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

Percentage Accurate: 99.9% → 99.9%

Time: 11.9s

Alternatives: 10

Speedup: 3.6×

• 28.3% 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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[Abs[N[(N[(N[Power[Pi, -0.5], $MachinePrecision] * x), $MachinePrecision] * N[(N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $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 99.4%

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

   1. clear-num 99.4%

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

   2. associate-/r/ 99.9%

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

   3. pow1/2 99.9%

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

   4. pow-flip 99.9%

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

   5. metadata-eval 99.9%

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

   6. add-sqr-sqrt 33.3%

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

   7. fabs-sqr 33.3%

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

   8. add-sqr-sqrt 99.9%

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

6. Applied egg-rr 99.9%

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

   1. metadata-eval 99.9%

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

   2. fma-undefine 99.9%

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

   3. metadata-eval 99.9%

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

8. Applied egg-rr 99.9%

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

Alternative 2: 99.3% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[Abs[N[(N[(N[Power[Pi, -0.5], $MachinePrecision] * x), $MachinePrecision] * N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $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 99.4%

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

   1. clear-num 99.4%

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

   2. associate-/r/ 99.9%

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

   3. pow1/2 99.9%

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

   4. pow-flip 99.9%

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

   5. metadata-eval 99.9%

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

   6. add-sqr-sqrt 33.3%

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

   7. fabs-sqr 33.3%

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

   8. add-sqr-sqrt 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around inf 99.5%

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

Alternative 3: 99.0% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[Abs[N[(x * N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision] + 2.0), $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 99.4%

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

5. Taylor expanded in x around inf 99.1%

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

6. Taylor expanded in x around 0 98.9%

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

   1. *-commutative 98.9%

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

   2. clear-num 98.9%

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

   3. un-div-inv 98.9%

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

   4. fma-define 98.9%

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

   5. add-sqr-sqrt 32.9%

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

   6. fabs-sqr 32.9%

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

   7. add-sqr-sqrt 98.9%

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

8. Applied egg-rr 98.9%

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

   1. associate-/r/ 99.3%

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

   2. *-commutative 99.3%

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

10. Simplified 99.3%

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

Alternative 4: 68.1% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around inf 99.1%

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

   6. Taylor expanded in x around 0 98.9%

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

      1. *-commutative 98.9%

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

      2. clear-num 98.9%

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

      3. un-div-inv 98.9%

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

      4. fma-define 98.9%

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

      5. add-sqr-sqrt 32.9%

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

      6. fabs-sqr 32.9%

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

      7. add-sqr-sqrt 98.9%

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

   8. Applied egg-rr 98.9%

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

      1. associate-/r/ 99.3%

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

      2. *-commutative 99.3%

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

   10. Simplified 99.3%

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

   11. Taylor expanded in x around 0 67.3%

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

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around inf 99.1%

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

   6. Taylor expanded in x around 0 98.9%

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

      1. *-commutative 98.9%

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

      2. clear-num 98.9%

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

      3. un-div-inv 98.9%

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

      4. fma-define 98.9%

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

      5. add-sqr-sqrt 32.9%

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

      6. fabs-sqr 32.9%

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

      7. add-sqr-sqrt 98.9%

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

   8. Applied egg-rr 98.9%

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

      1. associate-/r/ 99.3%

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

      2. *-commutative 99.3%

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

   10. Simplified 99.3%

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

   11. Taylor expanded in x around inf 37.8%

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

Alternative 5: 98.5% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around inf 99.1%

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

   1. expm1-log1p-u 98.8%

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

   2. expm1-undefine 38.8%

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

   3. add-sqr-sqrt 2.6%

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

   4. fabs-sqr 2.6%

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

   5. add-sqr-sqrt 5.1%

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

7. Applied egg-rr 5.1%

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

   1. log1p-undefine 5.1%

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

   2. rem-exp-log 39.2%

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

   3. +-commutative 39.2%

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

   4. associate--l+ 99.1%

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

   5. metadata-eval 99.1%

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

   6. +-commutative 99.1%

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

   7. +-lft-identity 99.1%

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

9. Simplified 99.1%

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

10. Taylor expanded in x around 0 98.9%

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

Alternative 6: 34.8% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(x * N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $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 99.9%

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

5. Taylor expanded in x around 0 99.5%

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

6. Taylor expanded in x around 0 92.4%

   \[\leadsto \left|x\right| \cdot \left|\frac{\color{blue}{0.2 \cdot {x}^{4}} + 2}{\sqrt{\pi}}\right| \]
7. Step-by-step derivation

   1. *-commutative 92.4%

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

   2. add-sqr-sqrt 91.3%

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

   3. fabs-sqr 91.3%

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

   4. add-sqr-sqrt 92.4%

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

   5. add-sqr-sqrt 33.0%

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

   6. fabs-sqr 33.0%

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

   7. add-sqr-sqrt 34.5%

      \[\leadsto \frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}} \cdot \color{blue}{x} \]

   8. associate-*l/ 34.3%

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

   9. fma-define 34.3%

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

8. Applied egg-rr 34.3%

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

   1. *-commutative 34.3%

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

   2. fma-undefine 34.3%

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

   3. distribute-rgt-in 34.3%

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

10. Applied egg-rr 34.3%

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

11. Final simplification 34.3%

    \[\leadsto \frac{x \cdot \left(0.2 \cdot {x}^{4}\right) + x \cdot 2}{\sqrt{\pi}} \]
12. Add Preprocessing

Alternative 7: 68.1% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{0.2 \cdot {x}^{5}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.8)
   (fabs (* x (/ 2.0 (sqrt PI))))
   (/ (* 0.2 (pow x 5.0)) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = (0.2 * pow(x, 5.0)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = (0.2 * Math.pow(x, 5.0)) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.8:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = (0.2 * math.pow(x, 5.0)) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.8)
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	else
		tmp = Float64(Float64(0.2 * (x ^ 5.0)) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.8)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = (0.2 * (x ^ 5.0)) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.8], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.80000000000000004

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around inf 99.1%

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

   6. Taylor expanded in x around 0 98.9%

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

      1. *-commutative 98.9%

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

      2. clear-num 98.9%

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

      3. un-div-inv 98.9%

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

      4. fma-define 98.9%

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

      5. add-sqr-sqrt 32.9%

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

      6. fabs-sqr 32.9%

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

      7. add-sqr-sqrt 98.9%

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

   8. Applied egg-rr 98.9%

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

      1. associate-/r/ 99.3%

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

      2. *-commutative 99.3%

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

   10. Simplified 99.3%

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

   11. Taylor expanded in x around 0 67.3%

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

   if 1.80000000000000004 < 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 99.9%

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

   5. Taylor expanded in x around 0 99.5%

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

   6. Taylor expanded in x around 0 92.4%

      \[\leadsto \left|x\right| \cdot \left|\frac{\color{blue}{0.2 \cdot {x}^{4}} + 2}{\sqrt{\pi}}\right| \]
   7. Step-by-step derivation

      1. *-commutative 92.4%

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

      2. add-sqr-sqrt 91.3%

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

      3. fabs-sqr 91.3%

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

      4. add-sqr-sqrt 92.4%

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

      5. add-sqr-sqrt 33.0%

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

      6. fabs-sqr 33.0%

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

      7. add-sqr-sqrt 34.5%

         \[\leadsto \frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}} \cdot \color{blue}{x} \]

      8. associate-*l/ 34.3%

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

      9. fma-define 34.3%

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

   8. Applied egg-rr 34.3%

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

   9. Taylor expanded in x around inf 3.7%

      \[\leadsto \frac{\color{blue}{0.2 \cdot {x}^{5}}}{\sqrt{\pi}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 34.8% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(x * N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $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 99.9%

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

5. Taylor expanded in x around 0 99.5%

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

6. Taylor expanded in x around 0 92.4%

   \[\leadsto \left|x\right| \cdot \left|\frac{\color{blue}{0.2 \cdot {x}^{4}} + 2}{\sqrt{\pi}}\right| \]
7. Step-by-step derivation

   1. *-commutative 92.4%

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

   2. add-sqr-sqrt 91.3%

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

   3. fabs-sqr 91.3%

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

   4. add-sqr-sqrt 92.4%

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

   5. add-sqr-sqrt 33.0%

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

   6. fabs-sqr 33.0%

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

   7. add-sqr-sqrt 34.5%

      \[\leadsto \frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}} \cdot \color{blue}{x} \]

   8. associate-*l/ 34.3%

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

   9. fma-define 34.3%

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

8. Applied egg-rr 34.3%

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

   1. fma-undefine 34.3%

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

10. Applied egg-rr 34.3%

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

11. Final simplification 34.3%

    \[\leadsto \frac{x \cdot \left(0.2 \cdot {x}^{4} + 2\right)}{\sqrt{\pi}} \]
12. Add Preprocessing

Alternative 9: 68.1% 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 99.8%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around inf 99.1%

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

6. Taylor expanded in x around 0 98.9%

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

   1. *-commutative 98.9%

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

   2. clear-num 98.9%

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

   3. un-div-inv 98.9%

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

   4. fma-define 98.9%

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

   5. add-sqr-sqrt 32.9%

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

   6. fabs-sqr 32.9%

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

   7. add-sqr-sqrt 98.9%

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

8. Applied egg-rr 98.9%

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

   1. associate-/r/ 99.3%

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

   2. *-commutative 99.3%

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

10. Simplified 99.3%

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

11. Taylor expanded in x around 0 67.3%

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

Alternative 10: 34.9% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(x * 2.0), $MachinePrecision] / N[Sqrt[Pi], $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 99.9%

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

5. Taylor expanded in x around 0 99.5%

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

6. Taylor expanded in x around 0 92.4%

   \[\leadsto \left|x\right| \cdot \left|\frac{\color{blue}{0.2 \cdot {x}^{4}} + 2}{\sqrt{\pi}}\right| \]
7. Step-by-step derivation

   1. *-commutative 92.4%

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

   2. add-sqr-sqrt 91.3%

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

   3. fabs-sqr 91.3%

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

   4. add-sqr-sqrt 92.4%

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

   5. add-sqr-sqrt 33.0%

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

   6. fabs-sqr 33.0%

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

   7. add-sqr-sqrt 34.5%

      \[\leadsto \frac{0.2 \cdot {x}^{4} + 2}{\sqrt{\pi}} \cdot \color{blue}{x} \]

   8. associate-*l/ 34.3%

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

   9. fma-define 34.3%

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

8. Applied egg-rr 34.3%

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

9. Taylor expanded in x around 0 34.3%

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

10. Final simplification 34.3%

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

Reproduce

herbie shell --seed 2024093 
(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)))))))