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

Percentage Accurate: 99.8% → 99.8%

Time: 11.1s

Alternatives: 11

Speedup: 3.0×

• 29.3% 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 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 99.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

   1. fma-undefine 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 3: 99.3% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around inf 99.5%

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

Alternative 4: 99.0% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around inf 99.5%

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

6. Taylor expanded in x around 0 99.0%

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

7. Final simplification 99.0%

   \[\leadsto \left|x\right| \cdot \left|\frac{2 + 0.047619047619047616 \cdot {x}^{6}}{\sqrt{\pi}}\right| \]
8. Add Preprocessing

Alternative 5: 98.6% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Taylor expanded in x around inf 99.5%

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

6. Taylor expanded in x around 0 99.0%

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

   1. add-sqr-sqrt 30.5%

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

   2. fabs-sqr 30.5%

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

   3. add-sqr-sqrt 99.0%

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

   4. clear-num 99.1%

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

   5. un-div-inv 98.7%

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

   6. fma-define 98.7%

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

8. Applied egg-rr 98.7%

   \[\leadsto \left|\color{blue}{\frac{x}{\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 2\right)}}}\right| \]
9. Add Preprocessing

Alternative 6: 89.5% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 90.3%

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

      1. +-commutative 90.3%

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

      2. fma-define 90.3%

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

      3. rem-square-sqrt 30.6%

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

      4. fabs-sqr 30.6%

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

      5. rem-square-sqrt 89.8%

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

      6. *-commutative 89.8%

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

      7. *-commutative 89.8%

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

      8. *-commutative 89.8%

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

      9. unpow2 89.8%

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

      10. rem-square-sqrt 30.8%

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

      11. fabs-sqr 30.8%

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

      12. rem-square-sqrt 90.3%

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

      13. cube-mult 90.3%

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

      14. associate-*l* 90.3%

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

      15. *-commutative 90.3%

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

   7. Simplified 90.3%

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

   8. Taylor expanded in x around 0 90.3%

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

      1. associate-*r* 90.3%

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

      2. distribute-rgt-out 90.3%

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

      3. fma-define 90.3%

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

   10. Simplified 90.3%

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

       1. fma-undefine 90.3%

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

   12. Applied egg-rr 90.3%

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

   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 99.8%

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

   5. Taylor expanded in x around inf 33.9%

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

      1. rem-square-sqrt 1.9%

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

      2. fabs-sqr 1.9%

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

      3. rem-square-sqrt 33.9%

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

      4. pow-plus 33.9%

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

      5. metadata-eval 33.9%

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

      6. rem-square-sqrt 1.9%

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

      7. fabs-sqr 1.9%

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

      8. rem-square-sqrt 33.9%

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

      9. associate-*l* 33.9%

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

      10. *-commutative 33.9%

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

      11. rem-square-sqrt 1.9%

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

      12. fabs-sqr 1.9%

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

      13. rem-square-sqrt 33.9%

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

   7. Simplified 33.9%

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

      1. inv-pow 33.9%

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

      2. sqrt-pow1 33.9%

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

      3. metadata-eval 33.9%

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

   9. Applied egg-rr 33.9%

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

4. Final simplification 90.3%

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

Alternative 7: 67.5% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.8600000000000001

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 70.9%

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

      1. rem-square-sqrt 30.5%

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

      2. fabs-sqr 30.5%

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

      3. rem-square-sqrt 70.9%

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

   7. Simplified 70.9%

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

      1. inv-pow 70.9%

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

      2. sqrt-pow1 70.9%

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

      3. metadata-eval 70.9%

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

   9. Applied egg-rr 70.9%

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

   if 1.8600000000000001 < 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 99.8%

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

   5. Taylor expanded in x around inf 33.9%

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

      1. rem-square-sqrt 1.9%

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

      2. fabs-sqr 1.9%

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

      3. rem-square-sqrt 33.9%

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

      4. pow-plus 33.9%

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

      5. metadata-eval 33.9%

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

      6. rem-square-sqrt 1.9%

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

      7. fabs-sqr 1.9%

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

      8. rem-square-sqrt 33.9%

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

      9. associate-*l* 33.9%

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

      10. *-commutative 33.9%

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

      11. rem-square-sqrt 1.9%

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

      12. fabs-sqr 1.9%

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

      13. rem-square-sqrt 33.9%

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

   7. Simplified 33.9%

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

      1. inv-pow 33.9%

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

      2. sqrt-pow1 33.9%

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

      3. metadata-eval 33.9%

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

   9. Applied egg-rr 33.9%

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

4. Final simplification 70.9%

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

Alternative 8: 67.5% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.8600000000000001

   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%

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

   5. Taylor expanded in x around 0 70.9%

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

      1. rem-square-sqrt 30.5%

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

      2. fabs-sqr 30.5%

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

      3. rem-square-sqrt 70.9%

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

   7. Simplified 70.9%

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

      1. inv-pow 70.9%

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

      2. sqrt-pow1 70.9%

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

      3. metadata-eval 70.9%

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

   9. Applied egg-rr 70.9%

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

   if 1.8600000000000001 < 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 99.8%

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

   5. Taylor expanded in x around inf 33.9%

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

      1. rem-square-sqrt 1.9%

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

      2. fabs-sqr 1.9%

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

      3. rem-square-sqrt 33.9%

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

      4. pow-plus 33.9%

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

      5. metadata-eval 33.9%

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

      6. rem-square-sqrt 1.9%

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

      7. fabs-sqr 1.9%

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

      8. rem-square-sqrt 33.9%

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

      9. associate-*l* 33.9%

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

      10. *-commutative 33.9%

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

      11. rem-square-sqrt 1.9%

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

      12. fabs-sqr 1.9%

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

      13. rem-square-sqrt 33.9%

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

   7. Simplified 33.9%

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

      1. associate-*r* 33.9%

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

      2. *-commutative 33.9%

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

      3. sqrt-div 33.9%

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

      4. metadata-eval 33.9%

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

      5. associate-*l/ 33.9%

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

      6. metadata-eval 33.9%

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

   9. Applied egg-rr 33.9%

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

4. Final simplification 70.9%

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

Alternative 9: 67.5% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-61}:\\ \;\;\;\;\left|2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|2 \cdot \sqrt{\frac{{x}^{2}}{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4e-61)
   (fabs (* 2.0 (* x (pow PI -0.5))))
   (fabs (* 2.0 (sqrt (/ (pow x 2.0) PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 4e-61) {
		tmp = fabs((2.0 * (x * pow(((double) M_PI), -0.5))));
	} else {
		tmp = fabs((2.0 * sqrt((pow(x, 2.0) / ((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 4e-61) {
		tmp = Math.abs((2.0 * (x * Math.pow(Math.PI, -0.5))));
	} else {
		tmp = Math.abs((2.0 * Math.sqrt((Math.pow(x, 2.0) / Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4e-61:
		tmp = math.fabs((2.0 * (x * math.pow(math.pi, -0.5))))
	else:
		tmp = math.fabs((2.0 * math.sqrt((math.pow(x, 2.0) / math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4e-61)
		tmp = abs(Float64(2.0 * Float64(x * (pi ^ -0.5))));
	else
		tmp = abs(Float64(2.0 * sqrt(Float64((x ^ 2.0) / pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4e-61)
		tmp = abs((2.0 * (x * (pi ^ -0.5))));
	else
		tmp = abs((2.0 * sqrt(((x ^ 2.0) / pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4e-61], N[Abs[N[(2.0 * N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(2.0 * N[Sqrt[N[(N[Power[x, 2.0], $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.0000000000000002e-61

   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%

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

   5. Taylor expanded in x around 0 68.9%

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

      1. rem-square-sqrt 25.7%

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

      2. fabs-sqr 25.7%

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

      3. rem-square-sqrt 68.9%

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

   7. Simplified 68.9%

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

      1. inv-pow 68.9%

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

      2. sqrt-pow1 68.9%

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

      3. metadata-eval 68.9%

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

   9. Applied egg-rr 68.9%

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

   if 4.0000000000000002e-61 < x

   1. Initial program 99.9%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 98.7%

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

      1. rem-square-sqrt 97.8%

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

      2. fabs-sqr 97.8%

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

      3. rem-square-sqrt 98.7%

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

   7. Simplified 98.7%

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

      1. *-un-lft-identity 98.7%

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

      2. *-commutative 98.7%

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

      3. sqrt-div 98.7%

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

      4. metadata-eval 98.7%

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

      5. div-inv 98.1%

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

   9. Applied egg-rr 98.1%

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

       1. *-lft-identity 98.1%

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

   11. Simplified 98.1%

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

       1. add-sqr-sqrt 97.8%

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

       2. sqrt-unprod 98.1%

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

       3. frac-times 98.1%

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

       4. unpow2 98.1%

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

       5. add-sqr-sqrt 98.4%

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

   13. Applied egg-rr 98.4%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-61}:\\ \;\;\;\;\left|2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|2 \cdot \sqrt{\frac{{x}^{2}}{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.5% accurate, 9.0× 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 99.8%

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

5. Taylor expanded in x around 0 70.9%

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

   1. rem-square-sqrt 30.5%

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

   2. fabs-sqr 30.5%

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

   3. rem-square-sqrt 70.9%

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

7. Simplified 70.9%

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

   1. inv-pow 70.9%

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

   2. sqrt-pow1 70.9%

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

   3. metadata-eval 70.9%

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

9. Applied egg-rr 70.9%

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

10. Final simplification 70.9%

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

Alternative 11: 67.1% accurate, 9.0× speedup

Math

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

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(2.0 * Float64(x / sqrt(pi))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 70.9%

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

   1. rem-square-sqrt 30.5%

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

   2. fabs-sqr 30.5%

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

   3. rem-square-sqrt 70.9%

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

7. Simplified 70.9%

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

   1. *-un-lft-identity 70.9%

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

   2. *-commutative 70.9%

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

   3. sqrt-div 70.9%

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

   4. metadata-eval 70.9%

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

   5. div-inv 70.5%

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

9. Applied egg-rr 70.5%

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

    1. *-lft-identity 70.5%

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

11. Simplified 70.5%

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

Reproduce

herbie shell --seed 2024096 
(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)))))))