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

Percentage Accurate: 99.8% → 99.8%

Time: 17.1s

Alternatives: 10

Speedup: 3.1×

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

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 3.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (* x (pow PI -0.5))
   (+
    (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(((x * pow(((double) M_PI), -0.5)) * (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(x * (pi ^ -0.5)) * 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[(x * N[Power[Pi, -0.5], $MachinePrecision]), $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 99.9%

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

   1. associate-*l/ 99.4%

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

3. Simplified 99.4%

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-udef 35.7%

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

   3. rem-sqrt-square 35.7%

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

   4. sqrt-prod 3.0%

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

   5. add-sqr-sqrt 5.9%

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

5. Applied egg-rr 5.9%

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

   1. expm1-def 69.4%

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

   2. expm1-log1p 99.4%

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

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. div-inv 99.9%

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

   2. pow1/2 99.9%

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

   3. pow-flip 99.9%

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

   4. metadata-eval 99.9%

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

10. Applied egg-rr 99.9%

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

11. Final simplification 99.9%

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

Alternative 3: 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 99.9%

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

   1. associate-*l/ 99.4%

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

3. Simplified 99.4%

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-udef 35.7%

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

   3. rem-sqrt-square 35.7%

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

   4. sqrt-prod 3.0%

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

   5. add-sqr-sqrt 5.9%

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

5. Applied egg-rr 5.9%

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

   1. expm1-def 69.4%

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

   2. expm1-log1p 99.4%

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

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. fma-udef 98.7%

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

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 4: 98.9% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return abs(Float64(Float64(x / sqrt(pi)) * Float64(Float64(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)) * ((0.047619047619047616 * (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[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. 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. expm1-log1p-u 99.2%

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

   2. expm1-udef 35.7%

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

   3. rem-sqrt-square 35.7%

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

   4. sqrt-prod 3.0%

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

   5. add-sqr-sqrt 5.9%

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

5. Applied egg-rr 5.9%

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

   1. expm1-def 69.4%

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

   2. expm1-log1p 99.4%

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

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. Step-by-step derivation

   1. fma-udef 98.7%

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

10. Applied egg-rr 98.7%

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

11. Final simplification 98.7%

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

Alternative 5: 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 99.9%

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

   1. associate-*l/ 99.4%

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

3. Simplified 99.4%

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-udef 35.7%

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

   3. rem-sqrt-square 35.7%

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

   4. sqrt-prod 3.0%

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

   5. add-sqr-sqrt 5.9%

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

5. Applied egg-rr 5.9%

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

   1. expm1-def 69.4%

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

   2. expm1-log1p 99.4%

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

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. Taylor expanded in x around 0 98.4%

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

10. Final simplification 98.4%

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

Alternative 6: 98.9% accurate, 4.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -1.85) {
		tmp = fabs((0.047619047619047616 * (pow(((double) M_PI), -0.5) * pow(x, 7.0))));
	} else {
		tmp = fabs((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.pow(Math.PI, -0.5) * 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.pow(math.pi, -0.5) * 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((pi ^ -0.5) * (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 * ((pi ^ -0.5) * (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[Power[Pi, -0.5], $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 99.9%

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

      1. associate-+l+ 99.8%

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

      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%

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

   4. Taylor expanded in x around inf 97.9%

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

      1. associate-*r* 97.9%

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

   6. Simplified 97.9%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-udef 0.0%

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

      3. associate-*l* 0.0%

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

      4. pow1/2 0.0%

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

      5. inv-pow 0.0%

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

      6. pow-pow 0.0%

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

      7. metadata-eval 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. expm1-def 0.0%

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

      2. expm1-log1p 97.9%

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

   10. Simplified 97.9%

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

   if -1.8500000000000001 < 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+ 99.9%

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

      2. +-commutative 99.9%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.2%

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

   6. Simplified 99.2%

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

      1. expm1-log1p-u 99.2%

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

      2. expm1-udef 8.1%

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

      3. *-commutative 8.1%

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

      4. pow1/2 8.1%

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

      5. inv-pow 8.1%

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

      6. pow-pow 8.1%

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

      7. metadata-eval 8.1%

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

   8. Applied egg-rr 8.1%

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

      1. expm1-def 99.2%

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

      2. expm1-log1p 99.2%

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

      3. *-commutative 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-*l* 99.2%

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

   10. Simplified 99.2%

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

       1. associate-*r* 99.2%

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

       2. *-commutative 99.2%

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

       3. *-commutative 99.2%

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

       4. expm1-log1p-u 99.2%

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

       5. expm1-udef 8.1%

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

   12. Applied egg-rr 8.1%

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

       1. expm1-def 98.6%

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

       2. expm1-log1p 98.6%

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

       3. associate-*l/ 98.6%

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

       4. *-lft-identity 98.6%

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

       5. times-frac 99.2%

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

       6. /-rgt-identity 99.2%

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

   14. Simplified 99.2%

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

4. Final simplification 98.8%

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

Alternative 7: 96.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85:\\ \;\;\;\;\left|\sqrt{\frac{{x}^{14}}{\pi} \cdot 0.0022675736961451248}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.85)
   (fabs (sqrt (* (/ (pow x 14.0) PI) 0.0022675736961451248)))
   (fabs (* x (/ 2.0 (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.85) {
		tmp = fabs(sqrt(((pow(x, 14.0) / ((double) M_PI)) * 0.0022675736961451248)));
	} 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(((Math.pow(x, 14.0) / Math.PI) * 0.0022675736961451248)));
	} 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(((math.pow(x, 14.0) / math.pi) * 0.0022675736961451248)))
	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(sqrt(Float64(Float64((x ^ 14.0) / pi) * 0.0022675736961451248)));
	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((((x ^ 14.0) / pi) * 0.0022675736961451248)));
	else
		tmp = abs((x * (2.0 / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.85], N[Abs[N[Sqrt[N[(N[(N[Power[x, 14.0], $MachinePrecision] / Pi), $MachinePrecision] * 0.0022675736961451248), $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 99.9%

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

      1. associate-+l+ 99.8%

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

      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%

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

   4. Taylor expanded in x around inf 97.9%

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

      1. associate-*r* 97.9%

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

   6. Simplified 97.9%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-udef 0.0%

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

      3. associate-*l* 0.0%

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

      4. pow1/2 0.0%

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

      5. inv-pow 0.0%

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

      6. pow-pow 0.0%

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

      7. metadata-eval 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. expm1-def 0.0%

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

      2. expm1-log1p 97.9%

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

   10. Simplified 97.9%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 93.1%

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

       3. *-commutative 93.1%

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

       4. *-commutative 93.1%

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

       5. swap-sqr 93.1%

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

       6. *-commutative 93.1%

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

       7. *-commutative 93.1%

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

       8. swap-sqr 93.1%

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

       9. pow-prod-up 93.1%

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

       10. metadata-eval 93.1%

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

       11. inv-pow 93.1%

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

       12. pow-prod-up 93.1%

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

       13. metadata-eval 93.1%

           \[\leadsto \left|\sqrt{\left(\frac{1}{\pi} \cdot {x}^{\color{blue}{14}}\right) \cdot \left(0.047619047619047616 \cdot 0.047619047619047616\right)}\right| \]

       14. metadata-eval 93.1%

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

   12. Applied egg-rr 93.1%

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

       1. associate-*l/ 93.1%

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

       2. *-lft-identity 93.1%

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

   14. Simplified 93.1%

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

   if -1.8500000000000001 < 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+ 99.9%

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

      2. +-commutative 99.9%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.2%

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

   6. Simplified 99.2%

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

      1. expm1-log1p-u 99.2%

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

      2. expm1-udef 8.1%

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

      3. *-commutative 8.1%

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

      4. pow1/2 8.1%

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

      5. inv-pow 8.1%

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

      6. pow-pow 8.1%

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

      7. metadata-eval 8.1%

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

   8. Applied egg-rr 8.1%

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

      1. expm1-def 99.2%

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

      2. expm1-log1p 99.2%

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

      3. *-commutative 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-*l* 99.2%

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

   10. Simplified 99.2%

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

       1. associate-*r* 99.2%

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

       2. *-commutative 99.2%

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

       3. *-commutative 99.2%

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

       4. expm1-log1p-u 99.2%

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

       5. expm1-udef 8.1%

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

   12. Applied egg-rr 8.1%

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

       1. expm1-def 98.6%

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

       2. expm1-log1p 98.6%

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

       3. associate-*l/ 98.6%

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

       4. *-lft-identity 98.6%

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

       5. times-frac 99.2%

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

       6. /-rgt-identity 99.2%

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

   14. Simplified 99.2%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85:\\ \;\;\;\;\left|\sqrt{\frac{{x}^{14}}{\pi} \cdot 0.0022675736961451248}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 8: 96.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85:\\ \;\;\;\;\left|0.047619047619047616 \cdot \sqrt{\frac{{x}^{14}}{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.85)
   (fabs (* 0.047619047619047616 (sqrt (/ (pow x 14.0) PI))))
   (fabs (* x (/ 2.0 (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.85) {
		tmp = fabs((0.047619047619047616 * sqrt((pow(x, 14.0) / ((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 <= -1.85) {
		tmp = Math.abs((0.047619047619047616 * Math.sqrt((Math.pow(x, 14.0) / Math.PI))));
	} 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((math.pow(x, 14.0) / math.pi))))
	else:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.85)
		tmp = abs(Float64(0.047619047619047616 * sqrt(Float64((x ^ 14.0) / 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 <= -1.85)
		tmp = abs((0.047619047619047616 * sqrt(((x ^ 14.0) / pi))));
	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[Sqrt[N[(N[Power[x, 14.0], $MachinePrecision] / Pi), $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 99.9%

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

      1. associate-+l+ 99.8%

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

      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%

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

   4. Taylor expanded in x around inf 97.9%

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

      1. associate-*r* 97.9%

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

   6. Simplified 97.9%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-udef 0.0%

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

      3. associate-*l* 0.0%

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

      4. pow1/2 0.0%

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

      5. inv-pow 0.0%

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

      6. pow-pow 0.0%

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

      7. metadata-eval 0.0%

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

   8. Applied egg-rr 0.0%

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

      1. expm1-def 0.0%

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

      2. expm1-log1p 97.9%

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

   10. Simplified 97.9%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 93.1%

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

       3. *-commutative 93.1%

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

       4. *-commutative 93.1%

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

       5. swap-sqr 93.1%

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

       6. pow-prod-up 93.2%

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

       7. metadata-eval 93.2%

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

       8. inv-pow 93.2%

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

       9. pow-prod-up 93.2%

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

       10. metadata-eval 93.2%

           \[\leadsto \left|0.047619047619047616 \cdot \sqrt{\frac{1}{\pi} \cdot {x}^{\color{blue}{14}}}\right| \]

   12. Applied egg-rr 93.2%

       \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\sqrt{\frac{1}{\pi} \cdot {x}^{14}}}\right| \]
   13. Step-by-step derivation

       1. associate-*l/ 93.2%

          \[\leadsto \left|0.047619047619047616 \cdot \sqrt{\color{blue}{\frac{1 \cdot {x}^{14}}{\pi}}}\right| \]

       2. *-lft-identity 93.2%

          \[\leadsto \left|0.047619047619047616 \cdot \sqrt{\frac{\color{blue}{{x}^{14}}}{\pi}}\right| \]

   14. Simplified 93.2%

       \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\sqrt{\frac{{x}^{14}}{\pi}}}\right| \]

   if -1.8500000000000001 < 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+ 99.9%

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

      2. +-commutative 99.9%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.2%

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

   6. Simplified 99.2%

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

      1. expm1-log1p-u 99.2%

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

      2. expm1-udef 8.1%

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

      3. *-commutative 8.1%

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

      4. pow1/2 8.1%

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

      5. inv-pow 8.1%

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

      6. pow-pow 8.1%

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

      7. metadata-eval 8.1%

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

   8. Applied egg-rr 8.1%

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

      1. expm1-def 99.2%

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

      2. expm1-log1p 99.2%

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

      3. *-commutative 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-*l* 99.2%

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

   10. Simplified 99.2%

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

       1. associate-*r* 99.2%

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

       2. *-commutative 99.2%

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

       3. *-commutative 99.2%

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

       4. expm1-log1p-u 99.2%

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

       5. expm1-udef 8.1%

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

   12. Applied egg-rr 8.1%

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

       1. expm1-def 98.6%

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

       2. expm1-log1p 98.6%

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

       3. associate-*l/ 98.6%

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

       4. *-lft-identity 98.6%

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

       5. times-frac 99.2%

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

       6. /-rgt-identity 99.2%

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

   14. Simplified 99.2%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85:\\ \;\;\;\;\left|0.047619047619047616 \cdot \sqrt{\frac{{x}^{14}}{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 9: 83.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-7}:\\ \;\;\;\;\left|\sqrt{\frac{x \cdot \left(x \cdot 4\right)}{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4e-7)
   (fabs (sqrt (/ (* x (* x 4.0)) PI)))
   (fabs (* x (/ 2.0 (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= -4e-7) {
		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 <= -4e-7) {
		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 <= -4e-7:
		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 <= -4e-7)
		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 <= -4e-7)
		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, -4e-7], 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 < -3.9999999999999998e-7

   1. Initial program 99.9%

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

      1. associate-+l+ 99.9%

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

      2. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 8.8%

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

      1. *-commutative 8.8%

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

      2. associate-*l* 8.8%

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

   6. Simplified 8.8%

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

      1. expm1-log1p-u 3.0%

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

      2. expm1-udef 2.7%

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

      3. *-commutative 2.7%

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

      4. pow1/2 2.7%

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

      5. inv-pow 2.7%

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

      6. pow-pow 2.7%

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

      7. metadata-eval 2.7%

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

   8. Applied egg-rr 2.7%

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

      1. expm1-def 3.0%

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

      2. expm1-log1p 8.8%

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

      3. *-commutative 8.8%

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

      4. *-commutative 8.8%

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

      5. associate-*l* 8.8%

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

   10. Simplified 8.8%

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

       1. associate-*r* 8.8%

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

       2. *-commutative 8.8%

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

       3. *-commutative 8.8%

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

       4. expm1-log1p-u 3.0%

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

       5. expm1-udef 2.7%

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

   12. Applied egg-rr 2.7%

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

       1. expm1-def 3.0%

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

       2. expm1-log1p 8.8%

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

       3. associate-*l/ 8.8%

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

       4. *-lft-identity 8.8%

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

       5. times-frac 8.8%

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

       6. /-rgt-identity 8.8%

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

   14. Simplified 8.8%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 51.6%

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

       3. swap-sqr 51.6%

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

       4. frac-times 51.6%

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

       5. metadata-eval 51.6%

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

       6. add-sqr-sqrt 51.6%

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

   16. Applied egg-rr 51.6%

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

       1. associate-*r/ 51.6%

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

       2. associate-*l* 51.6%

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

   18. Simplified 51.6%

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

   if -3.9999999999999998e-7 < 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+ 99.9%

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

      2. +-commutative 99.9%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.5%

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

   6. Simplified 99.5%

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

      1. expm1-log1p-u 99.5%

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

      2. expm1-udef 7.0%

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

      3. *-commutative 7.0%

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

      4. pow1/2 7.0%

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

      5. inv-pow 7.0%

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

      6. pow-pow 7.0%

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

      7. metadata-eval 7.0%

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

   8. Applied egg-rr 7.0%

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

      1. expm1-def 99.5%

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

      2. expm1-log1p 99.5%

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

      3. *-commutative 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-*l* 99.5%

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

   10. Simplified 99.5%

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

       1. associate-*r* 99.5%

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

       2. *-commutative 99.5%

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

       3. *-commutative 99.5%

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

       4. expm1-log1p-u 99.5%

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

       5. expm1-udef 7.0%

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

   12. Applied egg-rr 7.0%

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

       1. expm1-def 98.9%

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

       2. expm1-log1p 98.9%

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

       3. associate-*l/ 98.9%

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

       4. *-lft-identity 98.9%

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

       5. times-frac 99.5%

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

       6. /-rgt-identity 99.5%

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

   14. Simplified 99.5%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-7}:\\ \;\;\;\;\left|\sqrt{\frac{x \cdot \left(x \cdot 4\right)}{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 10: 67.7% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-+l+ 99.9%

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

   2. +-commutative 99.9%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around 0 71.2%

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

   1. *-commutative 71.2%

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

   2. associate-*l* 71.2%

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

6. Simplified 71.2%

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

   1. expm1-log1p-u 69.3%

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

   2. expm1-udef 5.7%

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

   3. *-commutative 5.7%

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

   4. pow1/2 5.7%

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

   5. inv-pow 5.7%

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

   6. pow-pow 5.7%

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

   7. metadata-eval 5.7%

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

8. Applied egg-rr 5.7%

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

   1. expm1-def 69.3%

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

   2. expm1-log1p 71.2%

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

   3. *-commutative 71.2%

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

   4. *-commutative 71.2%

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

   5. associate-*l* 71.2%

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

10. Simplified 71.2%

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

    1. associate-*r* 71.2%

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

    2. *-commutative 71.2%

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

    3. *-commutative 71.2%

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

    4. expm1-log1p-u 69.3%

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

    5. expm1-udef 5.7%

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

12. Applied egg-rr 5.7%

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

    1. expm1-def 68.9%

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

    2. expm1-log1p 70.7%

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

    3. associate-*l/ 70.7%

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

    4. *-lft-identity 70.7%

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

    5. times-frac 71.2%

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

    6. /-rgt-identity 71.2%

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

14. Simplified 71.2%

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

15. Final simplification 71.2%

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

Reproduce

herbie shell --seed 2023187 
(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)))))))