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

Percentage Accurate: 99.8% → 99.8%

Time: 9.6s

Alternatives: 12

Speedup: N/A×

• 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, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \left|\frac{\mathsf{fma}\left(2, x, \left(0.6666666666666666 \cdot {x}^{3} + 0.047619047619047616 \cdot {x}^{7}\right) + 0.2 \cdot {x}^{5}\right)}{\sqrt{\pi}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (/
   (fma
    2.0
    x
    (+
     (+
      (* 0.6666666666666666 (pow x 3.0))
      (* 0.047619047619047616 (pow x 7.0)))
     (* 0.2 (pow x 5.0))))
   (sqrt PI))))

C

double code(double x) {
	return fabs((fma(2.0, x, (((0.6666666666666666 * pow(x, 3.0)) + (0.047619047619047616 * pow(x, 7.0))) + (0.2 * pow(x, 5.0)))) / sqrt(((double) M_PI))));
}

Julia

function code(x)
	return abs(Float64(fma(2.0, x, Float64(Float64(Float64(0.6666666666666666 * (x ^ 3.0)) + Float64(0.047619047619047616 * (x ^ 7.0))) + Float64(0.2 * (x ^ 5.0)))) / sqrt(pi)))
end

Wolfram

code[x_] := N[Abs[N[(N[(2.0 * x + N[(N[(N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

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. Step-by-step derivation

   1. fma-udef 99.4%

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

   2. fma-udef 99.4%

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

   3. associate-+r+ 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 3: 99.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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 99.0%

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

   2. expm1-udef 34.8%

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

   3. add-sqr-sqrt 3.2%

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

   4. fabs-sqr 3.2%

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

   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 70.1%

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

   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. Step-by-step derivation

   1. fma-udef 99.4%

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

9. Applied egg-rr 99.4%

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

10. Final simplification 99.4%

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

Alternative 4: 98.6% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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 99.0%

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

   2. expm1-udef 34.8%

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

   3. add-sqr-sqrt 3.2%

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

   4. fabs-sqr 3.2%

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

   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 70.1%

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

   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 98.5%

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

9. Final simplification 98.5%

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

Alternative 5: 98.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \left|\frac{\mathsf{fma}\left(2, x, 0.047619047619047616 \cdot {x}^{7} + 0.2 \cdot {x}^{5}\right)}{\sqrt{\pi}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (/
   (fma 2.0 x (+ (* 0.047619047619047616 (pow x 7.0)) (* 0.2 (pow x 5.0))))
   (sqrt PI))))

C

double code(double x) {
	return fabs((fma(2.0, x, ((0.047619047619047616 * pow(x, 7.0)) + (0.2 * pow(x, 5.0)))) / sqrt(((double) M_PI))));
}

Julia

function code(x)
	return abs(Float64(fma(2.0, x, Float64(Float64(0.047619047619047616 * (x ^ 7.0)) + Float64(0.2 * (x ^ 5.0)))) / sqrt(pi)))
end

Wolfram

code[x_] := N[Abs[N[(N[(2.0 * x + N[(N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

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 inf 98.5%

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

5. Final simplification 98.5%

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

Alternative 6: 98.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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 inf 98.1%

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

   1. expm1-log1p-u 69.8%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(2, x, 0.047619047619047616 \cdot {x}^{7}\right)}{\sqrt{\pi}}\right)\right)}\right| \]

   2. expm1-udef 5.7%

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

6. Applied egg-rr 5.7%

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

   1. expm1-def 69.8%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(2, x, 0.047619047619047616 \cdot {x}^{7}\right)}{\sqrt{\pi}}\right)\right)}\right| \]

   2. expm1-log1p 98.1%

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

   3. fma-udef 98.1%

      \[\leadsto \left|\frac{\color{blue}{2 \cdot x + 0.047619047619047616 \cdot {x}^{7}}}{\sqrt{\pi}}\right| \]

   4. +-commutative 98.1%

      \[\leadsto \left|\frac{\color{blue}{0.047619047619047616 \cdot {x}^{7} + 2 \cdot x}}{\sqrt{\pi}}\right| \]

   5. fma-def 98.1%

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

   6. count-2 98.1%

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

8. Simplified 98.1%

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

9. Final simplification 98.1%

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

Alternative 7: 89.3% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 85.9%

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

      1. +-commutative 85.9%

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

      2. associate-*r* 85.9%

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

      3. associate-*r* 85.9%

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

      4. distribute-rgt-out 85.9%

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

      5. *-commutative 85.9%

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

   6. Simplified 85.9%

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

   if 2.2000000000000002 < x

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around inf 32.2%

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

      1. associate-*r* 32.2%

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

   6. Simplified 32.2%

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

      1. add-exp-log 3.6%

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

      2. log-pow 2.0%

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

   8. Applied egg-rr 2.0%

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

      1. expm1-log1p-u 2.0%

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

      2. expm1-udef 1.9%

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

      3. associate-*l* 1.9%

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

      4. sqrt-div 1.9%

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

      5. metadata-eval 1.9%

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

      6. un-div-inv 1.9%

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

      7. *-commutative 1.9%

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

      8. exp-to-pow 3.7%

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

   10. Applied egg-rr 3.7%

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

       1. expm1-def 3.9%

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

       2. expm1-log1p 32.2%

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

       3. associate-*r/ 32.2%

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

       4. *-commutative 32.2%

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

   12. Simplified 32.2%

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

4. Final simplification 85.9%

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

Alternative 8: 67.4% accurate, 4.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = fabs((2.0 * (x * pow(((double) M_PI), -0.5))));
	} else {
		tmp = fabs((0.047619047619047616 * (pow(x, 7.0) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = Math.abs((2.0 * (x * Math.pow(Math.PI, -0.5))));
	} else {
		tmp = Math.abs((0.047619047619047616 * (Math.pow(x, 7.0) / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = math.fabs((2.0 * (x * math.pow(math.pi, -0.5))))
	else:
		tmp = math.fabs((0.047619047619047616 * (math.pow(x, 7.0) / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = abs(Float64(2.0 * Float64(x * (pi ^ -0.5))));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64((x ^ 7.0) / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = abs((2.0 * (x * (pi ^ -0.5))));
	else
		tmp = abs((0.047619047619047616 * ((x ^ 7.0) / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[Abs[N[(2.0 * N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[Power[x, 7.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   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 71.9%

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

      1. associate-*r* 71.9%

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

   6. Simplified 71.9%

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

      1. expm1-log1p-u 70.3%

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

      2. expm1-udef 5.7%

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

      3. associate-*l* 5.7%

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

      4. pow1/2 5.7%

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

      5. inv-pow 5.7%

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

      6. pow-pow 5.7%

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

      7. metadata-eval 5.7%

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

   8. Applied egg-rr 5.7%

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

      1. expm1-def 70.3%

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

      2. expm1-log1p 71.9%

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

   10. Simplified 71.9%

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

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   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 inf 32.2%

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

      1. associate-*r* 32.2%

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

   6. Simplified 32.2%

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

      1. add-exp-log 3.6%

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

      2. log-pow 2.0%

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

   8. Applied egg-rr 2.0%

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

      1. expm1-log1p-u 2.0%

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

      2. expm1-udef 1.9%

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

      3. associate-*l* 1.9%

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

      4. sqrt-div 1.9%

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

      5. metadata-eval 1.9%

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

      6. un-div-inv 1.9%

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

      7. *-commutative 1.9%

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

      8. exp-to-pow 3.7%

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

   10. Applied egg-rr 3.7%

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

       1. expm1-def 3.9%

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

       2. expm1-log1p 32.2%

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

   12. Simplified 32.2%

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

4. Final simplification 71.9%

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

Alternative 9: 67.4% accurate, 4.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = fabs((2.0 * (x * pow(((double) M_PI), -0.5))));
	} else {
		tmp = fabs(((0.047619047619047616 * pow(x, 7.0)) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = Math.abs((2.0 * (x * Math.pow(Math.PI, -0.5))));
	} else {
		tmp = Math.abs(((0.047619047619047616 * Math.pow(x, 7.0)) / Math.sqrt(Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = math.fabs((2.0 * (x * math.pow(math.pi, -0.5))))
	else:
		tmp = math.fabs(((0.047619047619047616 * math.pow(x, 7.0)) / math.sqrt(math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = abs(Float64(2.0 * Float64(x * (pi ^ -0.5))));
	else
		tmp = abs(Float64(Float64(0.047619047619047616 * (x ^ 7.0)) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = abs((2.0 * (x * (pi ^ -0.5))));
	else
		tmp = abs(((0.047619047619047616 * (x ^ 7.0)) / sqrt(pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[Abs[N[(2.0 * N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   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 71.9%

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

      1. associate-*r* 71.9%

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

   6. Simplified 71.9%

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

      1. expm1-log1p-u 70.3%

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

      2. expm1-udef 5.7%

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

      3. associate-*l* 5.7%

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

      4. pow1/2 5.7%

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

      5. inv-pow 5.7%

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

      6. pow-pow 5.7%

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

      7. metadata-eval 5.7%

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

   8. Applied egg-rr 5.7%

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

      1. expm1-def 70.3%

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

      2. expm1-log1p 71.9%

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

   10. Simplified 71.9%

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

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   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 inf 32.2%

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

      1. associate-*r* 32.2%

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

   6. Simplified 32.2%

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

      1. add-exp-log 3.6%

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

      2. log-pow 2.0%

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

   8. Applied egg-rr 2.0%

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

      1. expm1-log1p-u 2.0%

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

      2. expm1-udef 1.9%

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

      3. associate-*l* 1.9%

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

      4. sqrt-div 1.9%

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

      5. metadata-eval 1.9%

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

      6. un-div-inv 1.9%

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

      7. *-commutative 1.9%

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

      8. exp-to-pow 3.7%

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

   10. Applied egg-rr 3.7%

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

       1. expm1-def 3.9%

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

       2. expm1-log1p 32.2%

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

       3. associate-*r/ 32.2%

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

       4. *-commutative 32.2%

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

   12. Simplified 32.2%

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

4. Final simplification 71.9%

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

Alternative 10: 67.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10000000000000:\\ \;\;\;\;\left|2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{x \cdot \left(x \cdot 4\right)}{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 10000000000000.0)
   (fabs (* 2.0 (* x (pow PI -0.5))))
   (fabs (sqrt (/ (* x (* x 4.0)) PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 10000000000000.0) {
		tmp = fabs((2.0 * (x * pow(((double) M_PI), -0.5))));
	} else {
		tmp = fabs(sqrt(((x * (x * 4.0)) / ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 10000000000000.0) {
		tmp = Math.abs((2.0 * (x * Math.pow(Math.PI, -0.5))));
	} else {
		tmp = Math.abs(Math.sqrt(((x * (x * 4.0)) / Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 10000000000000.0:
		tmp = math.fabs((2.0 * (x * math.pow(math.pi, -0.5))))
	else:
		tmp = math.fabs(math.sqrt(((x * (x * 4.0)) / math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 10000000000000.0)
		tmp = abs(Float64(2.0 * Float64(x * (pi ^ -0.5))));
	else
		tmp = abs(sqrt(Float64(Float64(x * Float64(x * 4.0)) / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 10000000000000.0)
		tmp = abs((2.0 * (x * (pi ^ -0.5))));
	else
		tmp = abs(sqrt(((x * (x * 4.0)) / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 10000000000000.0], N[Abs[N[(2.0 * N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[Sqrt[N[(N[(x * N[(x * 4.0), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e13

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 71.9%

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

      1. associate-*r* 71.9%

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

   6. Simplified 71.9%

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

      1. expm1-log1p-u 70.3%

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

      2. expm1-udef 5.7%

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

      3. associate-*l* 5.7%

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

      4. pow1/2 5.7%

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

      5. inv-pow 5.7%

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

      6. pow-pow 5.7%

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

      7. metadata-eval 5.7%

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

   8. Applied egg-rr 5.7%

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

      1. expm1-def 70.3%

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

      2. expm1-log1p 71.9%

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

   10. Simplified 71.9%

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

   if 1e13 < x

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 71.9%

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

      1. associate-*r* 71.9%

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

   6. Simplified 71.9%

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

      1. add-cube-cbrt 70.3%

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

      2. pow3 70.3%

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

      3. associate-*l* 70.3%

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

      4. pow1/2 70.3%

         \[\leadsto \left|{\left(\sqrt[3]{2 \cdot \left(x \cdot \color{blue}{{\left(\frac{1}{\pi}\right)}^{0.5}}\right)}\right)}^{3}\right| \]

      5. inv-pow 70.3%

         \[\leadsto \left|{\left(\sqrt[3]{2 \cdot \left(x \cdot {\color{blue}{\left({\pi}^{-1}\right)}}^{0.5}\right)}\right)}^{3}\right| \]

      6. pow-pow 70.3%

         \[\leadsto \left|{\left(\sqrt[3]{2 \cdot \left(x \cdot \color{blue}{{\pi}^{\left(-1 \cdot 0.5\right)}}\right)}\right)}^{3}\right| \]

      7. metadata-eval 70.3%

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

   8. Applied egg-rr 70.3%

      \[\leadsto \left|\color{blue}{{\left(\sqrt[3]{2 \cdot \left(x \cdot {\pi}^{-0.5}\right)}\right)}^{3}}\right| \]
   9. Step-by-step derivation

      1. rem-cube-cbrt 71.9%

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

      2. add-sqr-sqrt 35.0%

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

      3. sqrt-unprod 52.8%

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

      4. *-commutative 52.8%

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

      5. *-commutative 52.8%

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

      6. swap-sqr 52.6%

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

      7. pow2 52.6%

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

      8. metadata-eval 52.6%

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

      9. pow-flip 52.6%

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

      10. pow1/2 52.6%

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

      11. div-inv 52.4%

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

      12. pow2 52.4%

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

      13. frac-times 52.3%

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

      14. add-sqr-sqrt 52.6%

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

      15. metadata-eval 52.6%

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

   10. Applied egg-rr 52.6%

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

       1. associate-*l/ 52.7%

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

       2. associate-*l* 52.7%

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

   12. Simplified 52.7%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10000000000000:\\ \;\;\;\;\left|2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{x \cdot \left(x \cdot 4\right)}{\pi}}\right|\\ \end{array} \]

Alternative 11: 67.4% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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 71.9%

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

   1. associate-*r* 71.9%

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

6. Simplified 71.9%

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

   1. expm1-log1p-u 70.3%

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

   2. expm1-udef 5.7%

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

   3. associate-*l* 5.7%

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

   4. pow1/2 5.7%

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

   5. inv-pow 5.7%

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

   6. pow-pow 5.7%

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

   7. metadata-eval 5.7%

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

8. Applied egg-rr 5.7%

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

   1. expm1-def 70.3%

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

   2. expm1-log1p 71.9%

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

10. Simplified 71.9%

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

11. Final simplification 71.9%

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

Alternative 12: 66.9% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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 71.9%

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

   1. associate-*r* 71.9%

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

6. Simplified 71.9%

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

   1. add-cube-cbrt 70.3%

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

   2. pow3 70.3%

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

   3. associate-*l* 70.3%

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

   4. pow1/2 70.3%

      \[\leadsto \left|{\left(\sqrt[3]{2 \cdot \left(x \cdot \color{blue}{{\left(\frac{1}{\pi}\right)}^{0.5}}\right)}\right)}^{3}\right| \]

   5. inv-pow 70.3%

      \[\leadsto \left|{\left(\sqrt[3]{2 \cdot \left(x \cdot {\color{blue}{\left({\pi}^{-1}\right)}}^{0.5}\right)}\right)}^{3}\right| \]

   6. pow-pow 70.3%

      \[\leadsto \left|{\left(\sqrt[3]{2 \cdot \left(x \cdot \color{blue}{{\pi}^{\left(-1 \cdot 0.5\right)}}\right)}\right)}^{3}\right| \]

   7. metadata-eval 70.3%

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

8. Applied egg-rr 70.3%

   \[\leadsto \left|\color{blue}{{\left(\sqrt[3]{2 \cdot \left(x \cdot {\pi}^{-0.5}\right)}\right)}^{3}}\right| \]
9. Step-by-step derivation

   1. rem-cube-cbrt 71.9%

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

   2. expm1-log1p-u 70.3%

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

   3. expm1-udef 5.7%

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

   4. *-commutative 5.7%

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

   5. metadata-eval 5.7%

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

   6. pow-flip 5.7%

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

   7. pow1/2 5.7%

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

   8. div-inv 5.7%

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

10. Applied egg-rr 5.7%

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

    1. expm1-def 69.8%

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

    2. expm1-log1p 71.4%

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

    3. associate-*l/ 71.4%

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

12. Simplified 71.4%

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

13. Final simplification 71.4%

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

Reproduce

herbie shell --seed 2023217 
(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)))))))