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

Percentage Accurate: 99.8% → 99.8%

Time: 12.4s

Alternatives: 13

Speedup: 3.5×

• 27.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot \left(x \cdot x\right)\\ t_1 := \left|x\right| \cdot \left(\left|x\right| \cdot t\_0\right)\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + 0.6666666666666666 \cdot t\_0\right) + 0.2 \cdot t\_1\right) + 0.047619047619047616 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot t\_1\right)\right)\right)\right| \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return Math.abs(((x * (((0.2 * Math.pow(x, 4.0)) + (0.047619047619047616 * Math.pow(x, 6.0))) + (2.0 + (0.6666666666666666 * Math.pow(x, 2.0))))) * Math.pow(Math.PI, -0.5)));
}

Python

def code(x):
	return math.fabs(((x * (((0.2 * math.pow(x, 4.0)) + (0.047619047619047616 * math.pow(x, 6.0))) + (2.0 + (0.6666666666666666 * math.pow(x, 2.0))))) * math.pow(math.pi, -0.5)))

Julia

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

MATLAB

function tmp = code(x)
	tmp = abs(((x * (((0.2 * (x ^ 4.0)) + (0.047619047619047616 * (x ^ 6.0))) + (2.0 + (0.6666666666666666 * (x ^ 2.0))))) * (pi ^ -0.5)));
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

6. Applied egg-rr 99.9%

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

   1. fma-undefine 30.3%

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

8. Applied egg-rr 99.9%

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

   1. metadata-eval 99.9%

      \[\leadsto \left|\left(x \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{1}{5}}, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \left(0.6666666666666666 \cdot {x}^{2} + 2\right)\right)\right) \cdot {\pi}^{-0.5}\right| \]

   2. fma-undefine 99.9%

      \[\leadsto \left|\left(x \cdot \left(\color{blue}{\left(\frac{1}{5} \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)} + \left(0.6666666666666666 \cdot {x}^{2} + 2\right)\right)\right) \cdot {\pi}^{-0.5}\right| \]

   3. metadata-eval 99.9%

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

10. Applied egg-rr 99.9%

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

11. Final simplification 99.9%

    \[\leadsto \left|\left(x \cdot \left(\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right) \cdot {\pi}^{-0.5}\right| \]
12. Add Preprocessing

Alternative 3: 98.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;\left|x \cdot \left(t\_0 \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(t\_0 \cdot {x}^{7}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (if (<= (fabs x) 0.0001)
     (fabs (* x (* t_0 (fma 0.6666666666666666 (pow x 2.0) 2.0))))
     (fabs (* 0.047619047619047616 (* t_0 (pow x 7.0)))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double tmp;
	if (fabs(x) <= 0.0001) {
		tmp = fabs((x * (t_0 * fma(0.6666666666666666, pow(x, 2.0), 2.0))));
	} else {
		tmp = fabs((0.047619047619047616 * (t_0 * pow(x, 7.0))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / pi))
	tmp = 0.0
	if (abs(x) <= 0.0001)
		tmp = abs(Float64(x * Float64(t_0 * fma(0.6666666666666666, (x ^ 2.0), 2.0))));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64(t_0 * (x ^ 7.0))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.0001], N[Abs[N[(x * N[(t$95$0 * N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(t$95$0 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000005e-4

   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 99.2%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 99.9%

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

      1. associate-*r* 99.9%

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

      2. distribute-rgt-out 99.9%

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

      3. fma-define 99.9%

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

   9. Simplified 99.9%

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

   if 1.00000000000000005e-4 < (fabs.f64 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%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf 97.8%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;\left|x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{7}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0 99.3%

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

8. Final simplification 99.3%

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

Alternative 5: 99.2% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return Math.abs((Math.pow(Math.PI, -0.5) * (x * ((0.047619047619047616 * Math.pow(x, 6.0)) + (2.0 + (0.6666666666666666 * Math.pow(x, 2.0)))))));
}

Python

def code(x):
	return math.fabs((math.pow(math.pi, -0.5) * (x * ((0.047619047619047616 * math.pow(x, 6.0)) + (2.0 + (0.6666666666666666 * math.pow(x, 2.0)))))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = abs(((pi ^ -0.5) * (x * ((0.047619047619047616 * (x ^ 6.0)) + (2.0 + (0.6666666666666666 * (x ^ 2.0)))))));
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

6. Applied egg-rr 99.9%

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

   1. fma-undefine 30.3%

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

8. Applied egg-rr 99.9%

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

9. Taylor expanded in x around inf 99.1%

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

10. Final simplification 99.1%

    \[\leadsto \left|{\pi}^{-0.5} \cdot \left(x \cdot \left(0.047619047619047616 \cdot {x}^{6} + \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \]
11. Add Preprocessing

Alternative 6: 98.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{7}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.0001)
   (fabs (* x (/ 2.0 (sqrt PI))))
   (fabs (* 0.047619047619047616 (* (sqrt (/ 1.0 PI)) (pow x 7.0))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.0001) {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((0.047619047619047616 * (sqrt((1.0 / ((double) M_PI))) * pow(x, 7.0))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.0001) {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs((0.047619047619047616 * (Math.sqrt((1.0 / Math.PI)) * Math.pow(x, 7.0))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.0001:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs((0.047619047619047616 * (math.sqrt((1.0 / math.pi)) * math.pow(x, 7.0))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.0001)
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64(sqrt(Float64(1.0 / pi)) * (x ^ 7.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.0001)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = abs((0.047619047619047616 * (sqrt((1.0 / pi)) * (x ^ 7.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.0001], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000005e-4

   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 99.2%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 99.4%

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

      1. associate-*r* 99.4%

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

   9. Simplified 99.4%

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

       1. associate-*l* 99.4%

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

       2. sqrt-div 99.4%

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

       3. metadata-eval 99.4%

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

       4. div-inv 98.7%

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

       5. clear-num 98.8%

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

       6. un-div-inv 98.8%

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

   11. Applied egg-rr 98.8%

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

       1. associate-/r/ 99.4%

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

   13. Simplified 99.4%

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

   if 1.00000000000000005e-4 < (fabs.f64 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%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf 97.8%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{7}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.8% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Wolfram

code[x_] := N[(x / N[(N[Sqrt[Pi], $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 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 99.3%

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

   1. add-sqr-sqrt 28.9%

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

   2. fabs-sqr 28.9%

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

   3. add-sqr-sqrt 30.4%

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

   4. add-sqr-sqrt 29.8%

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

   5. fabs-sqr 29.8%

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

   6. add-sqr-sqrt 30.4%

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

   7. clear-num 30.4%

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

   8. un-div-inv 30.2%

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

   9. +-commutative 30.2%

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

7. Applied egg-rr 30.2%

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

8. Final simplification 30.2%

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

Alternative 8: 98.9% 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 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 99.3%

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

6. Taylor expanded in x around inf 98.8%

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

7. Final simplification 98.8%

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

Alternative 9: 68.9% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.75

   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 99.4%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 62.5%

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

      1. associate-*r* 62.5%

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

   9. Simplified 62.5%

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

       1. associate-*l* 62.5%

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

       2. sqrt-div 62.5%

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

       3. metadata-eval 62.5%

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

       4. div-inv 62.0%

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

       5. clear-num 62.1%

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

       6. un-div-inv 62.1%

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

   11. Applied egg-rr 62.1%

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

       1. associate-/r/ 62.5%

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

   13. Simplified 62.5%

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

   if 1.75 < 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 99.4%

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

   5. Taylor expanded in x around inf 28.4%

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

      1. associate-*r* 28.4%

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

      2. *-commutative 28.4%

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

      3. associate-*l* 28.4%

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

      4. unpow2 28.4%

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

      5. rem-square-sqrt 1.8%

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

      6. fabs-sqr 1.8%

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

      7. rem-square-sqrt 28.4%

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

      8. unpow3 28.4%

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

   7. Simplified 28.4%

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

      1. associate-*r* 28.4%

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

      2. sqrt-div 28.4%

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

      3. metadata-eval 28.4%

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

      4. un-div-inv 28.4%

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

   9. Applied egg-rr 28.4%

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

       1. *-un-lft-identity 28.4%

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

       2. add-sqr-sqrt 3.0%

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

       3. sqrt-unprod 35.9%

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

       4. frac-times 35.9%

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

       5. swap-sqr 35.9%

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

       6. pow-sqr 35.9%

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

       7. metadata-eval 35.9%

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

       8. metadata-eval 35.9%

          \[\leadsto \left|1 \cdot \sqrt{\frac{{x}^{6} \cdot \color{blue}{0.4444444444444444}}{\sqrt{\pi} \cdot \sqrt{\pi}}}\right| \]

       9. add-sqr-sqrt 35.9%

          \[\leadsto \left|1 \cdot \sqrt{\frac{{x}^{6} \cdot 0.4444444444444444}{\color{blue}{\pi}}}\right| \]

   11. Applied egg-rr 35.9%

       \[\leadsto \left|\color{blue}{1 \cdot \sqrt{\frac{{x}^{6} \cdot 0.4444444444444444}{\pi}}}\right| \]
   12. Step-by-step derivation

       1. *-lft-identity 35.9%

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

       2. *-commutative 35.9%

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

       3. associate-/l* 35.9%

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

   13. Simplified 35.9%

       \[\leadsto \left|\color{blue}{\sqrt{0.4444444444444444 \cdot \frac{{x}^{6}}{\pi}}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.5%

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

Alternative 10: 68.9% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.75

   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 99.4%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 62.5%

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

      1. associate-*r* 62.5%

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

   9. Simplified 62.5%

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

       1. associate-*l* 62.5%

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

       2. sqrt-div 62.5%

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

       3. metadata-eval 62.5%

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

       4. div-inv 62.0%

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

       5. clear-num 62.1%

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

       6. un-div-inv 62.1%

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

   11. Applied egg-rr 62.1%

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

       1. associate-/r/ 62.5%

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

   13. Simplified 62.5%

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

   if 1.75 < 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 99.9%

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

   5. Taylor expanded in x around inf 99.1%

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

      1. expm1-log1p-u 98.6%

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

      2. expm1-undefine 43.2%

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

   7. Applied egg-rr 3.9%

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

      1. sub-neg 3.9%

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

      2. log1p-undefine 3.9%

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

      3. rem-exp-log 3.9%

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

      4. +-commutative 3.9%

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

      5. metadata-eval 3.9%

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

      6. associate-+l+ 30.5%

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

      7. metadata-eval 30.5%

         \[\leadsto x \cdot \frac{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}{\sqrt{\pi}} + \color{blue}{0} \]

      8. metadata-eval 30.5%

         \[\leadsto x \cdot \frac{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}{\sqrt{\pi}} + \color{blue}{\left(-0\right)} \]

      9. sub-neg 30.5%

         \[\leadsto \color{blue}{x \cdot \frac{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}{\sqrt{\pi}} - 0} \]

      10. --rgt-identity 30.5%

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

      11. associate-*r/ 30.3%

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

      12. associate-*l/ 30.3%

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

      13. *-commutative 30.3%

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

   9. Simplified 30.3%

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

   10. Taylor expanded in x around 0 30.3%

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

   11. Taylor expanded in x around inf 3.5%

       \[\leadsto \color{blue}{\left(0.6666666666666666 \cdot {x}^{2}\right)} \cdot \frac{x}{\sqrt{\pi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.5%

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

Alternative 11: 35.0% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(x)
	return Float64(Float64(2.0 + Float64(0.6666666666666666 * (x ^ 2.0))) * Float64(x / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = (2.0 + (0.6666666666666666 * (x ^ 2.0))) * (x / sqrt(pi));
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around inf 99.1%

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

   1. expm1-log1p-u 98.6%

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

   2. expm1-undefine 43.2%

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

7. Applied egg-rr 3.9%

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

   1. sub-neg 3.9%

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

   2. log1p-undefine 3.9%

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

   3. rem-exp-log 3.9%

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

   4. +-commutative 3.9%

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

   5. metadata-eval 3.9%

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

   6. associate-+l+ 30.5%

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

   7. metadata-eval 30.5%

      \[\leadsto x \cdot \frac{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}{\sqrt{\pi}} + \color{blue}{0} \]

   8. metadata-eval 30.5%

      \[\leadsto x \cdot \frac{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}{\sqrt{\pi}} + \color{blue}{\left(-0\right)} \]

   9. sub-neg 30.5%

      \[\leadsto \color{blue}{x \cdot \frac{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}{\sqrt{\pi}} - 0} \]

   10. --rgt-identity 30.5%

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

   11. associate-*r/ 30.3%

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

   12. associate-*l/ 30.3%

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

   13. *-commutative 30.3%

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

9. Simplified 30.3%

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

10. Taylor expanded in x around 0 30.3%

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

    1. fma-undefine 30.3%

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

12. Applied egg-rr 30.3%

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

13. Final simplification 30.3%

    \[\leadsto \left(2 + 0.6666666666666666 \cdot {x}^{2}\right) \cdot \frac{x}{\sqrt{\pi}} \]
14. Add Preprocessing

Alternative 12: 68.9% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0 62.5%

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

   1. associate-*r* 62.5%

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

9. Simplified 62.5%

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

    1. associate-*l* 62.5%

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

    2. sqrt-div 62.5%

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

    3. metadata-eval 62.5%

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

    4. div-inv 62.0%

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

    5. clear-num 62.1%

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

    6. un-div-inv 62.1%

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

11. Applied egg-rr 62.1%

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

    1. associate-/r/ 62.5%

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

13. Simplified 62.5%

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

14. Final simplification 62.5%

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

Alternative 13: 34.9% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around inf 99.1%

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

   1. expm1-log1p-u 98.6%

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

   2. expm1-undefine 43.2%

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

7. Applied egg-rr 3.9%

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

   1. sub-neg 3.9%

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

   2. log1p-undefine 3.9%

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

   3. rem-exp-log 3.9%

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

   4. +-commutative 3.9%

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

   5. metadata-eval 3.9%

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

   6. associate-+l+ 30.5%

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

   7. metadata-eval 30.5%

      \[\leadsto x \cdot \frac{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}{\sqrt{\pi}} + \color{blue}{0} \]

   8. metadata-eval 30.5%

      \[\leadsto x \cdot \frac{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}{\sqrt{\pi}} + \color{blue}{\left(-0\right)} \]

   9. sub-neg 30.5%

      \[\leadsto \color{blue}{x \cdot \frac{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}{\sqrt{\pi}} - 0} \]

   10. --rgt-identity 30.5%

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

   11. associate-*r/ 30.3%

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

   12. associate-*l/ 30.3%

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

   13. *-commutative 30.3%

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

9. Simplified 30.3%

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

10. Taylor expanded in x around 0 30.3%

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

11. Taylor expanded in x around 0 30.3%

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

12. Final simplification 30.3%

    \[\leadsto 2 \cdot \frac{x}{\sqrt{\pi}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(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)))))))