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

Percentage Accurate: 99.8% → 99.9%

Time: 12.4s

Alternatives: 12

Speedup: 2.6×

• 29.2% 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, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 99.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. rem-square-sqrt 33.4%

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

   4. fabs-sqr 33.4%

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

   5. rem-square-sqrt 99.4%

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

   6. associate-+r+ 99.4%

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

   7. +-commutative 99.4%

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

   8. fma-define 99.4%

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

   9. rem-square-sqrt 33.6%

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

   10. fabs-sqr 33.6%

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

   11. rem-square-sqrt 99.4%

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

   12. +-commutative 99.4%

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

   13. fma-define 99.4%

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

   14. rem-square-sqrt 33.6%

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

   15. fabs-sqr 33.6%

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

   16. rem-square-sqrt 99.4%

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

7. Simplified 99.4%

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

8. Final simplification 99.4%

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

Alternative 3: 99.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 99.4%

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

   1. fma-undefine 99.4%

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

7. Applied egg-rr 99.4%

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

8. Final simplification 99.4%

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

Alternative 4: 33.2% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 99.4%

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

   1. fma-undefine 99.4%

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

7. Applied egg-rr 99.4%

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

   1. add-sqr-sqrt 33.4%

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

   2. fabs-sqr 33.4%

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

   3. add-sqr-sqrt 34.8%

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

   4. add-sqr-sqrt 34.1%

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

   5. fabs-sqr 34.1%

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

   6. add-sqr-sqrt 34.8%

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

   7. clear-num 34.8%

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

   8. un-div-inv 34.5%

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

   9. associate-+l+ 34.5%

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

   10. fma-define 34.5%

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

   11. fma-define 34.5%

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

9. Applied egg-rr 34.5%

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

10. Final simplification 34.5%

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

Alternative 5: 33.6% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \left(t\_0 \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot \left(t\_0 \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (if (<= x 1.85)
     (* x (* t_0 (fma 0.6666666666666666 (pow x 2.0) 2.0)))
     (* (pow x 7.0) (* t_0 (+ 0.047619047619047616 (/ 0.2 (pow x 2.0))))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double tmp;
	if (x <= 1.85) {
		tmp = x * (t_0 * fma(0.6666666666666666, pow(x, 2.0), 2.0));
	} else {
		tmp = pow(x, 7.0) * (t_0 * (0.047619047619047616 + (0.2 / pow(x, 2.0))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / pi))
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64(x * Float64(t_0 * fma(0.6666666666666666, (x ^ 2.0), 2.0)));
	else
		tmp = Float64((x ^ 7.0) * Float64(t_0 * Float64(0.047619047619047616 + Float64(0.2 / (x ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.85], N[(x * N[(t$95$0 * N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 7.0], $MachinePrecision] * N[(t$95$0 * N[(0.047619047619047616 + N[(0.2 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 34.5%

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

   7. Taylor expanded in x around 0 34.8%

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

      1. associate-*r* 34.8%

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

      2. distribute-rgt-out 34.8%

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

      3. fma-define 34.8%

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

   9. Simplified 34.8%

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

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 34.5%

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

   7. Taylor expanded in x around inf 1.6%

      \[\leadsto \color{blue}{{x}^{7} \cdot \left(0.047619047619047616 \cdot \sqrt{\frac{1}{\pi}} + 0.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 1.6%

         \[\leadsto {x}^{7} \cdot \left(0.047619047619047616 \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \sqrt{\frac{1}{\pi}}}\right) \]

      2. distribute-rgt-out 1.6%

         \[\leadsto {x}^{7} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + 0.2 \cdot \frac{1}{{x}^{2}}\right)\right)} \]

      3. associate-*r/ 1.6%

         \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \color{blue}{\frac{0.2 \cdot 1}{{x}^{2}}}\right)\right) \]

      4. metadata-eval 1.6%

         \[\leadsto {x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \frac{\color{blue}{0.2}}{{x}^{2}}\right)\right) \]

   9. Simplified 1.6%

      \[\leadsto \color{blue}{{x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 33.6% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{\pi} \cdot \left(\frac{-88.2}{{x}^{2}} + 21\right)}{{x}^{6}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (* x (* (sqrt (/ 1.0 PI)) (fma 0.6666666666666666 (pow x 2.0) 2.0)))
   (/ x (/ (* (sqrt PI) (+ (/ -88.2 (pow x 2.0)) 21.0)) (pow x 6.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = x * (sqrt((1.0 / ((double) M_PI))) * fma(0.6666666666666666, pow(x, 2.0), 2.0));
	} else {
		tmp = x / ((sqrt(((double) M_PI)) * ((-88.2 / pow(x, 2.0)) + 21.0)) / pow(x, 6.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = Float64(x * Float64(sqrt(Float64(1.0 / pi)) * fma(0.6666666666666666, (x ^ 2.0), 2.0)));
	else
		tmp = Float64(x / Float64(Float64(sqrt(pi) * Float64(Float64(-88.2 / (x ^ 2.0)) + 21.0)) / (x ^ 6.0)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[(x * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[(-88.2 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + 21.0), $MachinePrecision]), $MachinePrecision] / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 34.5%

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

   7. Taylor expanded in x around 0 34.8%

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

      1. associate-*r* 34.8%

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

      2. distribute-rgt-out 34.8%

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

      3. fma-define 34.8%

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

   9. Simplified 34.8%

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

   if 2.2000000000000002 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.4%

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

      1. fma-undefine 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. add-sqr-sqrt 33.4%

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

      2. fabs-sqr 33.4%

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

      3. add-sqr-sqrt 34.8%

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

      4. add-sqr-sqrt 34.1%

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

      5. fabs-sqr 34.1%

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

      6. add-sqr-sqrt 34.8%

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

      7. clear-num 34.8%

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

      8. un-div-inv 34.5%

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

      9. associate-+l+ 34.5%

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

      10. fma-define 34.5%

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

      11. fma-define 34.5%

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

   9. Applied egg-rr 34.5%

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

   10. Taylor expanded in x around inf 3.5%

       \[\leadsto \frac{x}{\color{blue}{\frac{-88.2 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\pi}\right) + 21 \cdot \sqrt{\pi}}{{x}^{6}}}} \]
   11. Step-by-step derivation

       1. associate-*r* 3.5%

          \[\leadsto \frac{x}{\frac{\color{blue}{\left(-88.2 \cdot \frac{1}{{x}^{2}}\right) \cdot \sqrt{\pi}} + 21 \cdot \sqrt{\pi}}{{x}^{6}}} \]

       2. distribute-rgt-out 3.5%

          \[\leadsto \frac{x}{\frac{\color{blue}{\sqrt{\pi} \cdot \left(-88.2 \cdot \frac{1}{{x}^{2}} + 21\right)}}{{x}^{6}}} \]

       3. associate-*r/ 3.5%

          \[\leadsto \frac{x}{\frac{\sqrt{\pi} \cdot \left(\color{blue}{\frac{-88.2 \cdot 1}{{x}^{2}}} + 21\right)}{{x}^{6}}} \]

       4. metadata-eval 3.5%

          \[\leadsto \frac{x}{\frac{\sqrt{\pi} \cdot \left(\frac{\color{blue}{-88.2}}{{x}^{2}} + 21\right)}{{x}^{6}}} \]

   12. Simplified 3.5%

       \[\leadsto \frac{x}{\color{blue}{\frac{\sqrt{\pi} \cdot \left(\frac{-88.2}{{x}^{2}} + 21\right)}{{x}^{6}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{\pi} \cdot \left(\frac{-88.2}{{x}^{2}} + 21\right)}{{x}^{6}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 33.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (* x (* (sqrt (/ 1.0 PI)) (fma 0.6666666666666666 (pow x 2.0) 2.0)))
   (/ x (* (sqrt PI) (/ 21.0 (pow x 6.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = x * (sqrt((1.0 / ((double) M_PI))) * fma(0.6666666666666666, pow(x, 2.0), 2.0));
	} else {
		tmp = x / (sqrt(((double) M_PI)) * (21.0 / pow(x, 6.0)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = Float64(x * Float64(sqrt(Float64(1.0 / pi)) * fma(0.6666666666666666, (x ^ 2.0), 2.0)));
	else
		tmp = Float64(x / Float64(sqrt(pi) * Float64(21.0 / (x ^ 6.0))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[(x * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[Sqrt[Pi], $MachinePrecision] * N[(21.0 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 34.5%

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

   7. Taylor expanded in x around 0 34.8%

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

      1. associate-*r* 34.8%

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

      2. distribute-rgt-out 34.8%

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

      3. fma-define 34.8%

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

   9. Simplified 34.8%

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

   if 2.2000000000000002 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.4%

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

      1. fma-undefine 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. add-sqr-sqrt 33.4%

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

      2. fabs-sqr 33.4%

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

      3. add-sqr-sqrt 34.8%

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

      4. add-sqr-sqrt 34.1%

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

      5. fabs-sqr 34.1%

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

      6. add-sqr-sqrt 34.8%

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

      7. clear-num 34.8%

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

      8. un-div-inv 34.5%

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

      9. associate-+l+ 34.5%

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

      10. fma-define 34.5%

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

      11. fma-define 34.5%

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

   9. Applied egg-rr 34.5%

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

   10. Taylor expanded in x around inf 3.5%

       \[\leadsto \frac{x}{\color{blue}{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}} \]
   11. Step-by-step derivation

       1. associate-*r* 3.5%

          \[\leadsto \frac{x}{\color{blue}{\left(21 \cdot \frac{1}{{x}^{6}}\right) \cdot \sqrt{\pi}}} \]

       2. *-commutative 3.5%

          \[\leadsto \frac{x}{\color{blue}{\sqrt{\pi} \cdot \left(21 \cdot \frac{1}{{x}^{6}}\right)}} \]

       3. associate-*r/ 3.5%

          \[\leadsto \frac{x}{\sqrt{\pi} \cdot \color{blue}{\frac{21 \cdot 1}{{x}^{6}}}} \]

       4. metadata-eval 3.5%

          \[\leadsto \frac{x}{\sqrt{\pi} \cdot \frac{\color{blue}{21}}{{x}^{6}}} \]

   12. Simplified 3.5%

       \[\leadsto \frac{x}{\color{blue}{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 33.5% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;2 \cdot \left(x \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left(t\_0 \cdot {x}^{7}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (if (<= x 1.9)
     (* 2.0 (* x t_0))
     (* 0.047619047619047616 (* t_0 (pow x 7.0))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double tmp;
	if (x <= 1.9) {
		tmp = 2.0 * (x * t_0);
	} else {
		tmp = 0.047619047619047616 * (t_0 * pow(x, 7.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	double tmp;
	if (x <= 1.9) {
		tmp = 2.0 * (x * t_0);
	} else {
		tmp = 0.047619047619047616 * (t_0 * Math.pow(x, 7.0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / math.pi))
	tmp = 0
	if x <= 1.9:
		tmp = 2.0 * (x * t_0)
	else:
		tmp = 0.047619047619047616 * (t_0 * math.pow(x, 7.0))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / pi))
	tmp = 0.0
	if (x <= 1.9)
		tmp = Float64(2.0 * Float64(x * t_0));
	else
		tmp = Float64(0.047619047619047616 * Float64(t_0 * (x ^ 7.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 / pi));
	tmp = 0.0;
	if (x <= 1.9)
		tmp = 2.0 * (x * t_0);
	else
		tmp = 0.047619047619047616 * (t_0 * (x ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.9], N[(2.0 * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(t$95$0 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 34.5%

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

   7. Taylor expanded in x around 0 34.8%

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

   if 1.8999999999999999 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 34.5%

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

   7. Taylor expanded in x around inf 3.5%

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

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.5% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.9)
   (* 2.0 (* x (sqrt (/ 1.0 PI))))
   (/ x (* (sqrt PI) (/ 21.0 (pow x 6.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = 2.0 * (x * sqrt((1.0 / ((double) M_PI))));
	} else {
		tmp = x / (sqrt(((double) M_PI)) * (21.0 / pow(x, 6.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = 2.0 * (x * Math.sqrt((1.0 / Math.PI)));
	} else {
		tmp = x / (Math.sqrt(Math.PI) * (21.0 / Math.pow(x, 6.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.9:
		tmp = 2.0 * (x * math.sqrt((1.0 / math.pi)))
	else:
		tmp = x / (math.sqrt(math.pi) * (21.0 / math.pow(x, 6.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.9)
		tmp = Float64(2.0 * Float64(x * sqrt(Float64(1.0 / pi))));
	else
		tmp = Float64(x / Float64(sqrt(pi) * Float64(21.0 / (x ^ 6.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.9)
		tmp = 2.0 * (x * sqrt((1.0 / pi)));
	else
		tmp = x / (sqrt(pi) * (21.0 / (x ^ 6.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.9], N[(2.0 * N[(x * N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[Sqrt[Pi], $MachinePrecision] * N[(21.0 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 34.5%

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

   7. Taylor expanded in x around 0 34.8%

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

   if 1.8999999999999999 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.4%

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

      1. fma-undefine 99.4%

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

   7. Applied egg-rr 99.4%

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

      1. add-sqr-sqrt 33.4%

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

      2. fabs-sqr 33.4%

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

      3. add-sqr-sqrt 34.8%

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

      4. add-sqr-sqrt 34.1%

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

      5. fabs-sqr 34.1%

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

      6. add-sqr-sqrt 34.8%

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

      7. clear-num 34.8%

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

      8. un-div-inv 34.5%

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

      9. associate-+l+ 34.5%

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

      10. fma-define 34.5%

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

      11. fma-define 34.5%

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

   9. Applied egg-rr 34.5%

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

   10. Taylor expanded in x around inf 3.5%

       \[\leadsto \frac{x}{\color{blue}{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}} \]
   11. Step-by-step derivation

       1. associate-*r* 3.5%

          \[\leadsto \frac{x}{\color{blue}{\left(21 \cdot \frac{1}{{x}^{6}}\right) \cdot \sqrt{\pi}}} \]

       2. *-commutative 3.5%

          \[\leadsto \frac{x}{\color{blue}{\sqrt{\pi} \cdot \left(21 \cdot \frac{1}{{x}^{6}}\right)}} \]

       3. associate-*r/ 3.5%

          \[\leadsto \frac{x}{\sqrt{\pi} \cdot \color{blue}{\frac{21 \cdot 1}{{x}^{6}}}} \]

       4. metadata-eval 3.5%

          \[\leadsto \frac{x}{\sqrt{\pi} \cdot \frac{\color{blue}{21}}{{x}^{6}}} \]

   12. Simplified 3.5%

       \[\leadsto \frac{x}{\color{blue}{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 33.5% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

6. Applied egg-rr 34.5%

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

7. Taylor expanded in x around 0 34.8%

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

8. Final simplification 34.8%

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

Alternative 11: 33.3% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\sqrt{\pi} \cdot 0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x (* (sqrt PI) 0.5)))

C

double code(double x) {
	return x / (sqrt(((double) M_PI)) * 0.5);
}

Java

public static double code(double x) {
	return x / (Math.sqrt(Math.PI) * 0.5);
}

Python

def code(x):
	return x / (math.sqrt(math.pi) * 0.5)

Julia

function code(x)
	return Float64(x / Float64(sqrt(pi) * 0.5))
end

MATLAB

function tmp = code(x)
	tmp = x / (sqrt(pi) * 0.5);
end

Wolfram

code[x_] := N[(x / N[(N[Sqrt[Pi], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 99.4%

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

   1. fma-undefine 99.4%

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

7. Applied egg-rr 99.4%

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

   1. add-sqr-sqrt 33.4%

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

   2. fabs-sqr 33.4%

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

   3. add-sqr-sqrt 34.8%

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

   4. add-sqr-sqrt 34.1%

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

   5. fabs-sqr 34.1%

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

   6. add-sqr-sqrt 34.8%

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

   7. clear-num 34.8%

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

   8. un-div-inv 34.5%

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

   9. associate-+l+ 34.5%

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

   10. fma-define 34.5%

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

   11. fma-define 34.5%

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

9. Applied egg-rr 34.5%

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

10. Taylor expanded in x around 0 34.6%

    \[\leadsto \frac{x}{\color{blue}{0.5 \cdot \sqrt{\pi}}} \]

11. Final simplification 34.6%

    \[\leadsto \frac{x}{\sqrt{\pi} \cdot 0.5} \]
12. Add Preprocessing

Alternative 12: 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 99.9%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. rem-square-sqrt 33.4%

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

   4. fabs-sqr 33.4%

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

   5. rem-square-sqrt 99.4%

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

   6. associate-+r+ 99.4%

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

   7. +-commutative 99.4%

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

   8. fma-define 99.4%

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

   9. rem-square-sqrt 33.6%

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

   10. fabs-sqr 33.6%

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

   11. rem-square-sqrt 99.4%

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

   12. +-commutative 99.4%

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

   13. fma-define 99.4%

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

   14. rem-square-sqrt 33.6%

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

   15. fabs-sqr 33.6%

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

   16. rem-square-sqrt 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in x around 0 68.5%

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

   1. associate-*r* 68.5%

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

10. Simplified 68.5%

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

    1. expm1-log1p-u 66.5%

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

    2. expm1-undefine 4.7%

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

    3. *-commutative 4.7%

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

    4. associate-*l* 4.7%

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

    5. sqrt-div 4.7%

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

    6. metadata-eval 4.7%

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

    7. un-div-inv 4.7%

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

12. Applied egg-rr 4.7%

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

    1. sub-neg 4.7%

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

    2. metadata-eval 4.7%

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

    3. +-commutative 4.7%

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

    4. log1p-undefine 4.7%

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

    5. rem-exp-log 6.7%

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

    6. +-commutative 6.7%

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

    7. fma-define 6.7%

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

14. Simplified 6.7%

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

15. Taylor expanded in x around 0 4.0%

    \[\leadsto \left|-1 + \color{blue}{1}\right| \]

16. Final simplification 4.0%

    \[\leadsto 0 \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024089 
(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)))))))