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

Percentage Accurate: 99.8% → 99.9%

Time: 11.1s

Alternatives: 10

Speedup: 3.5×

• 28.9% 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: 99.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fabs x)
  (fabs
   (*
    (sqrt (/ 1.0 PI))
    (+
     2.0
     (+
      (* 0.047619047619047616 (pow x 6.0))
      (+ (* 0.2 (pow x 4.0)) (* 0.6666666666666666 (* x x)))))))))

C

double code(double x) {
	return fabs(x) * fabs((sqrt((1.0 / ((double) M_PI))) * (2.0 + ((0.047619047619047616 * pow(x, 6.0)) + ((0.2 * pow(x, 4.0)) + (0.6666666666666666 * (x * x)))))));
}

Java

public static double code(double x) {
	return Math.abs(x) * Math.abs((Math.sqrt((1.0 / Math.PI)) * (2.0 + ((0.047619047619047616 * Math.pow(x, 6.0)) + ((0.2 * Math.pow(x, 4.0)) + (0.6666666666666666 * (x * x)))))));
}

Python

def code(x):
	return math.fabs(x) * math.fabs((math.sqrt((1.0 / math.pi)) * (2.0 + ((0.047619047619047616 * math.pow(x, 6.0)) + ((0.2 * math.pow(x, 4.0)) + (0.6666666666666666 * (x * x)))))))

Julia

function code(x)
	return Float64(abs(x) * abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(2.0 + Float64(Float64(0.047619047619047616 * (x ^ 6.0)) + Float64(Float64(0.2 * (x ^ 4.0)) + Float64(0.6666666666666666 * Float64(x * x))))))))
end

MATLAB

function tmp = code(x)
	tmp = abs(x) * abs((sqrt((1.0 / pi)) * (2.0 + ((0.047619047619047616 * (x ^ 6.0)) + ((0.2 * (x ^ 4.0)) + (0.6666666666666666 * (x * x)))))));
end

Wolfram

code[x_] := N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(2.0 + N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 99.9%

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

   1. pow2 99.9%

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

7. Applied egg-rr 99.9%

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

8. Final simplification 99.9%

   \[\leadsto \left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right| \]
9. Add Preprocessing

Alternative 2: 34.8% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * N[(x * N[Power[Pi, -0.5], $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 99.9%

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

   1. add-sqr-sqrt 98.7%

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

   2. fabs-sqr 98.7%

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

   3. add-sqr-sqrt 99.9%

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

   4. clear-num 99.8%

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

   5. un-div-inv 99.4%

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

   6. add-sqr-sqrt 33.2%

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

   7. fabs-sqr 33.2%

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

   8. add-sqr-sqrt 34.5%

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

   9. pow2 34.5%

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

6. Applied egg-rr 34.5%

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

   1. associate-/r/ 34.5%

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

   2. *-commutative 34.5%

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

8. Simplified 34.5%

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

   1. pow2 99.9%

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

10. Applied egg-rr 34.5%

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

    1. div-inv 34.5%

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

    2. pow1/2 34.5%

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

    3. pow-flip 34.5%

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

    4. metadata-eval 34.5%

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

12. Applied egg-rr 34.7%

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

13. Final simplification 34.7%

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

Alternative 3: 34.5% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

   1. add-sqr-sqrt 98.7%

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

   2. fabs-sqr 98.7%

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

   3. add-sqr-sqrt 99.9%

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

   4. clear-num 99.8%

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

   5. un-div-inv 99.4%

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

   6. add-sqr-sqrt 33.2%

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

   7. fabs-sqr 33.2%

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

   8. add-sqr-sqrt 34.5%

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

   9. pow2 34.5%

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

6. Applied egg-rr 34.5%

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

   1. associate-/r/ 34.5%

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

   2. *-commutative 34.5%

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

8. Simplified 34.5%

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

   1. pow2 99.9%

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

10. Applied egg-rr 34.5%

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

    1. fma-undefine 34.5%

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

12. Applied egg-rr 34.5%

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

13. Final simplification 34.5%

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

Alternative 4: 34.6% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 2.0], $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 99.9%

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

   1. add-sqr-sqrt 98.7%

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

   2. fabs-sqr 98.7%

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

   3. add-sqr-sqrt 99.9%

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

   4. clear-num 99.8%

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

   5. un-div-inv 99.4%

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

   6. add-sqr-sqrt 33.2%

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

   7. fabs-sqr 33.2%

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

   8. add-sqr-sqrt 34.5%

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

   9. pow2 34.5%

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

6. Applied egg-rr 34.5%

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

   1. associate-/r/ 34.5%

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

   2. *-commutative 34.5%

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

8. Simplified 34.5%

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

9. Taylor expanded in x around inf 34.3%

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

    1. div-inv 34.5%

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

    2. pow1/2 34.5%

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

    3. pow-flip 34.5%

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

    4. metadata-eval 34.5%

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

11. Applied egg-rr 34.5%

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

12. Final simplification 34.5%

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

Alternative 5: 34.4% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $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 99.9%

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

   1. add-sqr-sqrt 98.7%

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

   2. fabs-sqr 98.7%

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

   3. add-sqr-sqrt 99.9%

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

   4. clear-num 99.8%

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

   5. un-div-inv 99.4%

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

   6. add-sqr-sqrt 33.2%

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

   7. fabs-sqr 33.2%

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

   8. add-sqr-sqrt 34.5%

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

   9. pow2 34.5%

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

6. Applied egg-rr 34.5%

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

   1. associate-/r/ 34.5%

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

   2. *-commutative 34.5%

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

8. Simplified 34.5%

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

   1. pow2 99.9%

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

10. Applied egg-rr 34.5%

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

11. Taylor expanded in x around inf 34.3%

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

12. Final simplification 34.3%

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

Alternative 6: 34.5% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;t\_0 \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.2 \cdot \left(t\_0 \cdot {x}^{4}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (if (<= x 1.75) (* t_0 (* x 2.0)) (* x (* 0.2 (* t_0 (pow x 4.0)))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double tmp;
	if (x <= 1.75) {
		tmp = t_0 * (x * 2.0);
	} else {
		tmp = x * (0.2 * (t_0 * pow(x, 4.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	double tmp;
	if (x <= 1.75) {
		tmp = t_0 * (x * 2.0);
	} else {
		tmp = x * (0.2 * (t_0 * Math.pow(x, 4.0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / math.pi))
	tmp = 0
	if x <= 1.75:
		tmp = t_0 * (x * 2.0)
	else:
		tmp = x * (0.2 * (t_0 * math.pow(x, 4.0)))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / pi))
	tmp = 0.0
	if (x <= 1.75)
		tmp = Float64(t_0 * Float64(x * 2.0));
	else
		tmp = Float64(x * Float64(0.2 * Float64(t_0 * (x ^ 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 / pi));
	tmp = 0.0;
	if (x <= 1.75)
		tmp = t_0 * (x * 2.0);
	else
		tmp = x * (0.2 * (t_0 * (x ^ 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.75], N[(t$95$0 * N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.2 * N[(t$95$0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.75

   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 93.9%

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

      1. pow1 93.9%

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

      2. add-sqr-sqrt 33.2%

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

      3. fabs-sqr 33.2%

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

      4. add-sqr-sqrt 34.6%

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

      5. add-sqr-sqrt 34.0%

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

      6. fabs-sqr 34.0%

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

      7. add-sqr-sqrt 34.6%

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

      8. fma-define 34.6%

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

      9. pow2 34.6%

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

   7. Applied egg-rr 34.6%

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

      1. unpow1 34.6%

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

   9. Simplified 34.6%

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

   10. Taylor expanded in x around 0 34.5%

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

       1. associate-*r* 34.5%

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

   12. Simplified 34.5%

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

   if 1.75 < 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 93.9%

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

      1. pow1 93.9%

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

      2. add-sqr-sqrt 33.2%

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

      3. fabs-sqr 33.2%

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

      4. add-sqr-sqrt 34.6%

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

      5. add-sqr-sqrt 34.0%

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

      6. fabs-sqr 34.0%

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

      7. add-sqr-sqrt 34.6%

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

      8. fma-define 34.6%

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

      9. pow2 34.6%

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

   7. Applied egg-rr 34.6%

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

      1. unpow1 34.6%

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

   9. Simplified 34.6%

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

   10. Taylor expanded in x around inf 3.4%

       \[\leadsto x \cdot \color{blue}{\left(0.2 \cdot \left({x}^{4} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{4}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.5% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;t\_0 \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.6666666666666666 \cdot \left(t\_0 \cdot {x}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (if (<= x 1.75)
     (* t_0 (* x 2.0))
     (* 0.6666666666666666 (* t_0 (pow x 3.0))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double tmp;
	if (x <= 1.75) {
		tmp = t_0 * (x * 2.0);
	} else {
		tmp = 0.6666666666666666 * (t_0 * pow(x, 3.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	double tmp;
	if (x <= 1.75) {
		tmp = t_0 * (x * 2.0);
	} else {
		tmp = 0.6666666666666666 * (t_0 * Math.pow(x, 3.0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / math.pi))
	tmp = 0
	if x <= 1.75:
		tmp = t_0 * (x * 2.0)
	else:
		tmp = 0.6666666666666666 * (t_0 * math.pow(x, 3.0))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / pi))
	tmp = 0.0
	if (x <= 1.75)
		tmp = Float64(t_0 * Float64(x * 2.0));
	else
		tmp = Float64(0.6666666666666666 * Float64(t_0 * (x ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 / pi));
	tmp = 0.0;
	if (x <= 1.75)
		tmp = t_0 * (x * 2.0);
	else
		tmp = 0.6666666666666666 * (t_0 * (x ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.75], N[(t$95$0 * N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.6666666666666666 * N[(t$95$0 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.75

   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 93.9%

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

      1. pow1 93.9%

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

      2. add-sqr-sqrt 33.2%

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

      3. fabs-sqr 33.2%

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

      4. add-sqr-sqrt 34.6%

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

      5. add-sqr-sqrt 34.0%

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

      6. fabs-sqr 34.0%

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

      7. add-sqr-sqrt 34.6%

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

      8. fma-define 34.6%

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

      9. pow2 34.6%

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

   7. Applied egg-rr 34.6%

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

      1. unpow1 34.6%

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

   9. Simplified 34.6%

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

   10. Taylor expanded in x around 0 34.5%

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

       1. associate-*r* 34.5%

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

   12. Simplified 34.5%

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

   if 1.75 < 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 93.9%

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

      1. pow1 93.9%

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

      2. add-sqr-sqrt 33.2%

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

      3. fabs-sqr 33.2%

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

      4. add-sqr-sqrt 34.6%

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

      5. add-sqr-sqrt 34.0%

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

      6. fabs-sqr 34.0%

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

      7. add-sqr-sqrt 34.6%

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

      8. fma-define 34.6%

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

      9. pow2 34.6%

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

   7. Applied egg-rr 34.6%

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

      1. unpow1 34.6%

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

   9. Simplified 34.6%

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

   10. Taylor expanded in x around 0 34.6%

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

   11. Taylor expanded in x around inf 3.5%

       \[\leadsto \color{blue}{0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.2% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(0.047619047619047616 * N[Power[x, 6.0], $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 99.9%

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

   1. add-sqr-sqrt 98.7%

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

   2. fabs-sqr 98.7%

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

   3. add-sqr-sqrt 99.9%

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

   4. clear-num 99.8%

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

   5. un-div-inv 99.4%

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

   6. add-sqr-sqrt 33.2%

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

   7. fabs-sqr 33.2%

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

   8. add-sqr-sqrt 34.5%

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

   9. pow2 34.5%

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

6. Applied egg-rr 34.5%

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

   1. associate-/r/ 34.5%

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

   2. *-commutative 34.5%

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

8. Simplified 34.5%

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

9. Taylor expanded in x around inf 34.3%

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

10. Taylor expanded in x around 0 34.2%

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

11. Final simplification 34.2%

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

Alternative 9: 34.5% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (sqrt (/ 1.0 PI)) (* x 2.0)))

C

double code(double x) {
	return sqrt((1.0 / ((double) M_PI))) * (x * 2.0);
}

Java

public static double code(double x) {
	return Math.sqrt((1.0 / Math.PI)) * (x * 2.0);
}

Python

def code(x):
	return math.sqrt((1.0 / math.pi)) * (x * 2.0)

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 / pi)) * Float64(x * 2.0))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 / pi)) * (x * 2.0);
end

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x * 2.0), $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 93.9%

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

   1. pow1 93.9%

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

   2. add-sqr-sqrt 33.2%

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

   3. fabs-sqr 33.2%

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

   4. add-sqr-sqrt 34.6%

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

   5. add-sqr-sqrt 34.0%

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

   6. fabs-sqr 34.0%

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

   7. add-sqr-sqrt 34.6%

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

   8. fma-define 34.6%

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

   9. pow2 34.6%

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

7. Applied egg-rr 34.6%

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

   1. unpow1 34.6%

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

9. Simplified 34.6%

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

10. Taylor expanded in x around 0 34.5%

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

    1. associate-*r* 34.5%

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

12. Simplified 34.5%

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

13. Final simplification 34.5%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2\right) \]
14. Add Preprocessing

Alternative 10: 4.1% accurate, 1849.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

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%

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

5. Taylor expanded in x around inf 27.2%

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

   1. expm1-log1p-u 27.2%

      \[\leadsto \left|0.6666666666666666 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({x}^{2} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)}\right| \]

   2. expm1-undefine 26.7%

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

   3. *-commutative 26.7%

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

   4. inv-pow 26.7%

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

   5. sqrt-pow1 26.7%

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

   6. metadata-eval 26.7%

      \[\leadsto \left|0.6666666666666666 \cdot \left(e^{\mathsf{log1p}\left({\pi}^{\color{blue}{-0.5}} \cdot \left({x}^{2} \cdot \left|x\right|\right)\right)} - 1\right)\right| \]

   7. add-sqr-sqrt 1.8%

      \[\leadsto \left|0.6666666666666666 \cdot \left(e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left({x}^{2} \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)\right)} - 1\right)\right| \]

   8. fabs-sqr 1.8%

      \[\leadsto \left|0.6666666666666666 \cdot \left(e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left({x}^{2} \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right)\right)} - 1\right)\right| \]

   9. add-sqr-sqrt 3.5%

      \[\leadsto \left|0.6666666666666666 \cdot \left(e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left({x}^{2} \cdot \color{blue}{x}\right)\right)} - 1\right)\right| \]

   10. pow-plus 3.5%

       \[\leadsto \left|0.6666666666666666 \cdot \left(e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \color{blue}{{x}^{\left(2 + 1\right)}}\right)} - 1\right)\right| \]

   11. metadata-eval 3.5%

       \[\leadsto \left|0.6666666666666666 \cdot \left(e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot {x}^{\color{blue}{3}}\right)} - 1\right)\right| \]

7. Applied egg-rr 3.5%

   \[\leadsto \left|0.6666666666666666 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot {x}^{3}\right)} - 1\right)}\right| \]
8. Step-by-step derivation

   1. sub-neg 3.5%

      \[\leadsto \left|0.6666666666666666 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot {x}^{3}\right)} + \left(-1\right)\right)}\right| \]

   2. log1p-undefine 3.5%

      \[\leadsto \left|0.6666666666666666 \cdot \left(e^{\color{blue}{\log \left(1 + {\pi}^{-0.5} \cdot {x}^{3}\right)}} + \left(-1\right)\right)\right| \]

   3. rem-exp-log 26.7%

      \[\leadsto \left|0.6666666666666666 \cdot \left(\color{blue}{\left(1 + {\pi}^{-0.5} \cdot {x}^{3}\right)} + \left(-1\right)\right)\right| \]

   4. +-commutative 26.7%

      \[\leadsto \left|0.6666666666666666 \cdot \left(\color{blue}{\left({\pi}^{-0.5} \cdot {x}^{3} + 1\right)} + \left(-1\right)\right)\right| \]

   5. fma-define 26.7%

      \[\leadsto \left|0.6666666666666666 \cdot \left(\color{blue}{\mathsf{fma}\left({\pi}^{-0.5}, {x}^{3}, 1\right)} + \left(-1\right)\right)\right| \]

   6. metadata-eval 26.7%

      \[\leadsto \left|0.6666666666666666 \cdot \left(\mathsf{fma}\left({\pi}^{-0.5}, {x}^{3}, 1\right) + \color{blue}{-1}\right)\right| \]

9. Simplified 26.7%

   \[\leadsto \left|0.6666666666666666 \cdot \color{blue}{\left(\mathsf{fma}\left({\pi}^{-0.5}, {x}^{3}, 1\right) + -1\right)}\right| \]

10. Taylor expanded in x around 0 3.9%

    \[\leadsto \left|0.6666666666666666 \cdot \left(\color{blue}{1} + -1\right)\right| \]

11. Final simplification 3.9%

    \[\leadsto 0 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024044 
(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)))))))