

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

   4. Taylor expanded in x around 0 99.8%

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

      1. unpow2 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-*r* 99.8%

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

      4. rem-square-sqrt 55.1%

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

      5. fabs-sqr 55.1%

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

      6. rem-square-sqrt 98.0%

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

      7. associate-*r* 98.0%

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

      8. unpow3 98.0%

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

   6. Simplified 98.0%

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

   7. Taylor expanded in x around 0 97.8%

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

      1. associate-*r* 97.8%

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

      2. *-commutative 97.8%

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

   9. Simplified 97.8%

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

       1. expm1-log1p-u 97.8%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot \left|x\right|\right)\right)\right)}\right| \]

       2. expm1-udef 8.2%

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

       3. associate-*r* 8.2%

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

       4. add-sqr-sqrt 4.2%

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

       5. fabs-sqr 4.2%

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

       6. add-sqr-sqrt 8.2%

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

       7. *-commutative 8.2%

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

       8. sqrt-div 8.2%

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

       9. metadata-eval 8.2%

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

       10. associate-*l/ 8.2%

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

       11. metadata-eval 8.2%

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

   11. Applied egg-rr 8.2%

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

       1. expm1-def 97.8%

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

       2. expm1-log1p 97.8%

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

   13. Simplified 97.8%

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

   if 1 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 100.0%

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

      1. add-log-exp 90.6%

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

      2. *-un-lft-identity 90.6%

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

      3. log-prod 90.6%

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

      4. metadata-eval 90.6%

         \[\leadsto \left|\left(\color{blue}{0} + \log \left(e^{\frac{\left|x\right|}{\sqrt{\pi}}}\right)\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

      5. add-log-exp 100.0%

         \[\leadsto \left|\left(0 + \color{blue}{\frac{\left|x\right|}{\sqrt{\pi}}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

      6. add-sqr-sqrt 0.0%

         \[\leadsto \left|\left(0 + \frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\sqrt{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

      7. fabs-sqr 0.0%

         \[\leadsto \left|\left(0 + \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

      8. add-sqr-sqrt 100.0%

         \[\leadsto \left|\left(0 + \frac{\color{blue}{x}}{\sqrt{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   5. Applied egg-rr 100.0%

      \[\leadsto \left|\color{blue}{\left(0 + \frac{x}{\sqrt{\pi}}\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]
   6. Step-by-step derivation

      1. +-lft-identity 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 81.6%

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

   9. Taylor expanded in x around 0 81.6%

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

   10. Taylor expanded in x around inf 81.6%

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

4. Final simplification 92.8%

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

Alternative 4: 98.9% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(x)
	return abs(Float64(Float64((pi ^ -0.5) * x) * Float64(2.0 + Float64(0.047619047619047616 * (x ^ 6.0)))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around 0 98.0%

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

5. Taylor expanded in x around inf 97.9%

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

   1. div-inv 99.9%

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

   2. add-sqr-sqrt 37.6%

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

   3. fabs-sqr 37.6%

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

   4. add-sqr-sqrt 99.9%

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

   5. *-commutative 99.9%

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

   6. pow1/2 99.9%

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

   7. pow-flip 99.9%

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

   8. metadata-eval 99.9%

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

7. Applied egg-rr 98.3%

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

8. Final simplification 98.3%

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

Alternative 5: 98.4% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around 0 98.0%

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

5. Taylor expanded in x around inf 97.9%

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

   1. add-log-exp 35.0%

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

   2. *-un-lft-identity 35.0%

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

   3. log-prod 35.0%

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

   4. metadata-eval 35.0%

      \[\leadsto \left|\left(\color{blue}{0} + \log \left(e^{\frac{\left|x\right|}{\sqrt{\pi}}}\right)\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   5. add-log-exp 99.4%

      \[\leadsto \left|\left(0 + \color{blue}{\frac{\left|x\right|}{\sqrt{\pi}}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   6. add-sqr-sqrt 37.5%

      \[\leadsto \left|\left(0 + \frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\sqrt{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   7. fabs-sqr 37.5%

      \[\leadsto \left|\left(0 + \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   8. add-sqr-sqrt 99.4%

      \[\leadsto \left|\left(0 + \frac{\color{blue}{x}}{\sqrt{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

7. Applied egg-rr 97.9%

   \[\leadsto \left|\color{blue}{\left(0 + \frac{x}{\sqrt{\pi}}\right)} \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]
8. Step-by-step derivation

   1. +-lft-identity 99.4%

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

9. Simplified 97.9%

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

10. Final simplification 97.9%

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

Alternative 6: 83.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-49}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\left(x \cdot x\right) \cdot \frac{4}{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 2e-49)
   (fabs (* x (/ 2.0 (sqrt PI))))
   (fabs (sqrt (* (* x x) (/ 4.0 PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 2e-49) {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs(sqrt(((x * x) * (4.0 / ((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 2e-49) {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs(Math.sqrt(((x * x) * (4.0 / Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 2e-49:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs(math.sqrt(((x * x) * (4.0 / math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 2e-49)
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(sqrt(Float64(Float64(x * x) * Float64(4.0 / pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 2e-49)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = abs(sqrt(((x * x) * (4.0 / pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 2e-49], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[Sqrt[N[(N[(x * x), $MachinePrecision] * N[(4.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.99999999999999987e-49

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. unpow2 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. rem-square-sqrt 57.5%

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

      5. fabs-sqr 57.5%

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

      6. rem-square-sqrt 99.9%

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

      7. associate-*r* 99.9%

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

      8. unpow3 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around 0 99.9%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

   9. Simplified 99.9%

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

       1. expm1-log1p-u 99.9%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot \left|x\right|\right)\right)\right)}\right| \]

       2. expm1-udef 5.9%

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

       3. associate-*r* 5.9%

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

       4. add-sqr-sqrt 3.2%

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

       5. fabs-sqr 3.2%

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

       6. add-sqr-sqrt 5.9%

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

       7. *-commutative 5.9%

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

       8. sqrt-div 5.9%

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

       9. metadata-eval 5.9%

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

       10. associate-*l/ 5.9%

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

       11. metadata-eval 5.9%

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

   11. Applied egg-rr 5.9%

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

       1. expm1-def 99.9%

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

       2. expm1-log1p 99.9%

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

   13. Simplified 99.9%

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

   if 1.99999999999999987e-49 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. unpow2 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. rem-square-sqrt 11.8%

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

      5. fabs-sqr 11.8%

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

      6. rem-square-sqrt 49.6%

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

      7. associate-*r* 49.6%

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

      8. unpow3 49.6%

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

   6. Simplified 49.6%

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

   7. Taylor expanded in x around 0 28.1%

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

      1. associate-*r* 28.1%

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

      2. *-commutative 28.1%

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

   9. Simplified 28.1%

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

       1. expm1-log1p-u 28.1%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot \left|x\right|\right)\right)\right)}\right| \]

       2. expm1-udef 9.5%

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

       3. associate-*r* 9.5%

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

       4. add-sqr-sqrt 2.5%

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

       5. fabs-sqr 2.5%

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

       6. add-sqr-sqrt 5.4%

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

       7. *-commutative 5.4%

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

       8. sqrt-div 5.4%

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

       9. metadata-eval 5.4%

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

       10. associate-*l/ 5.4%

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

       11. metadata-eval 5.4%

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

   11. Applied egg-rr 5.4%

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

       1. expm1-def 24.0%

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

       2. expm1-log1p 28.1%

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

   13. Simplified 28.1%

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

       1. add-sqr-sqrt 11.4%

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

       2. sqrt-unprod 64.5%

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

       3. swap-sqr 64.4%

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

       4. frac-times 64.3%

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

       5. metadata-eval 64.3%

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

       6. add-sqr-sqrt 64.5%

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

   15. Applied egg-rr 64.5%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-49}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\left(x \cdot x\right) \cdot \frac{4}{\pi}}\right|\\ \end{array} \]

Alternative 7: 67.9% accurate, 6.4× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (fabs (* x (/ 2.0 (sqrt PI)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 99.9%

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

   1. unpow2 99.9%

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

   2. *-commutative 99.9%

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

   3. associate-*r* 99.9%

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

   4. rem-square-sqrt 37.9%

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

   5. fabs-sqr 37.9%

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

   6. rem-square-sqrt 78.3%

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

   7. associate-*r* 78.3%

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

   8. unpow3 78.3%

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

6. Simplified 78.3%

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

7. Taylor expanded in x around 0 69.0%

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

   1. associate-*r* 69.0%

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

   2. *-commutative 69.0%

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

9. Simplified 69.0%

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

    1. expm1-log1p-u 69.0%

       \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot \left|x\right|\right)\right)\right)}\right| \]

    2. expm1-udef 7.4%

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

    3. associate-*r* 7.4%

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

    4. add-sqr-sqrt 2.9%

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

    5. fabs-sqr 2.9%

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

    6. add-sqr-sqrt 5.6%

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

    7. *-commutative 5.6%

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

    8. sqrt-div 5.6%

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

    9. metadata-eval 5.6%

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

    10. associate-*l/ 5.6%

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

    11. metadata-eval 5.6%

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

11. Applied egg-rr 5.6%

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

    1. expm1-def 67.3%

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

    2. expm1-log1p 69.0%

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

13. Simplified 69.0%

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

14. Final simplification 69.0%

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

Reproduce

herbie shell --seed 2023199 
(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)))))))