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

Percentage Accurate: 99.8% → 99.8%

Time: 9.3s

Alternatives: 11

Speedup: 3.7×

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

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + \left(\left(e^{\mathsf{log1p}\left(0.2 \cdot {x}^{4}\right)} + -1\right) + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (* x (pow PI -0.5))
   (+
    (+ 2.0 (* 0.6666666666666666 (* x x)))
    (+
     (+ (exp (log1p (* 0.2 (pow x 4.0)))) -1.0)
     (* 0.047619047619047616 (pow x 6.0)))))))

C

double code(double x) {
	return fabs(((x * pow(((double) M_PI), -0.5)) * ((2.0 + (0.6666666666666666 * (x * x))) + ((exp(log1p((0.2 * pow(x, 4.0)))) + -1.0) + (0.047619047619047616 * pow(x, 6.0))))));
}

Java

public static double code(double x) {
	return Math.abs(((x * Math.pow(Math.PI, -0.5)) * ((2.0 + (0.6666666666666666 * (x * x))) + ((Math.exp(Math.log1p((0.2 * Math.pow(x, 4.0)))) + -1.0) + (0.047619047619047616 * Math.pow(x, 6.0))))));
}

Python

def code(x):
	return math.fabs(((x * math.pow(math.pi, -0.5)) * ((2.0 + (0.6666666666666666 * (x * x))) + ((math.exp(math.log1p((0.2 * math.pow(x, 4.0)))) + -1.0) + (0.047619047619047616 * math.pow(x, 6.0))))))

Julia

function code(x)
	return abs(Float64(Float64(x * (pi ^ -0.5)) * Float64(Float64(2.0 + Float64(0.6666666666666666 * Float64(x * x))) + Float64(Float64(exp(log1p(Float64(0.2 * (x ^ 4.0)))) + -1.0) + Float64(0.047619047619047616 * (x ^ 6.0))))))
end

Wolfram

code[x_] := N[Abs[N[(N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Exp[N[Log[1 + N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $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 99.5%

   \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|} \]
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. rem-sqrt-square 69.7%

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

   3. sqrt-prod 31.8%

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

   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. pow1/2 99.9%

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

   6. pow-flip 99.9%

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

   7. metadata-eval 99.9%

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

5. Applied egg-rr 99.9%

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

6. Taylor expanded in x around 0 99.9%

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

   1. metadata-eval 95.1%

      \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{2}{3}}, x \cdot x, 2\right) + 0.2 \cdot {x}^{4}\right)\right| \]

   2. fma-udef 95.1%

      \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\color{blue}{\left(\frac{2}{3} \cdot \left(x \cdot x\right) + 2\right)} + 0.2 \cdot {x}^{4}\right)\right| \]

   3. metadata-eval 95.1%

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

8. Applied egg-rr 99.9%

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

   1. expm1-log1p-u 99.8%

      \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right) + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.2 \cdot {x}^{4}\right)\right)} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   2. expm1-udef 99.9%

      \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right) + \left(\color{blue}{\left(e^{\mathsf{log1p}\left(0.2 \cdot {x}^{4}\right)} - 1\right)} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

10. Applied egg-rr 99.9%

    \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right) + \left(\color{blue}{\left(e^{\mathsf{log1p}\left(0.2 \cdot {x}^{4}\right)} - 1\right)} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

11. Final simplification 99.9%

    \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + \left(\left(e^{\mathsf{log1p}\left(0.2 \cdot {x}^{4}\right)} + -1\right) + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

Alternative 3: 99.8% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $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 99.5%

   \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|} \]
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. rem-sqrt-square 69.7%

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

   3. sqrt-prod 31.8%

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

   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. pow1/2 99.9%

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

   6. pow-flip 99.9%

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

   7. metadata-eval 99.9%

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

5. Applied egg-rr 99.9%

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

6. Taylor expanded in x around 0 99.9%

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

   1. metadata-eval 95.1%

      \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{2}{3}}, x \cdot x, 2\right) + 0.2 \cdot {x}^{4}\right)\right| \]

   2. fma-udef 95.1%

      \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\color{blue}{\left(\frac{2}{3} \cdot \left(x \cdot x\right) + 2\right)} + 0.2 \cdot {x}^{4}\right)\right| \]

   3. metadata-eval 95.1%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 4: 99.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[Abs[N[(N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $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 99.5%

   \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|} \]
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. rem-sqrt-square 69.7%

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

   3. sqrt-prod 31.8%

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

   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. pow1/2 99.9%

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

   6. pow-flip 99.9%

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

   7. metadata-eval 99.9%

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

5. Applied egg-rr 99.9%

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

6. Taylor expanded in x around inf 98.4%

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

7. Final simplification 98.4%

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

Alternative 5: 98.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \left|\frac{\mathsf{fma}\left(2, x, 0.047619047619047616 \cdot {x}^{7}\right)}{\sqrt{\pi}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs (/ (fma 2.0 x (* 0.047619047619047616 (pow x 7.0))) (sqrt PI))))

C

double code(double x) {
	return fabs((fma(2.0, x, (0.047619047619047616 * pow(x, 7.0))) / sqrt(((double) M_PI))));
}

Julia

function code(x)
	return abs(Float64(fma(2.0, x, Float64(0.047619047619047616 * (x ^ 7.0))) / sqrt(pi)))
end

Wolfram

code[x_] := N[Abs[N[(N[(2.0 * x + N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $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 99.5%

   \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

4. Taylor expanded in x around inf 97.6%

   \[\leadsto \left|\frac{\mathsf{fma}\left(2, x, \color{blue}{0.047619047619047616 \cdot {x}^{7}}\right)}{\sqrt{\pi}}\right| \]

5. Final simplification 97.6%

   \[\leadsto \left|\frac{\mathsf{fma}\left(2, x, 0.047619047619047616 \cdot {x}^{7}\right)}{\sqrt{\pi}}\right| \]

Alternative 6: 93.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7:\\ \;\;\;\;\left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {x}^{4}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.7)
   (fabs
    (*
     (* x (pow PI -0.5))
     (+ (+ 2.0 (* 0.6666666666666666 (* x x))) (* 0.2 (pow x 4.0)))))
   (fabs (* 0.047619047619047616 (* (pow PI -0.5) (pow x 7.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.7) {
		tmp = fabs(((x * pow(((double) M_PI), -0.5)) * ((2.0 + (0.6666666666666666 * (x * x))) + (0.2 * pow(x, 4.0)))));
	} else {
		tmp = fabs((0.047619047619047616 * (pow(((double) M_PI), -0.5) * pow(x, 7.0))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.7) {
		tmp = Math.abs(((x * Math.pow(Math.PI, -0.5)) * ((2.0 + (0.6666666666666666 * (x * x))) + (0.2 * Math.pow(x, 4.0)))));
	} else {
		tmp = Math.abs((0.047619047619047616 * (Math.pow(Math.PI, -0.5) * Math.pow(x, 7.0))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.7:
		tmp = math.fabs(((x * math.pow(math.pi, -0.5)) * ((2.0 + (0.6666666666666666 * (x * x))) + (0.2 * math.pow(x, 4.0)))))
	else:
		tmp = math.fabs((0.047619047619047616 * (math.pow(math.pi, -0.5) * math.pow(x, 7.0))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.7)
		tmp = abs(Float64(Float64(x * (pi ^ -0.5)) * Float64(Float64(2.0 + Float64(0.6666666666666666 * Float64(x * x))) + Float64(0.2 * (x ^ 4.0)))));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64((pi ^ -0.5) * (x ^ 7.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.7)
		tmp = abs(((x * (pi ^ -0.5)) * ((2.0 + (0.6666666666666666 * (x * x))) + (0.2 * (x ^ 4.0)))));
	else
		tmp = abs((0.047619047619047616 * ((pi ^ -0.5) * (x ^ 7.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.7], N[Abs[N[(N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7000000000000002

   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 99.5%

      \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|} \]
   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. rem-sqrt-square 69.7%

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

      3. sqrt-prod 31.8%

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

      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. pow1/2 99.9%

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

      6. pow-flip 99.9%

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

      7. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 95.1%

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

      1. metadata-eval 95.1%

         \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{2}{3}}, x \cdot x, 2\right) + 0.2 \cdot {x}^{4}\right)\right| \]

      2. fma-udef 95.1%

         \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\color{blue}{\left(\frac{2}{3} \cdot \left(x \cdot x\right) + 2\right)} + 0.2 \cdot {x}^{4}\right)\right| \]

      3. metadata-eval 95.1%

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

   8. Applied egg-rr 95.1%

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

   if 2.7000000000000002 < 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 99.5%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around inf 37.1%

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

      1. associate-*r* 37.1%

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

   6. Simplified 37.1%

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

      1. expm1-log1p-u 4.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)}\right| \]

      2. expm1-udef 3.9%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}\right)} - 1}\right| \]

      3. associate-*l* 3.9%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)}\right)} - 1\right| \]

      4. inv-pow 3.9%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\color{blue}{{\pi}^{-1}}}\right)\right)} - 1\right| \]

      5. sqrt-pow1 3.9%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot \color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}\right)\right)} - 1\right| \]

      6. metadata-eval 3.9%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{\color{blue}{-0.5}}\right)\right)} - 1\right| \]

   8. Applied egg-rr 3.9%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)\right)} - 1}\right| \]
   9. Step-by-step derivation

      1. expm1-def 4.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)\right)\right)}\right| \]

      2. expm1-log1p 37.1%

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

   10. Simplified 37.1%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7:\\ \;\;\;\;\left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {x}^{4}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)\right|\\ \end{array} \]

Alternative 7: 89.3% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (fabs
    (* (sqrt (/ 1.0 PI)) (+ (* 2.0 x) (* 0.6666666666666666 (pow x 3.0)))))
   (fabs (* 0.047619047619047616 (* (pow PI -0.5) (pow x 7.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * ((2.0 * x) + (0.6666666666666666 * pow(x, 3.0)))));
	} else {
		tmp = fabs((0.047619047619047616 * (pow(((double) M_PI), -0.5) * pow(x, 7.0))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * ((2.0 * x) + (0.6666666666666666 * Math.pow(x, 3.0)))));
	} else {
		tmp = Math.abs((0.047619047619047616 * (Math.pow(Math.PI, -0.5) * Math.pow(x, 7.0))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.2:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * ((2.0 * x) + (0.6666666666666666 * math.pow(x, 3.0)))))
	else:
		tmp = math.fabs((0.047619047619047616 * (math.pow(math.pi, -0.5) * math.pow(x, 7.0))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(2.0 * x) + Float64(0.6666666666666666 * (x ^ 3.0)))));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64((pi ^ -0.5) * (x ^ 7.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.2)
		tmp = abs((sqrt((1.0 / pi)) * ((2.0 * x) + (0.6666666666666666 * (x ^ 3.0)))));
	else
		tmp = abs((0.047619047619047616 * ((pi ^ -0.5) * (x ^ 7.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(2.0 * x), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.9%

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

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around 0 89.0%

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

      1. +-commutative 89.0%

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

      2. associate-*r* 89.1%

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

      3. associate-*r* 89.1%

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

      4. distribute-rgt-out 89.1%

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

      5. *-commutative 89.1%

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

   6. Simplified 89.1%

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

   if 2.2000000000000002 < 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 99.5%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around inf 37.1%

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

      1. associate-*r* 37.1%

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

   6. Simplified 37.1%

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

      1. expm1-log1p-u 4.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)}\right| \]

      2. expm1-udef 3.9%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}\right)} - 1}\right| \]

      3. associate-*l* 3.9%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)}\right)} - 1\right| \]

      4. inv-pow 3.9%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\color{blue}{{\pi}^{-1}}}\right)\right)} - 1\right| \]

      5. sqrt-pow1 3.9%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot \color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}\right)\right)} - 1\right| \]

      6. metadata-eval 3.9%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{\color{blue}{-0.5}}\right)\right)} - 1\right| \]

   8. Applied egg-rr 3.9%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)\right)} - 1}\right| \]
   9. Step-by-step derivation

      1. expm1-def 4.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)\right)\right)}\right| \]

      2. expm1-log1p 37.1%

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

   10. Simplified 37.1%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)\right|\\ \end{array} \]

Alternative 8: 67.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.86:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.86)
   (fabs (* x (/ 2.0 (sqrt PI))))
   (fabs (* 0.047619047619047616 (* (pow PI -0.5) (pow x 7.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.86) {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((0.047619047619047616 * (pow(((double) M_PI), -0.5) * pow(x, 7.0))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.86) {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs((0.047619047619047616 * (Math.pow(Math.PI, -0.5) * Math.pow(x, 7.0))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.86:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs((0.047619047619047616 * (math.pow(math.pi, -0.5) * math.pow(x, 7.0))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.86)
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64((pi ^ -0.5) * (x ^ 7.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.86)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = abs((0.047619047619047616 * ((pi ^ -0.5) * (x ^ 7.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.86], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8600000000000001

   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 99.5%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around 0 66.8%

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

      1. associate-*r* 66.9%

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

   6. Simplified 66.9%

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

      1. expm1-log1p-u 64.9%

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

      2. expm1-udef 5.6%

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

      3. associate-*l* 5.6%

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

      4. inv-pow 5.6%

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

      5. sqrt-pow1 5.6%

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

      6. metadata-eval 5.6%

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

   8. Applied egg-rr 5.6%

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

      1. expm1-def 64.9%

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

      2. expm1-log1p 66.8%

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

      3. *-commutative 66.8%

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

      4. *-commutative 66.8%

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

      5. associate-*l* 66.9%

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

   10. Simplified 66.9%

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

       1. *-commutative 66.9%

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

       2. *-commutative 66.9%

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

       3. associate-*r* 66.8%

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

       4. add-sqr-sqrt 30.9%

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

       5. sqrt-unprod 53.9%

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

       6. swap-sqr 53.8%

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

       7. metadata-eval 53.8%

          \[\leadsto \left|\sqrt{\color{blue}{4} \cdot \left(\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(x \cdot {\pi}^{-0.5}\right)\right)}\right| \]

       8. swap-sqr 53.7%

          \[\leadsto \left|\sqrt{4 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)\right)}}\right| \]

       9. pow-prod-up 53.8%

          \[\leadsto \left|\sqrt{4 \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{{\pi}^{\left(-0.5 + -0.5\right)}}\right)}\right| \]

       10. metadata-eval 53.8%

           \[\leadsto \left|\sqrt{4 \cdot \left(\left(x \cdot x\right) \cdot {\pi}^{\color{blue}{-1}}\right)}\right| \]

   12. Applied egg-rr 53.8%

       \[\leadsto \left|\color{blue}{\sqrt{4 \cdot \left(\left(x \cdot x\right) \cdot {\pi}^{-1}\right)}}\right| \]
   13. Step-by-step derivation

       1. associate-*r* 53.8%

          \[\leadsto \left|\sqrt{\color{blue}{\left(4 \cdot \left(x \cdot x\right)\right) \cdot {\pi}^{-1}}}\right| \]

       2. *-commutative 53.8%

          \[\leadsto \left|\sqrt{\color{blue}{\left(\left(x \cdot x\right) \cdot 4\right)} \cdot {\pi}^{-1}}\right| \]

       3. metadata-eval 53.8%

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

       4. swap-sqr 53.9%

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

       5. unpow-1 53.9%

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

       6. associate-*r/ 53.9%

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

       7. *-rgt-identity 53.9%

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

       8. swap-sqr 53.9%

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

       9. metadata-eval 53.9%

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

       10. *-commutative 53.9%

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

   14. Simplified 53.9%

       \[\leadsto \left|\color{blue}{\sqrt{\frac{4 \cdot \left(x \cdot x\right)}{\pi}}}\right| \]
   15. Step-by-step derivation

       1. expm1-log1p-u 53.9%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{4 \cdot \left(x \cdot x\right)}{\pi}}\right)\right)}\right| \]

       2. expm1-udef 24.7%

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

       3. sqrt-div 24.7%

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

       4. *-commutative 24.7%

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

       5. metadata-eval 24.7%

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

       6. swap-sqr 24.7%

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

       7. sqrt-unprod 3.1%

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

       8. add-sqr-sqrt 5.6%

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

   16. Applied egg-rr 5.6%

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

       1. expm1-def 64.5%

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

       2. expm1-log1p 66.5%

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

       3. *-rgt-identity 66.5%

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

       4. associate-*r/ 66.9%

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

       5. associate-*l* 66.9%

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

       6. associate-*r/ 66.9%

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

       7. metadata-eval 66.9%

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

   18. Simplified 66.9%

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

   if 1.8600000000000001 < 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 99.5%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around inf 37.1%

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

      1. associate-*r* 37.1%

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

   6. Simplified 37.1%

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

      1. expm1-log1p-u 4.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)}\right| \]

      2. expm1-udef 3.9%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}\right)} - 1}\right| \]

      3. associate-*l* 3.9%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)}\right)} - 1\right| \]

      4. inv-pow 3.9%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\color{blue}{{\pi}^{-1}}}\right)\right)} - 1\right| \]

      5. sqrt-pow1 3.9%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot \color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}\right)\right)} - 1\right| \]

      6. metadata-eval 3.9%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{\color{blue}{-0.5}}\right)\right)} - 1\right| \]

   8. Applied egg-rr 3.9%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)\right)} - 1}\right| \]
   9. Step-by-step derivation

      1. expm1-def 4.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)\right)\right)}\right| \]

      2. expm1-log1p 37.1%

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

   10. Simplified 37.1%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.86:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)\right|\\ \end{array} \]

Alternative 9: 67.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.86:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{0.0022675736961451248 \cdot {x}^{14}}{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.86)
   (fabs (* x (/ 2.0 (sqrt PI))))
   (fabs (sqrt (/ (* 0.0022675736961451248 (pow x 14.0)) PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.86) {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs(sqrt(((0.0022675736961451248 * pow(x, 14.0)) / ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.86) {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs(Math.sqrt(((0.0022675736961451248 * Math.pow(x, 14.0)) / Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.86:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs(math.sqrt(((0.0022675736961451248 * math.pow(x, 14.0)) / math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.86)
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(sqrt(Float64(Float64(0.0022675736961451248 * (x ^ 14.0)) / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.86)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = abs(sqrt(((0.0022675736961451248 * (x ^ 14.0)) / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.86], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[Sqrt[N[(N[(0.0022675736961451248 * N[Power[x, 14.0], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8600000000000001

   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 99.5%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around 0 66.8%

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

      1. associate-*r* 66.9%

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

   6. Simplified 66.9%

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

      1. expm1-log1p-u 64.9%

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

      2. expm1-udef 5.6%

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

      3. associate-*l* 5.6%

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

      4. inv-pow 5.6%

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

      5. sqrt-pow1 5.6%

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

      6. metadata-eval 5.6%

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

   8. Applied egg-rr 5.6%

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

      1. expm1-def 64.9%

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

      2. expm1-log1p 66.8%

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

      3. *-commutative 66.8%

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

      4. *-commutative 66.8%

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

      5. associate-*l* 66.9%

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

   10. Simplified 66.9%

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

       1. *-commutative 66.9%

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

       2. *-commutative 66.9%

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

       3. associate-*r* 66.8%

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

       4. add-sqr-sqrt 30.9%

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

       5. sqrt-unprod 53.9%

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

       6. swap-sqr 53.8%

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

       7. metadata-eval 53.8%

          \[\leadsto \left|\sqrt{\color{blue}{4} \cdot \left(\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(x \cdot {\pi}^{-0.5}\right)\right)}\right| \]

       8. swap-sqr 53.7%

          \[\leadsto \left|\sqrt{4 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)\right)}}\right| \]

       9. pow-prod-up 53.8%

          \[\leadsto \left|\sqrt{4 \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{{\pi}^{\left(-0.5 + -0.5\right)}}\right)}\right| \]

       10. metadata-eval 53.8%

           \[\leadsto \left|\sqrt{4 \cdot \left(\left(x \cdot x\right) \cdot {\pi}^{\color{blue}{-1}}\right)}\right| \]

   12. Applied egg-rr 53.8%

       \[\leadsto \left|\color{blue}{\sqrt{4 \cdot \left(\left(x \cdot x\right) \cdot {\pi}^{-1}\right)}}\right| \]
   13. Step-by-step derivation

       1. associate-*r* 53.8%

          \[\leadsto \left|\sqrt{\color{blue}{\left(4 \cdot \left(x \cdot x\right)\right) \cdot {\pi}^{-1}}}\right| \]

       2. *-commutative 53.8%

          \[\leadsto \left|\sqrt{\color{blue}{\left(\left(x \cdot x\right) \cdot 4\right)} \cdot {\pi}^{-1}}\right| \]

       3. metadata-eval 53.8%

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

       4. swap-sqr 53.9%

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

       5. unpow-1 53.9%

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

       6. associate-*r/ 53.9%

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

       7. *-rgt-identity 53.9%

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

       8. swap-sqr 53.9%

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

       9. metadata-eval 53.9%

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

       10. *-commutative 53.9%

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

   14. Simplified 53.9%

       \[\leadsto \left|\color{blue}{\sqrt{\frac{4 \cdot \left(x \cdot x\right)}{\pi}}}\right| \]
   15. Step-by-step derivation

       1. expm1-log1p-u 53.9%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{4 \cdot \left(x \cdot x\right)}{\pi}}\right)\right)}\right| \]

       2. expm1-udef 24.7%

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

       3. sqrt-div 24.7%

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

       4. *-commutative 24.7%

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

       5. metadata-eval 24.7%

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

       6. swap-sqr 24.7%

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

       7. sqrt-unprod 3.1%

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

       8. add-sqr-sqrt 5.6%

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

   16. Applied egg-rr 5.6%

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

       1. expm1-def 64.5%

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

       2. expm1-log1p 66.5%

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

       3. *-rgt-identity 66.5%

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

       4. associate-*r/ 66.9%

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

       5. associate-*l* 66.9%

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

       6. associate-*r/ 66.9%

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

       7. metadata-eval 66.9%

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

   18. Simplified 66.9%

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

   if 1.8600000000000001 < 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 99.5%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around inf 37.1%

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

      1. associate-*r* 37.1%

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

   6. Simplified 37.1%

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

      1. expm1-log1p-u 4.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)}\right| \]

      2. expm1-udef 3.9%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}\right)} - 1}\right| \]

      3. associate-*l* 3.9%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)}\right)} - 1\right| \]

      4. inv-pow 3.9%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\color{blue}{{\pi}^{-1}}}\right)\right)} - 1\right| \]

      5. sqrt-pow1 3.9%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot \color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}\right)\right)} - 1\right| \]

      6. metadata-eval 3.9%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{\color{blue}{-0.5}}\right)\right)} - 1\right| \]

   8. Applied egg-rr 3.9%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)\right)} - 1}\right| \]
   9. Step-by-step derivation

      1. expm1-def 4.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)\right)\right)}\right| \]

      2. expm1-log1p 37.1%

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

      3. associate-*r* 37.1%

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

   10. Simplified 37.1%

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

       1. add-sqr-sqrt 3.7%

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

       2. sqrt-unprod 35.7%

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

       3. swap-sqr 35.7%

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

       4. swap-sqr 35.7%

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

       5. metadata-eval 35.7%

          \[\leadsto \left|\sqrt{\left(\color{blue}{0.0022675736961451248} \cdot \left({x}^{7} \cdot {x}^{7}\right)\right) \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)}\right| \]

       6. pow-prod-up 35.7%

          \[\leadsto \left|\sqrt{\left(0.0022675736961451248 \cdot \color{blue}{{x}^{\left(7 + 7\right)}}\right) \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)}\right| \]

       7. metadata-eval 35.7%

          \[\leadsto \left|\sqrt{\left(0.0022675736961451248 \cdot {x}^{\color{blue}{14}}\right) \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)}\right| \]

       8. pow-prod-up 35.6%

          \[\leadsto \left|\sqrt{\left(0.0022675736961451248 \cdot {x}^{14}\right) \cdot \color{blue}{{\pi}^{\left(-0.5 + -0.5\right)}}}\right| \]

       9. metadata-eval 35.6%

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

       10. inv-pow 35.6%

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

   12. Applied egg-rr 35.6%

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

       1. associate-*r/ 35.7%

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

       2. *-rgt-identity 35.7%

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

   14. Simplified 35.7%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.86:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{0.0022675736961451248 \cdot {x}^{14}}{\pi}}\right|\\ \end{array} \]

Alternative 10: 67.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-63}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\left(x \cdot x\right) \cdot \frac{4}{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e-63)
   (fabs (* x (/ 2.0 (sqrt PI))))
   (fabs (sqrt (* (* x x) (/ 4.0 PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 1e-63) {
		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 (x <= 1e-63) {
		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 x <= 1e-63:
		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 (x <= 1e-63)
		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 (x <= 1e-63)
		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[x, 1e-63], 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 x < 1.00000000000000007e-63

   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 99.5%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around 0 65.6%

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

      1. associate-*r* 65.6%

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

   6. Simplified 65.6%

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

      1. expm1-log1p-u 63.5%

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

      2. expm1-udef 4.4%

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

      3. associate-*l* 4.4%

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

      4. inv-pow 4.4%

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

      5. sqrt-pow1 4.4%

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

      6. metadata-eval 4.4%

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

   8. Applied egg-rr 4.4%

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

      1. expm1-def 63.5%

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

      2. expm1-log1p 65.6%

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

      3. *-commutative 65.6%

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

      4. *-commutative 65.6%

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

      5. associate-*l* 65.6%

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

   10. Simplified 65.6%

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

       1. *-commutative 65.6%

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

       2. *-commutative 65.6%

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

       3. associate-*r* 65.6%

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

       4. add-sqr-sqrt 27.3%

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

       5. sqrt-unprod 51.7%

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

       6. swap-sqr 51.7%

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

       7. metadata-eval 51.7%

          \[\leadsto \left|\sqrt{\color{blue}{4} \cdot \left(\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(x \cdot {\pi}^{-0.5}\right)\right)}\right| \]

       8. swap-sqr 51.6%

          \[\leadsto \left|\sqrt{4 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)\right)}}\right| \]

       9. pow-prod-up 51.7%

          \[\leadsto \left|\sqrt{4 \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{{\pi}^{\left(-0.5 + -0.5\right)}}\right)}\right| \]

       10. metadata-eval 51.7%

           \[\leadsto \left|\sqrt{4 \cdot \left(\left(x \cdot x\right) \cdot {\pi}^{\color{blue}{-1}}\right)}\right| \]

   12. Applied egg-rr 51.7%

       \[\leadsto \left|\color{blue}{\sqrt{4 \cdot \left(\left(x \cdot x\right) \cdot {\pi}^{-1}\right)}}\right| \]
   13. Step-by-step derivation

       1. associate-*r* 51.7%

          \[\leadsto \left|\sqrt{\color{blue}{\left(4 \cdot \left(x \cdot x\right)\right) \cdot {\pi}^{-1}}}\right| \]

       2. *-commutative 51.7%

          \[\leadsto \left|\sqrt{\color{blue}{\left(\left(x \cdot x\right) \cdot 4\right)} \cdot {\pi}^{-1}}\right| \]

       3. metadata-eval 51.7%

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

       4. swap-sqr 51.7%

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

       5. unpow-1 51.7%

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

       6. associate-*r/ 51.7%

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

       7. *-rgt-identity 51.7%

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

       8. swap-sqr 51.7%

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

       9. metadata-eval 51.7%

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

       10. *-commutative 51.7%

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

   14. Simplified 51.7%

       \[\leadsto \left|\color{blue}{\sqrt{\frac{4 \cdot \left(x \cdot x\right)}{\pi}}}\right| \]
   15. Step-by-step derivation

       1. expm1-log1p-u 51.7%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{4 \cdot \left(x \cdot x\right)}{\pi}}\right)\right)}\right| \]

       2. expm1-udef 24.7%

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

       3. sqrt-div 24.7%

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

       4. *-commutative 24.7%

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

       5. metadata-eval 24.7%

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

       6. swap-sqr 24.7%

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

       7. sqrt-unprod 1.7%

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

       8. add-sqr-sqrt 4.4%

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

   16. Applied egg-rr 4.4%

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

       1. expm1-def 63.1%

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

       2. expm1-log1p 65.2%

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

       3. *-rgt-identity 65.2%

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

       4. associate-*r/ 65.6%

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

       5. associate-*l* 65.6%

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

       6. associate-*r/ 65.6%

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

       7. metadata-eval 65.6%

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

   18. Simplified 65.6%

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

   if 1.00000000000000007e-63 < 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 99.2%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around 0 86.0%

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

      1. associate-*r* 86.0%

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

   6. Simplified 86.0%

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

      1. expm1-log1p-u 86.0%

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

      2. expm1-udef 24.6%

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

      3. associate-*l* 24.6%

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

      4. inv-pow 24.6%

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

      5. sqrt-pow1 24.6%

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

      6. metadata-eval 24.6%

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

   8. Applied egg-rr 24.6%

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

      1. expm1-def 86.0%

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

      2. expm1-log1p 86.0%

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

      3. *-commutative 86.0%

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

      4. *-commutative 86.0%

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

      5. associate-*l* 86.0%

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

   10. Simplified 86.0%

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

       1. *-commutative 86.0%

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

       2. *-commutative 86.0%

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

       3. associate-*r* 86.0%

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

       4. add-sqr-sqrt 85.7%

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

       5. sqrt-unprod 86.0%

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

       6. swap-sqr 86.0%

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

       7. metadata-eval 86.0%

          \[\leadsto \left|\sqrt{\color{blue}{4} \cdot \left(\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(x \cdot {\pi}^{-0.5}\right)\right)}\right| \]

       8. swap-sqr 85.3%

          \[\leadsto \left|\sqrt{4 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)\right)}}\right| \]

       9. pow-prod-up 86.1%

          \[\leadsto \left|\sqrt{4 \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{{\pi}^{\left(-0.5 + -0.5\right)}}\right)}\right| \]

       10. metadata-eval 86.1%

           \[\leadsto \left|\sqrt{4 \cdot \left(\left(x \cdot x\right) \cdot {\pi}^{\color{blue}{-1}}\right)}\right| \]

   12. Applied egg-rr 86.1%

       \[\leadsto \left|\color{blue}{\sqrt{4 \cdot \left(\left(x \cdot x\right) \cdot {\pi}^{-1}\right)}}\right| \]
   13. Step-by-step derivation

       1. associate-*r* 86.1%

          \[\leadsto \left|\sqrt{\color{blue}{\left(4 \cdot \left(x \cdot x\right)\right) \cdot {\pi}^{-1}}}\right| \]

       2. *-commutative 86.1%

          \[\leadsto \left|\sqrt{\color{blue}{\left(\left(x \cdot x\right) \cdot 4\right)} \cdot {\pi}^{-1}}\right| \]

       3. metadata-eval 86.1%

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

       4. swap-sqr 86.1%

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

       5. unpow-1 86.1%

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

       6. associate-*r/ 86.1%

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

       7. *-rgt-identity 86.1%

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

       8. swap-sqr 86.1%

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

       9. metadata-eval 86.1%

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

       10. *-commutative 86.1%

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

   14. Simplified 86.1%

       \[\leadsto \left|\color{blue}{\sqrt{\frac{4 \cdot \left(x \cdot x\right)}{\pi}}}\right| \]
   15. Step-by-step derivation

       1. associate-/l* 86.1%

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

       2. associate-/r/ 86.1%

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

   16. Applied egg-rr 86.1%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-63}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\left(x \cdot x\right) \cdot \frac{4}{\pi}}\right|\\ \end{array} \]

Alternative 11: 67.7% 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 99.5%

   \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

4. Taylor expanded in x around 0 66.8%

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

   1. associate-*r* 66.9%

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

6. Simplified 66.9%

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

   1. expm1-log1p-u 64.9%

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

   2. expm1-udef 5.6%

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

   3. associate-*l* 5.6%

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

   4. inv-pow 5.6%

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

   5. sqrt-pow1 5.6%

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

   6. metadata-eval 5.6%

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

8. Applied egg-rr 5.6%

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

   1. expm1-def 64.9%

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

   2. expm1-log1p 66.8%

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

   3. *-commutative 66.8%

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

   4. *-commutative 66.8%

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

   5. associate-*l* 66.9%

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

10. Simplified 66.9%

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

    1. *-commutative 66.9%

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

    2. *-commutative 66.9%

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

    3. associate-*r* 66.8%

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

    4. add-sqr-sqrt 30.9%

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

    5. sqrt-unprod 53.9%

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

    6. swap-sqr 53.8%

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

    7. metadata-eval 53.8%

       \[\leadsto \left|\sqrt{\color{blue}{4} \cdot \left(\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(x \cdot {\pi}^{-0.5}\right)\right)}\right| \]

    8. swap-sqr 53.7%

       \[\leadsto \left|\sqrt{4 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)\right)}}\right| \]

    9. pow-prod-up 53.8%

       \[\leadsto \left|\sqrt{4 \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{{\pi}^{\left(-0.5 + -0.5\right)}}\right)}\right| \]

    10. metadata-eval 53.8%

        \[\leadsto \left|\sqrt{4 \cdot \left(\left(x \cdot x\right) \cdot {\pi}^{\color{blue}{-1}}\right)}\right| \]

12. Applied egg-rr 53.8%

    \[\leadsto \left|\color{blue}{\sqrt{4 \cdot \left(\left(x \cdot x\right) \cdot {\pi}^{-1}\right)}}\right| \]
13. Step-by-step derivation

    1. associate-*r* 53.8%

       \[\leadsto \left|\sqrt{\color{blue}{\left(4 \cdot \left(x \cdot x\right)\right) \cdot {\pi}^{-1}}}\right| \]

    2. *-commutative 53.8%

       \[\leadsto \left|\sqrt{\color{blue}{\left(\left(x \cdot x\right) \cdot 4\right)} \cdot {\pi}^{-1}}\right| \]

    3. metadata-eval 53.8%

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

    4. swap-sqr 53.9%

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

    5. unpow-1 53.9%

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

    6. associate-*r/ 53.9%

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

    7. *-rgt-identity 53.9%

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

    8. swap-sqr 53.9%

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

    9. metadata-eval 53.9%

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

    10. *-commutative 53.9%

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

14. Simplified 53.9%

    \[\leadsto \left|\color{blue}{\sqrt{\frac{4 \cdot \left(x \cdot x\right)}{\pi}}}\right| \]
15. Step-by-step derivation

    1. expm1-log1p-u 53.9%

       \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{4 \cdot \left(x \cdot x\right)}{\pi}}\right)\right)}\right| \]

    2. expm1-udef 24.7%

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

    3. sqrt-div 24.7%

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

    4. *-commutative 24.7%

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

    5. metadata-eval 24.7%

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

    6. swap-sqr 24.7%

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

    7. sqrt-unprod 3.1%

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

    8. add-sqr-sqrt 5.6%

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

16. Applied egg-rr 5.6%

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

    1. expm1-def 64.5%

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

    2. expm1-log1p 66.5%

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

    3. *-rgt-identity 66.5%

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

    4. associate-*r/ 66.9%

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

    5. associate-*l* 66.9%

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

    6. associate-*r/ 66.9%

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

    7. metadata-eval 66.9%

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

18. Simplified 66.9%

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

19. Final simplification 66.9%

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

Reproduce

herbie shell --seed 2023230 
(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)))))))