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

Percentage Accurate: 99.8% → 99.9%

Time: 8.9s

Alternatives: 11

Speedup: 3.1×

• 28.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 99.4%

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

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. Step-by-step derivation

   1. div-inv 99.9%

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

5. Applied egg-rr 99.9%

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

   1. associate-*r/ 99.4%

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

   2. *-rgt-identity 99.4%

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

   3. unpow1 99.4%

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

   4. sqr-pow 31.0%

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

   5. fabs-sqr 31.0%

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

   6. sqr-pow 99.4%

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

   7. unpow1 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in x around 0 99.4%

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

   1. clear-num 99.3%

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

   2. associate-/r/ 99.9%

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

   3. pow1/2 99.9%

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

   4. pow-flip 99.9%

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

   5. metadata-eval 99.9%

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

10. Applied egg-rr 99.9%

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

11. Final simplification 99.9%

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

Alternative 2: 99.4% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. div-inv 99.9%

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

5. Applied egg-rr 99.9%

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

   1. associate-*r/ 99.4%

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

   2. *-rgt-identity 99.4%

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

   3. unpow1 99.4%

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

   4. sqr-pow 31.0%

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

   5. fabs-sqr 31.0%

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

   6. sqr-pow 99.4%

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

   7. unpow1 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in x around 0 99.4%

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

   1. metadata-eval 99.4%

      \[\leadsto \left|\frac{x}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{2}{3}}, x \cdot x, 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   2. fma-udef 99.4%

      \[\leadsto \left|\frac{x}{\sqrt{\pi}} \cdot \left(\color{blue}{\left(\frac{2}{3} \cdot \left(x \cdot x\right) + 2\right)} + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   3. metadata-eval 99.4%

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

10. Applied egg-rr 99.4%

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

11. Final simplification 99.4%

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

Alternative 3: 99.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \left|\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + 0.047619047619047616 \cdot {x}^{6}\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (* x (sqrt (/ 1.0 PI)))
   (+
    (fma 0.6666666666666666 (* x x) 2.0)
    (* 0.047619047619047616 (pow x 6.0))))))

C

double code(double x) {
	return fabs(((x * sqrt((1.0 / ((double) M_PI)))) * (fma(0.6666666666666666, (x * x), 2.0) + (0.047619047619047616 * pow(x, 6.0)))));
}

Julia

function code(x)
	return abs(Float64(Float64(x * sqrt(Float64(1.0 / pi))) * 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[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $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 99.8%

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

   1. associate-*l/ 99.4%

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

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 0 99.9%

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

   1. unpow1 99.9%

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

   2. sqr-pow 31.0%

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

   3. fabs-sqr 31.0%

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

   4. sqr-pow 99.9%

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

   5. unpow1 99.9%

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

6. Simplified 99.9%

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

7. Taylor expanded in x around inf 99.1%

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

8. Final simplification 99.1%

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

Alternative 4: 98.8% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return abs(Float64(Float64(x / sqrt(pi)) * 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[Sqrt[Pi], $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 99.8%

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

   1. associate-*l/ 99.4%

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

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. Step-by-step derivation

   1. div-inv 99.9%

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

5. Applied egg-rr 99.9%

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

   1. associate-*r/ 99.4%

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

   2. *-rgt-identity 99.4%

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

   3. unpow1 99.4%

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

   4. sqr-pow 31.0%

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

   5. fabs-sqr 31.0%

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

   6. sqr-pow 99.4%

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

   7. unpow1 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in x around inf 98.7%

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

9. Final simplification 98.7%

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

Alternative 5: 99.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{7}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.85)
   (fabs (* 0.047619047619047616 (* (sqrt (/ 1.0 PI)) (pow x 7.0))))
   (fabs (* x (/ 2.0 (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.85) {
		tmp = fabs((0.047619047619047616 * (sqrt((1.0 / ((double) M_PI))) * pow(x, 7.0))));
	} else {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.85) {
		tmp = Math.abs((0.047619047619047616 * (Math.sqrt((1.0 / Math.PI)) * Math.pow(x, 7.0))));
	} else {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.85:
		tmp = math.fabs((0.047619047619047616 * (math.sqrt((1.0 / math.pi)) * math.pow(x, 7.0))))
	else:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.85)
		tmp = abs(Float64(0.047619047619047616 * Float64(sqrt(Float64(1.0 / pi)) * (x ^ 7.0))));
	else
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.85)
		tmp = abs((0.047619047619047616 * (sqrt((1.0 / pi)) * (x ^ 7.0))));
	else
		tmp = abs((x * (2.0 / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.85], N[Abs[N[(0.047619047619047616 * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8500000000000001

   1. Initial program 99.8%

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

      1. associate-*l/ 99.8%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. unpow1 99.9%

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

      2. sqr-pow 0.0%

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

      3. fabs-sqr 0.0%

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

      4. sqr-pow 99.9%

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

      5. unpow1 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around inf 97.8%

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

   8. Taylor expanded in x around inf 97.8%

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

   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/ 99.2%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around 0 99.8%

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

      1. unpow1 99.8%

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

      2. sqr-pow 48.4%

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

      3. fabs-sqr 48.4%

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

      4. sqr-pow 99.8%

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

      5. unpow1 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in x around inf 99.8%

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

   8. Taylor expanded in x around 0 98.8%

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

      1. associate-*r* 98.8%

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

   10. Simplified 98.8%

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

       1. expm1-log1p-u 98.8%

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

       2. expm1-udef 8.6%

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

   12. Applied egg-rr 8.6%

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

       1. expm1-def 98.0%

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

       2. expm1-log1p 98.0%

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

       3. associate-/r/ 98.8%

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

   14. Simplified 98.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{7}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 6: 99.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.85)
   (fabs (* (sqrt (/ 1.0 PI)) (* 0.047619047619047616 (pow x 7.0))))
   (fabs (* x (/ 2.0 (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.85) {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * (0.047619047619047616 * pow(x, 7.0))));
	} else {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.85) {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * (0.047619047619047616 * Math.pow(x, 7.0))));
	} else {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.85:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * (0.047619047619047616 * math.pow(x, 7.0))))
	else:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.85)
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(0.047619047619047616 * (x ^ 7.0))));
	else
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.85)
		tmp = abs((sqrt((1.0 / pi)) * (0.047619047619047616 * (x ^ 7.0))));
	else
		tmp = abs((x * (2.0 / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.85], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8500000000000001

   1. Initial program 99.8%

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

      1. associate-*l/ 99.8%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. unpow1 99.9%

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

      2. sqr-pow 0.0%

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

      3. fabs-sqr 0.0%

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

      4. sqr-pow 99.9%

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

      5. unpow1 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around inf 97.8%

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

   8. Taylor expanded in x around inf 97.8%

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

      1. associate-*r* 97.9%

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

   10. Simplified 97.9%

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

   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/ 99.2%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around 0 99.8%

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

      1. unpow1 99.8%

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

      2. sqr-pow 48.4%

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

      3. fabs-sqr 48.4%

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

      4. sqr-pow 99.8%

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

      5. unpow1 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in x around inf 99.8%

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

   8. Taylor expanded in x around 0 98.8%

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

      1. associate-*r* 98.8%

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

   10. Simplified 98.8%

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

       1. expm1-log1p-u 98.8%

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

       2. expm1-udef 8.6%

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

   12. Applied egg-rr 8.6%

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

       1. expm1-def 98.0%

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

       2. expm1-log1p 98.0%

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

       3. associate-/r/ 98.8%

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

   14. Simplified 98.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 7: 98.5% 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 99.8%

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

   1. associate-*l/ 99.4%

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

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 0 98.3%

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

5. Taylor expanded in x around inf 98.0%

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

   1. div-inv 99.9%

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

7. Applied egg-rr 98.4%

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

   1. associate-*r/ 99.4%

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

   2. *-rgt-identity 99.4%

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

   3. unpow1 99.4%

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

   4. sqr-pow 31.0%

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

   5. fabs-sqr 31.0%

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

   6. sqr-pow 99.4%

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

   7. unpow1 99.4%

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

9. Simplified 98.0%

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

10. Final simplification 98.0%

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

Alternative 8: 94.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75:\\ \;\;\;\;\left|\sqrt{\frac{{x}^{6}}{\pi} \cdot 0.4444444444444444}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.75)
   (fabs (sqrt (* (/ (pow x 6.0) PI) 0.4444444444444444)))
   (fabs (* x (/ 2.0 (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.75) {
		tmp = fabs(sqrt(((pow(x, 6.0) / ((double) M_PI)) * 0.4444444444444444)));
	} else {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.75) {
		tmp = Math.abs(Math.sqrt(((Math.pow(x, 6.0) / Math.PI) * 0.4444444444444444)));
	} else {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.75:
		tmp = math.fabs(math.sqrt(((math.pow(x, 6.0) / math.pi) * 0.4444444444444444)))
	else:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.75)
		tmp = abs(sqrt(Float64(Float64((x ^ 6.0) / pi) * 0.4444444444444444)));
	else
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.75)
		tmp = abs(sqrt((((x ^ 6.0) / pi) * 0.4444444444444444)));
	else
		tmp = abs((x * (2.0 / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.75], N[Abs[N[Sqrt[N[(N[(N[Power[x, 6.0], $MachinePrecision] / Pi), $MachinePrecision] * 0.4444444444444444), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   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 99.9%

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

   4. Taylor expanded in x around inf 72.8%

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

      1. add-sqr-sqrt 72.8%

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

      2. sqrt-unprod 82.8%

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

      3. *-commutative 82.8%

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

      4. *-commutative 82.8%

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

      5. swap-sqr 82.8%

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

      6. add-sqr-sqrt 82.8%

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

      7. pow2 82.8%

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

      8. add-sqr-sqrt 0.0%

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

      9. fabs-sqr 0.0%

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

      10. add-sqr-sqrt 82.8%

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

      11. unpow2 82.8%

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

      12. cube-mult 82.8%

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

      13. pow2 82.8%

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

      14. pow-sqr 82.8%

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

      15. metadata-eval 82.8%

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

   6. Applied egg-rr 82.8%

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

      1. associate-*l/ 82.8%

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

      2. *-lft-identity 82.8%

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

   8. Simplified 82.8%

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

      1. div-inv 82.8%

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

      2. sqrt-prod 82.8%

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

      3. sqrt-pow1 72.8%

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

      4. metadata-eval 72.8%

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

      5. metadata-eval 72.8%

         \[\leadsto \left|0.6666666666666666 \cdot \left({x}^{\color{blue}{\left(1.5 + 1.5\right)}} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      6. pow-prod-up 0.0%

         \[\leadsto \left|0.6666666666666666 \cdot \left(\color{blue}{\left({x}^{1.5} \cdot {x}^{1.5}\right)} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      7. unpow-prod-down 72.8%

         \[\leadsto \left|0.6666666666666666 \cdot \left(\color{blue}{{\left(x \cdot x\right)}^{1.5}} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      8. expm1-log1p-u 72.8%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.6666666666666666 \cdot \left({\left(x \cdot x\right)}^{1.5} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)}\right| \]

      9. expm1-udef 72.8%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(0.6666666666666666 \cdot \left({\left(x \cdot x\right)}^{1.5} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} - 1}\right| \]

   10. Applied egg-rr 82.8%

       \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{{x}^{6}}{\pi} \cdot 0.4444444444444444}\right)} - 1}\right| \]
   11. Step-by-step derivation

       1. expm1-def 82.8%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{{x}^{6}}{\pi} \cdot 0.4444444444444444}\right)\right)}\right| \]

       2. expm1-log1p 82.8%

          \[\leadsto \left|\color{blue}{\sqrt{\frac{{x}^{6}}{\pi} \cdot 0.4444444444444444}}\right| \]

   12. Simplified 82.8%

       \[\leadsto \left|\color{blue}{\sqrt{\frac{{x}^{6}}{\pi} \cdot 0.4444444444444444}}\right| \]

   if -1.75 < x

   1. Initial program 99.8%

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

      1. associate-*l/ 99.2%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around 0 99.8%

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

      1. unpow1 99.8%

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

      2. sqr-pow 48.4%

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

      3. fabs-sqr 48.4%

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

      4. sqr-pow 99.8%

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

      5. unpow1 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in x around inf 99.8%

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

   8. Taylor expanded in x around 0 98.8%

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

      1. associate-*r* 98.8%

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

   10. Simplified 98.8%

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

       1. expm1-log1p-u 98.8%

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

       2. expm1-udef 8.6%

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

   12. Applied egg-rr 8.6%

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

       1. expm1-def 98.0%

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

       2. expm1-log1p 98.0%

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

       3. associate-/r/ 98.8%

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

   14. Simplified 98.8%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75:\\ \;\;\;\;\left|\sqrt{\frac{{x}^{6}}{\pi} \cdot 0.4444444444444444}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 9: 89.0% accurate, 6.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.75)
   (fabs (* 0.6666666666666666 (* (sqrt (/ 1.0 PI)) (* x (* x x)))))
   (fabs (* x (/ 2.0 (sqrt PI))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, -1.75], N[Abs[N[(0.6666666666666666 * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]
   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 99.9%

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

   4. Taylor expanded in x around inf 72.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. fabs-sqr 0.0%

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

      3. add-sqr-sqrt 72.8%

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

      4. unpow2 72.8%

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

      5. cube-mult 72.8%

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

      6. sqr-pow 0.0%

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

      7. pow-prod-down 72.8%

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

      8. metadata-eval 72.8%

         \[\leadsto \left|0.6666666666666666 \cdot \left({\left(x \cdot x\right)}^{\color{blue}{1.5}} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

   6. Applied egg-rr 72.8%

      \[\leadsto \left|0.6666666666666666 \cdot \left(\color{blue}{{\left(x \cdot x\right)}^{1.5}} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]
   7. Step-by-step derivation

      1. pow2 72.8%

         \[\leadsto \left|0.6666666666666666 \cdot \left({\color{blue}{\left({x}^{2}\right)}}^{1.5} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      2. pow-pow 72.8%

         \[\leadsto \left|0.6666666666666666 \cdot \left(\color{blue}{{x}^{\left(2 \cdot 1.5\right)}} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      3. metadata-eval 72.8%

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

      4. pow3 72.8%

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

   8. Applied egg-rr 72.8%

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

   if -1.75 < x

   1. Initial program 99.8%

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

      1. associate-*l/ 99.2%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around 0 99.8%

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

      1. unpow1 99.8%

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

      2. sqr-pow 48.4%

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

      3. fabs-sqr 48.4%

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

      4. sqr-pow 99.8%

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

      5. unpow1 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in x around inf 99.8%

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

   8. Taylor expanded in x around 0 98.8%

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

      1. associate-*r* 98.8%

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

   10. Simplified 98.8%

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

       1. expm1-log1p-u 98.8%

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

       2. expm1-udef 8.6%

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

   12. Applied egg-rr 8.6%

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

       1. expm1-def 98.0%

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

       2. expm1-log1p 98.0%

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

       3. associate-/r/ 98.8%

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

   14. Simplified 98.8%

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

4. Final simplification 89.5%

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

Alternative 10: 83.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-8}:\\ \;\;\;\;\left|\sqrt{\frac{x \cdot \left(x \cdot 4\right)}{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-8)
   (fabs (sqrt (/ (* x (* x 4.0)) PI)))
   (fabs (* x (/ 2.0 (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= -2e-8) {
		tmp = fabs(sqrt(((x * (x * 4.0)) / ((double) M_PI))));
	} else {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -2e-8) {
		tmp = Math.abs(Math.sqrt(((x * (x * 4.0)) / Math.PI)));
	} else {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2e-8:
		tmp = math.fabs(math.sqrt(((x * (x * 4.0)) / math.pi)))
	else:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-8)
		tmp = abs(sqrt(Float64(Float64(x * Float64(x * 4.0)) / pi)));
	else
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2e-8)
		tmp = abs(sqrt(((x * (x * 4.0)) / pi)));
	else
		tmp = abs((x * (2.0 / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2e-8], N[Abs[N[Sqrt[N[(N[(x * N[(x * 4.0), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2e-8

   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%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. unpow1 99.9%

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

      2. sqr-pow 0.0%

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

      3. fabs-sqr 0.0%

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

      4. sqr-pow 99.9%

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

      5. unpow1 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around inf 98.0%

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

   8. Taylor expanded in x around 0 10.2%

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

      1. associate-*r* 10.2%

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

   10. Simplified 10.2%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 56.0%

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

       3. sqrt-div 56.0%

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

       4. metadata-eval 56.0%

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

       5. un-div-inv 56.0%

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

       6. associate-*r/ 56.0%

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

       7. add-sqr-sqrt 0.0%

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

       8. fabs-sqr 0.0%

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

       9. add-sqr-sqrt 0.0%

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

   12. Applied egg-rr 56.0%

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

       1. associate-*r/ 56.0%

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

       2. associate-*r* 56.0%

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

   14. Simplified 56.0%

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

   if -2e-8 < x

   1. Initial program 99.9%

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

      1. associate-*l/ 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. unpow1 99.9%

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

      2. sqr-pow 50.2%

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

      3. fabs-sqr 50.2%

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

      4. sqr-pow 99.9%

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

      5. unpow1 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around inf 99.9%

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

   8. Taylor expanded in x around 0 99.5%

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

      1. associate-*r* 99.5%

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

   10. Simplified 99.5%

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

       1. expm1-log1p-u 99.5%

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

       2. expm1-udef 6.4%

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

   12. Applied egg-rr 6.4%

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

       1. expm1-def 98.7%

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

       2. expm1-log1p 98.7%

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

       3. associate-/r/ 99.5%

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

   14. Simplified 99.5%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-8}:\\ \;\;\;\;\left|\sqrt{\frac{x \cdot \left(x \cdot 4\right)}{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 11: 68.1% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 99.4%

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

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 0 99.9%

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

   1. unpow1 99.9%

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

   2. sqr-pow 31.0%

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

   3. fabs-sqr 31.0%

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

   4. sqr-pow 99.9%

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

   5. unpow1 99.9%

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

6. Simplified 99.9%

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

7. Taylor expanded in x around inf 99.1%

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

8. Taylor expanded in x around 0 65.3%

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

   1. associate-*r* 65.3%

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

10. Simplified 65.3%

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

    1. expm1-log1p-u 63.3%

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

    2. expm1-udef 5.5%

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

12. Applied egg-rr 5.5%

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

    1. expm1-def 62.8%

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

    2. expm1-log1p 64.8%

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

    3. associate-/r/ 65.3%

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

14. Simplified 65.3%

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

15. Final simplification 65.3%

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

Reproduce

herbie shell --seed 2023178 
(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)))))))