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

Percentage Accurate: 99.9% → 99.8%

Time: 8.8s

Alternatives: 11

Speedup: N/A×

• Unsound rule application detected in e-graph. Results may not be sound.

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

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. *-rgt-identity 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 99.2% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. associate-*l/ 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around inf 99.3%

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

   1. div-inv 99.8%

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

   2. add-sqr-sqrt 37.6%

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

   3. fabs-sqr 37.6%

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

   4. add-sqr-sqrt 99.8%

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

   5. pow1/2 99.8%

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

   6. pow-flip 99.8%

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

   7. metadata-eval 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 3: 98.8% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. associate-*l/ 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around inf 99.3%

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

   1. expm1-log1p-u 99.1%

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

   2. expm1-udef 38.1%

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

   3. add-sqr-sqrt 2.8%

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

   4. fabs-sqr 2.8%

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

   5. add-sqr-sqrt 5.6%

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

6. Applied egg-rr 5.6%

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

   1. expm1-def 66.5%

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

   2. expm1-log1p 99.1%

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

8. Simplified 99.3%

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

9. Final simplification 99.3%

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

Alternative 4: 98.4% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. associate-*l/ 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around inf 99.3%

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

5. Taylor expanded in x around 0 99.1%

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

   1. expm1-log1p-u 99.1%

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

   2. expm1-udef 38.1%

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

   3. add-sqr-sqrt 2.8%

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

   4. fabs-sqr 2.8%

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

   5. add-sqr-sqrt 5.6%

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

7. Applied egg-rr 5.6%

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

   1. expm1-def 66.5%

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

   2. expm1-log1p 99.1%

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

9. Simplified 99.1%

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

10. Final simplification 99.1%

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

Alternative 5: 98.9% accurate, 4.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.85)
   (fabs (* 0.047619047619047616 (* (pow PI -0.5) (pow x 7.0))))
   (fabs (* (pow PI -0.5) (+ x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.85) {
		tmp = fabs((0.047619047619047616 * (pow(((double) M_PI), -0.5) * pow(x, 7.0))));
	} else {
		tmp = fabs((pow(((double) M_PI), -0.5) * (x + x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.85) {
		tmp = Math.abs((0.047619047619047616 * (Math.pow(Math.PI, -0.5) * Math.pow(x, 7.0))));
	} else {
		tmp = Math.abs((Math.pow(Math.PI, -0.5) * (x + x)));
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.85)
		tmp = abs(Float64(0.047619047619047616 * Float64((pi ^ -0.5) * (x ^ 7.0))));
	else
		tmp = abs(Float64((pi ^ -0.5) * Float64(x + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.85)
		tmp = abs((0.047619047619047616 * ((pi ^ -0.5) * (x ^ 7.0))));
	else
		tmp = abs(((pi ^ -0.5) * (x + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.85], N[Abs[N[(0.047619047619047616 * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8500000000000001

   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. Step-by-step derivation

      1. associate-+l+ 99.9%

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

      2. +-commutative 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 99.9%

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

      3. rem-square-sqrt 0.0%

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

      4. fabs-sqr 0.0%

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

      5. rem-square-sqrt 99.9%

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

      6. *-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. associate-+l+ 99.9%

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

      9. *-commutative 99.9%

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

      10. fma-def 99.9%

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

      11. rem-square-sqrt 0.0%

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

      12. fabs-sqr 0.0%

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

      13. rem-square-sqrt 99.9%

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

      14. fma-def 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around inf 99.6%

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

      1. associate-*r* 99.6%

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

      2. *-commutative 99.6%

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

   9. Simplified 99.6%

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

       1. expm1-log1p-u 0.0%

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

       2. expm1-udef 0.0%

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

       3. pow1/2 0.0%

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

       4. inv-pow 0.0%

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

       5. pow-pow 0.0%

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

       6. metadata-eval 0.0%

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

   11. Applied egg-rr 0.0%

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

       1. expm1-def 0.0%

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

       2. expm1-log1p 99.6%

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

       3. *-commutative 99.6%

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

       4. associate-*l* 99.6%

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

   13. Simplified 99.6%

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

   if -1.8500000000000001 < x

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Step-by-step derivation

      1. associate-+l+ 99.8%

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

      2. +-commutative 99.8%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l* 99.6%

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

      3. rem-square-sqrt 56.0%

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

      4. fabs-sqr 56.0%

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

      5. rem-square-sqrt 99.6%

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

      6. *-commutative 99.6%

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

      7. +-commutative 99.6%

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

      8. associate-+l+ 99.6%

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

      9. *-commutative 99.6%

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

      10. fma-def 99.6%

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

      11. rem-square-sqrt 56.3%

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

      12. fabs-sqr 56.3%

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

      13. rem-square-sqrt 99.6%

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

      14. fma-def 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around 0 99.6%

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

      1. associate-*r* 99.6%

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

      2. *-commutative 99.6%

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

   9. Simplified 99.6%

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

       1. expm1-log1p-u 99.6%

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

       2. expm1-udef 8.4%

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

       3. pow1/2 8.4%

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

       4. inv-pow 8.4%

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

       5. pow-pow 8.4%

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

       6. metadata-eval 8.4%

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

       7. *-commutative 8.4%

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

   11. Applied egg-rr 8.4%

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

       1. expm1-def 99.6%

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

       2. expm1-log1p 99.6%

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

       3. *-commutative 99.6%

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

       4. rem-log-exp 8.4%

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

       5. log-pow 8.3%

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

       6. exp-prod 8.4%

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

       7. exp-lft-sqr 8.3%

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

       8. log-prod 8.4%

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

       9. rem-log-exp 20.8%

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

       10. rem-log-exp 99.6%

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

   13. Simplified 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x + x\right)\right|\\ \end{array} \]

Alternative 6: 94.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.72:\\ \;\;\;\;\left|0.6666666666666666 \cdot \sqrt{\frac{{x}^{6}}{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x + x\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.72)
   (fabs (* 0.6666666666666666 (sqrt (/ (pow x 6.0) PI))))
   (fabs (* (pow PI -0.5) (+ x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.72) {
		tmp = fabs((0.6666666666666666 * sqrt((pow(x, 6.0) / ((double) M_PI)))));
	} else {
		tmp = fabs((pow(((double) M_PI), -0.5) * (x + x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.72) {
		tmp = Math.abs((0.6666666666666666 * Math.sqrt((Math.pow(x, 6.0) / Math.PI))));
	} else {
		tmp = Math.abs((Math.pow(Math.PI, -0.5) * (x + x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.72:
		tmp = math.fabs((0.6666666666666666 * math.sqrt((math.pow(x, 6.0) / math.pi))))
	else:
		tmp = math.fabs((math.pow(math.pi, -0.5) * (x + x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.72)
		tmp = abs(Float64(0.6666666666666666 * sqrt(Float64((x ^ 6.0) / pi))));
	else
		tmp = abs(Float64((pi ^ -0.5) * Float64(x + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.72)
		tmp = abs((0.6666666666666666 * sqrt(((x ^ 6.0) / pi))));
	else
		tmp = abs(((pi ^ -0.5) * (x + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.72], N[Abs[N[(0.6666666666666666 * N[Sqrt[N[(N[Power[x, 6.0], $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.71999999999999997

   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. Step-by-step derivation

      1. associate-+l+ 99.9%

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

      2. +-commutative 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 75.5%

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

      1. unpow2 75.5%

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

   6. Simplified 75.5%

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

      1. add-sqr-sqrt 75.5%

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

      2. sqrt-unprod 92.1%

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

      3. *-commutative 92.1%

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

      4. *-commutative 92.1%

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

      5. swap-sqr 92.1%

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

      6. add-sqr-sqrt 92.1%

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

      7. pow2 92.1%

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

      8. add-sqr-sqrt 0.0%

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

      9. fabs-sqr 0.0%

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

      10. add-sqr-sqrt 92.1%

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

      11. cube-mult 92.1%

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

      12. pow2 92.1%

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

      13. pow-sqr 92.1%

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

      14. metadata-eval 92.1%

          \[\leadsto \left|0.6666666666666666 \cdot \sqrt{\frac{1}{\pi} \cdot {x}^{\color{blue}{6}}}\right| \]

   8. Applied egg-rr 92.1%

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

      1. associate-*l/ 92.1%

         \[\leadsto \left|0.6666666666666666 \cdot \sqrt{\color{blue}{\frac{1 \cdot {x}^{6}}{\pi}}}\right| \]

      2. *-lft-identity 92.1%

         \[\leadsto \left|0.6666666666666666 \cdot \sqrt{\frac{\color{blue}{{x}^{6}}}{\pi}}\right| \]

   10. Simplified 92.1%

       \[\leadsto \left|0.6666666666666666 \cdot \color{blue}{\sqrt{\frac{{x}^{6}}{\pi}}}\right| \]

   if -1.71999999999999997 < x

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Step-by-step derivation

      1. associate-+l+ 99.8%

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

      2. +-commutative 99.8%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l* 99.6%

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

      3. rem-square-sqrt 56.0%

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

      4. fabs-sqr 56.0%

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

      5. rem-square-sqrt 99.6%

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

      6. *-commutative 99.6%

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

      7. +-commutative 99.6%

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

      8. associate-+l+ 99.6%

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

      9. *-commutative 99.6%

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

      10. fma-def 99.6%

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

      11. rem-square-sqrt 56.3%

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

      12. fabs-sqr 56.3%

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

      13. rem-square-sqrt 99.6%

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

      14. fma-def 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around 0 99.6%

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

      1. associate-*r* 99.6%

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

      2. *-commutative 99.6%

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

   9. Simplified 99.6%

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

       1. expm1-log1p-u 99.6%

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

       2. expm1-udef 8.4%

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

       3. pow1/2 8.4%

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

       4. inv-pow 8.4%

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

       5. pow-pow 8.4%

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

       6. metadata-eval 8.4%

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

       7. *-commutative 8.4%

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

   11. Applied egg-rr 8.4%

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

       1. expm1-def 99.6%

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

       2. expm1-log1p 99.6%

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

       3. *-commutative 99.6%

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

       4. rem-log-exp 8.4%

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

       5. log-pow 8.3%

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

       6. exp-prod 8.4%

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

       7. exp-lft-sqr 8.3%

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

       8. log-prod 8.4%

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

       9. rem-log-exp 20.8%

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

       10. rem-log-exp 99.6%

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

   13. Simplified 99.6%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.72:\\ \;\;\;\;\left|0.6666666666666666 \cdot \sqrt{\frac{{x}^{6}}{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x + x\right)\right|\\ \end{array} \]

Alternative 7: 95.0% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+23}:\\ \;\;\;\;\left|\frac{1}{\sqrt{\pi} \cdot 0}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x + x\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -9.2e+23)
   (fabs (/ 1.0 (* (sqrt PI) 0.0)))
   (fabs (* (pow PI -0.5) (+ x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -9.2e+23) {
		tmp = fabs((1.0 / (sqrt(((double) M_PI)) * 0.0)));
	} else {
		tmp = fabs((pow(((double) M_PI), -0.5) * (x + x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -9.2e+23) {
		tmp = Math.abs((1.0 / (Math.sqrt(Math.PI) * 0.0)));
	} else {
		tmp = Math.abs((Math.pow(Math.PI, -0.5) * (x + x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -9.2e+23:
		tmp = math.fabs((1.0 / (math.sqrt(math.pi) * 0.0)))
	else:
		tmp = math.fabs((math.pow(math.pi, -0.5) * (x + x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -9.2e+23)
		tmp = abs(Float64(1.0 / Float64(sqrt(pi) * 0.0)));
	else
		tmp = abs(Float64((pi ^ -0.5) * Float64(x + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -9.2e+23)
		tmp = abs((1.0 / (sqrt(pi) * 0.0)));
	else
		tmp = abs(((pi ^ -0.5) * (x + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -9.2e+23], N[Abs[N[(1.0 / N[(N[Sqrt[Pi], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.2000000000000002e23

   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. Step-by-step derivation

      1. associate-+l+ 99.9%

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

      2. +-commutative 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 76.3%

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

      1. associate-*r* 76.3%

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

      2. *-commutative 76.3%

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

      3. unpow2 76.3%

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

      4. *-commutative 76.3%

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

      5. associate-*r* 76.3%

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

      6. rem-square-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. rem-square-sqrt 76.3%

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

      9. associate-*r* 76.3%

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

      10. unpow3 76.3%

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

   6. Simplified 76.3%

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

      1. *-commutative 76.3%

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

      2. cube-mult 76.3%

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

      3. add-sqr-sqrt 0.0%

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

      4. fabs-sqr 0.0%

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

      5. add-sqr-sqrt 76.3%

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

      6. associate-*l* 76.3%

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

      7. *-commutative 76.3%

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

      8. *-commutative 76.3%

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

      9. expm1-log1p-u 76.3%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.6666666666666666 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)}\right| \]

      10. expm1-udef 76.3%

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

   8. Applied egg-rr 0.0%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(0.6666666666666666 \cdot \frac{{x}^{3}}{\sqrt{\pi}}\right)} - 1}\right| \]
   9. Step-by-step derivation

      1. expm1-def 0.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.6666666666666666 \cdot \frac{{x}^{3}}{\sqrt{\pi}}\right)\right)}\right| \]

      2. expm1-log1p 76.3%

         \[\leadsto \left|\color{blue}{0.6666666666666666 \cdot \frac{{x}^{3}}{\sqrt{\pi}}}\right| \]

      3. associate-*r/ 76.3%

         \[\leadsto \left|\color{blue}{\frac{0.6666666666666666 \cdot {x}^{3}}{\sqrt{\pi}}}\right| \]

   10. Simplified 76.3%

       \[\leadsto \left|\color{blue}{\frac{0.6666666666666666 \cdot {x}^{3}}{\sqrt{\pi}}}\right| \]
   11. Step-by-step derivation

       1. expm1-log1p-u 0.0%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.6666666666666666 \cdot {x}^{3}}{\sqrt{\pi}}\right)\right)}\right| \]

       2. expm1-udef 0.0%

          \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{0.6666666666666666 \cdot {x}^{3}}{\sqrt{\pi}}\right)} - 1}\right| \]

       3. associate-/l* 0.0%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{0.6666666666666666}{\frac{\sqrt{\pi}}{{x}^{3}}}}\right)} - 1\right| \]

   12. Applied egg-rr 0.0%

       \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{0.6666666666666666}{\frac{\sqrt{\pi}}{{x}^{3}}}\right)} - 1}\right| \]
   13. Step-by-step derivation

       1. expm1-def 0.0%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.6666666666666666}{\frac{\sqrt{\pi}}{{x}^{3}}}\right)\right)}\right| \]

       2. expm1-log1p 76.3%

          \[\leadsto \left|\color{blue}{\frac{0.6666666666666666}{\frac{\sqrt{\pi}}{{x}^{3}}}}\right| \]

       3. associate-/r/ 76.3%

          \[\leadsto \left|\color{blue}{\frac{0.6666666666666666}{\sqrt{\pi}} \cdot {x}^{3}}\right| \]

       4. *-commutative 76.3%

          \[\leadsto \left|\color{blue}{{x}^{3} \cdot \frac{0.6666666666666666}{\sqrt{\pi}}}\right| \]

   14. Simplified 76.3%

       \[\leadsto \left|\color{blue}{{x}^{3} \cdot \frac{0.6666666666666666}{\sqrt{\pi}}}\right| \]
   15. Step-by-step derivation

       1. expm1-log1p-u 0.0%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{3} \cdot \frac{0.6666666666666666}{\sqrt{\pi}}\right)\right)}\right| \]

       2. expm1-udef 0.0%

          \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left({x}^{3} \cdot \frac{0.6666666666666666}{\sqrt{\pi}}\right)} - 1}\right| \]

       3. clear-num 0.0%

          \[\leadsto \left|e^{\mathsf{log1p}\left({x}^{3} \cdot \color{blue}{\frac{1}{\frac{\sqrt{\pi}}{0.6666666666666666}}}\right)} - 1\right| \]

       4. un-div-inv 0.0%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{{x}^{3}}{\frac{\sqrt{\pi}}{0.6666666666666666}}}\right)} - 1\right| \]

       5. div-inv 0.0%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\frac{{x}^{3}}{\color{blue}{\sqrt{\pi} \cdot \frac{1}{0.6666666666666666}}}\right)} - 1\right| \]

       6. metadata-eval 0.0%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\frac{{x}^{3}}{\sqrt{\pi} \cdot \color{blue}{1.5}}\right)} - 1\right| \]

   16. Applied egg-rr 0.0%

       \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{{x}^{3}}{\sqrt{\pi} \cdot 1.5}\right)} - 1}\right| \]

   18. Simplified 95.5%

       \[\leadsto \left|\color{blue}{\frac{{x}^{0}}{0 \cdot \sqrt{\pi}}}\right| \]

   if -9.2000000000000002e23 < x

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   2. Step-by-step derivation

      1. associate-+l+ 99.8%

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

      2. +-commutative 99.8%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l* 99.6%

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

      3. rem-square-sqrt 55.7%

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

      4. fabs-sqr 55.7%

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

      5. rem-square-sqrt 99.6%

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

      6. *-commutative 99.6%

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

      7. +-commutative 99.6%

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

      8. associate-+l+ 99.6%

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

      9. *-commutative 99.6%

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

      10. fma-def 99.6%

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

      11. rem-square-sqrt 56.0%

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

      12. fabs-sqr 56.0%

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

      13. rem-square-sqrt 99.6%

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

      14. fma-def 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around 0 99.0%

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

      1. associate-*r* 99.0%

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

      2. *-commutative 99.0%

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

   9. Simplified 99.0%

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

       1. expm1-log1p-u 99.0%

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

       2. expm1-udef 8.3%

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

       3. pow1/2 8.3%

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

       4. inv-pow 8.3%

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

       5. pow-pow 8.3%

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

       6. metadata-eval 8.3%

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

       7. *-commutative 8.3%

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

   11. Applied egg-rr 8.3%

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

       1. expm1-def 99.0%

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

       2. expm1-log1p 99.0%

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

       3. *-commutative 99.0%

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

       4. rem-log-exp 8.3%

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

       5. log-pow 8.3%

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

       6. exp-prod 8.4%

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

       7. exp-lft-sqr 8.3%

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

       8. log-prod 8.3%

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

       9. rem-log-exp 20.7%

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

       10. rem-log-exp 99.0%

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

   13. Simplified 99.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+23}:\\ \;\;\;\;\left|\frac{1}{\sqrt{\pi} \cdot 0}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x + x\right)\right|\\ \end{array} \]

Alternative 8: 68.0% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x + x), $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. Step-by-step derivation

   1. associate-+l+ 99.9%

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

   2. +-commutative 99.9%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around 0 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*l* 99.7%

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

   3. rem-square-sqrt 37.6%

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

   4. fabs-sqr 37.6%

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

   5. rem-square-sqrt 99.7%

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

   6. *-commutative 99.7%

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

   7. +-commutative 99.7%

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

   8. associate-+l+ 99.7%

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

   9. *-commutative 99.7%

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

   10. fma-def 99.7%

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

   11. rem-square-sqrt 37.8%

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

   12. fabs-sqr 37.8%

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

   13. rem-square-sqrt 99.7%

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

   14. fma-def 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around 0 68.8%

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

   1. associate-*r* 68.8%

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

   2. *-commutative 68.8%

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

9. Simplified 68.8%

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

    1. expm1-log1p-u 66.9%

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

    2. expm1-udef 5.6%

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

    3. pow1/2 5.6%

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

    4. inv-pow 5.6%

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

    5. pow-pow 5.6%

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

    6. metadata-eval 5.6%

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

    7. *-commutative 5.6%

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

11. Applied egg-rr 5.6%

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

    1. expm1-def 66.9%

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

    2. expm1-log1p 68.8%

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

    3. *-commutative 68.8%

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

    4. rem-log-exp 36.6%

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

    5. log-pow 36.6%

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

    6. exp-prod 36.6%

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

    7. exp-lft-sqr 36.6%

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

    8. log-prod 36.6%

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

    9. rem-log-exp 45.0%

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

    10. rem-log-exp 68.8%

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

13. Simplified 68.8%

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

14. Final simplification 68.8%

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

Alternative 9: 67.5% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(2.0 / N[(N[Sqrt[Pi], $MachinePrecision] / x), $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. Step-by-step derivation

   1. associate-+l+ 99.9%

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

   2. +-commutative 99.9%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around 0 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*l* 99.7%

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

   3. rem-square-sqrt 37.6%

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

   4. fabs-sqr 37.6%

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

   5. rem-square-sqrt 99.7%

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

   6. *-commutative 99.7%

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

   7. +-commutative 99.7%

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

   8. associate-+l+ 99.7%

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

   9. *-commutative 99.7%

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

   10. fma-def 99.7%

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

   11. rem-square-sqrt 37.8%

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

   12. fabs-sqr 37.8%

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

   13. rem-square-sqrt 99.7%

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

   14. fma-def 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around 0 68.8%

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

   1. associate-*r* 68.8%

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

   2. *-commutative 68.8%

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

9. Simplified 68.8%

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

    1. expm1-log1p-u 66.9%

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

    2. expm1-udef 5.6%

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

    3. pow1/2 5.6%

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

    4. inv-pow 5.6%

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

    5. pow-pow 5.6%

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

    6. metadata-eval 5.6%

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

    7. *-commutative 5.6%

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

11. Applied egg-rr 5.6%

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

    1. expm1-def 66.9%

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

    2. expm1-log1p 68.8%

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

    3. *-commutative 68.8%

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

    4. rem-log-exp 36.6%

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

    5. log-pow 36.6%

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

    6. exp-prod 36.6%

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

    7. exp-lft-sqr 36.6%

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

    8. log-prod 36.6%

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

    9. rem-log-exp 45.0%

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

    10. rem-log-exp 68.8%

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

13. Simplified 68.8%

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

    1. *-commutative 68.8%

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

    2. count-2 68.8%

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

    3. associate-*r* 68.8%

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

    4. metadata-eval 68.8%

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

    5. pow-flip 68.8%

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

    6. pow1/2 68.8%

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

    7. div-inv 68.4%

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

    8. add-sqr-sqrt 37.4%

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

    9. fabs-sqr 37.4%

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

    10. add-sqr-sqrt 68.4%

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

    11. clear-num 68.3%

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

    12. un-div-inv 68.3%

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

    13. add-sqr-sqrt 37.5%

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

    14. fabs-sqr 37.5%

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

    15. add-sqr-sqrt 68.3%

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

15. Applied egg-rr 68.3%

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

16. Final simplification 68.3%

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

Alternative 10: 14.4% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
2. Step-by-step derivation

   1. associate-+l+ 99.9%

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

   2. +-commutative 99.9%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around 0 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*l* 99.7%

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

   3. rem-square-sqrt 37.6%

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

   4. fabs-sqr 37.6%

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

   5. rem-square-sqrt 99.7%

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

   6. *-commutative 99.7%

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

   7. +-commutative 99.7%

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

   8. associate-+l+ 99.7%

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

   9. *-commutative 99.7%

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

   10. fma-def 99.7%

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

   11. rem-square-sqrt 37.8%

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

   12. fabs-sqr 37.8%

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

   13. rem-square-sqrt 99.7%

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

   14. fma-def 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around 0 68.8%

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

   1. associate-*r* 68.8%

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

   2. *-commutative 68.8%

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

9. Simplified 68.8%

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

    1. expm1-log1p-u 66.9%

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

    2. expm1-udef 5.6%

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

    3. pow1/2 5.6%

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

    4. inv-pow 5.6%

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

    5. pow-pow 5.6%

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

    6. metadata-eval 5.6%

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

    7. *-commutative 5.6%

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

11. Applied egg-rr 5.6%

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

    1. expm1-def 66.9%

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

    2. expm1-log1p 68.8%

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

    3. *-commutative 68.8%

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

    4. rem-log-exp 36.6%

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

    5. log-pow 36.6%

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

    6. exp-prod 36.6%

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

    7. exp-lft-sqr 36.6%

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

    8. log-prod 36.6%

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

    9. rem-log-exp 45.0%

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

    10. rem-log-exp 68.8%

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

13. Simplified 68.8%

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

    1. distribute-rgt-in 68.8%

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

    2. metadata-eval 68.8%

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

    3. pow-flip 68.8%

       \[\leadsto \left|x \cdot \color{blue}{\frac{1}{{\pi}^{0.5}}} + x \cdot {\pi}^{-0.5}\right| \]

    4. pow1/2 68.8%

       \[\leadsto \left|x \cdot \frac{1}{\color{blue}{\sqrt{\pi}}} + x \cdot {\pi}^{-0.5}\right| \]

    5. div-inv 68.5%

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

    6. metadata-eval 68.5%

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

    7. pow-flip 68.5%

       \[\leadsto \left|\frac{x}{\sqrt{\pi}} + x \cdot \color{blue}{\frac{1}{{\pi}^{0.5}}}\right| \]

    8. pow1/2 68.5%

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

    9. div-inv 68.4%

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

15. Applied egg-rr 68.4%

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

17. Simplified 14.5%

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

18. Final simplification 14.5%

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

Alternative 11: 4.1% accurate, 1948.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

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. Step-by-step derivation

   1. associate-+l+ 99.9%

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

   2. +-commutative 99.9%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around 0 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*l* 99.7%

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

   3. rem-square-sqrt 37.6%

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

   4. fabs-sqr 37.6%

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

   5. rem-square-sqrt 99.7%

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

   6. *-commutative 99.7%

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

   7. +-commutative 99.7%

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

   8. associate-+l+ 99.7%

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

   9. *-commutative 99.7%

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

   10. fma-def 99.7%

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

   11. rem-square-sqrt 37.8%

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

   12. fabs-sqr 37.8%

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

   13. rem-square-sqrt 99.7%

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

   14. fma-def 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around 0 68.8%

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

   1. associate-*r* 68.8%

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

   2. *-commutative 68.8%

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

9. Simplified 68.8%

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

    1. expm1-log1p-u 66.9%

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

    2. expm1-udef 5.6%

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

    3. pow1/2 5.6%

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

    4. inv-pow 5.6%

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

    5. pow-pow 5.6%

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

    6. metadata-eval 5.6%

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

    7. *-commutative 5.6%

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

11. Applied egg-rr 5.6%

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

    1. expm1-def 66.9%

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

    2. expm1-log1p 68.8%

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

    3. *-commutative 68.8%

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

    4. rem-log-exp 36.6%

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

    5. log-pow 36.6%

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

    6. exp-prod 36.6%

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

    7. exp-lft-sqr 36.6%

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

    8. log-prod 36.6%

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

    9. rem-log-exp 45.0%

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

    10. rem-log-exp 68.8%

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

13. Simplified 68.8%

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

    1. distribute-lft-in 68.8%

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

15. Applied egg-rr 68.8%

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

17. Simplified 4.2%

    \[\leadsto \left|\color{blue}{0}\right| \]

18. Final simplification 4.2%

    \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023175 
(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)))))))