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

Percentage Accurate: 99.8% → 99.9%

Time: 12.9s

Alternatives: 13

Speedup: 2.6×

• 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, 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 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 34.8% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(x * N[(N[(2.0 + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision] + N[(N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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 99.8%

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

5. Taylor expanded in x around 0 99.8%

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

   1. expm1-log1p-u 99.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)\right)} \]

   2. expm1-undefine 41.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)} - 1} \]

7. Applied egg-rr 4.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} - 1} \]
8. Step-by-step derivation

   1. sub-neg 4.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} + \left(-1\right)} \]

   2. +-commutative 4.1%

      \[\leadsto \color{blue}{\left(-1\right) + e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)}} \]

   3. log1p-undefine 4.1%

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

   4. rem-exp-log 4.1%

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

   5. associate-+r+ 33.6%

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

   6. metadata-eval 33.6%

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

   7. metadata-eval 33.6%

      \[\leadsto \color{blue}{0} + x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \]

   8. *-commutative 33.6%

      \[\leadsto 0 + \color{blue}{\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \cdot x} \]

   9. cancel-sign-sub 33.6%

      \[\leadsto \color{blue}{0 - \left(-\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right) \cdot x} \]

9. Simplified 33.6%

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

    1. fma-undefine 33.6%

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

11. Applied egg-rr 33.6%

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

12. Final simplification 33.6%

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

Alternative 3: 34.6% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 1.0], N[(x * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(2.0 + N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 7.0], $MachinePrecision] * N[(N[(0.2 * N[Power[x, -2.0], $MachinePrecision] + 0.047619047619047616), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. expm1-log1p-u 99.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)\right)} \]

      2. expm1-undefine 9.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)} - 1} \]

   7. Applied egg-rr 6.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} - 1} \]
   8. Step-by-step derivation

      1. sub-neg 6.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} + \left(-1\right)} \]

      2. +-commutative 6.3%

         \[\leadsto \color{blue}{\left(-1\right) + e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)}} \]

      3. log1p-undefine 6.3%

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

      4. rem-exp-log 6.3%

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

      5. associate-+r+ 52.3%

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

      6. metadata-eval 52.3%

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

      7. metadata-eval 52.3%

         \[\leadsto \color{blue}{0} + x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \]

      8. *-commutative 52.3%

         \[\leadsto 0 + \color{blue}{\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \cdot x} \]

      9. cancel-sign-sub 52.3%

         \[\leadsto \color{blue}{0 - \left(-\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right) \cdot x} \]

   9. Simplified 52.3%

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

   10. Taylor expanded in x around 0 52.3%

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

       1. associate-*r* 52.3%

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

       2. *-commutative 52.3%

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

       3. distribute-rgt-out 52.3%

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

       4. *-commutative 52.3%

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

   12. Simplified 52.3%

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

   if 1 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. expm1-log1p-u 97.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)\right)} \]

      2. expm1-undefine 97.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)} - 1} \]

   7. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} - 1} \]
   8. Step-by-step derivation

      1. sub-neg 0.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} + \left(-1\right)} \]

      2. +-commutative 0.0%

         \[\leadsto \color{blue}{\left(-1\right) + e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)}} \]

      3. log1p-undefine 0.0%

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

      4. rem-exp-log 0.2%

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

      5. associate-+r+ 0.2%

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

      6. metadata-eval 0.2%

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

      7. metadata-eval 0.2%

         \[\leadsto \color{blue}{0} + x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \]

      8. *-commutative 0.2%

         \[\leadsto 0 + \color{blue}{\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \cdot x} \]

      9. cancel-sign-sub 0.2%

         \[\leadsto \color{blue}{0 - \left(-\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right) \cdot x} \]

   9. Simplified 0.2%

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

   10. Taylor expanded in x around inf 0.2%

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

       1. associate-*r* 0.2%

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

       2. distribute-rgt-out 0.2%

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

       3. associate-*r/ 0.2%

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

       4. metadata-eval 0.2%

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

   12. Simplified 0.2%

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

       1. *-commutative 0.2%

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

       2. sqrt-div 0.2%

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

       3. metadata-eval 0.2%

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

       4. un-div-inv 0.2%

          \[\leadsto {x}^{7} \cdot \color{blue}{\frac{0.047619047619047616 + \frac{0.2}{{x}^{2}}}{\sqrt{\pi}}} \]

       5. +-commutative 0.2%

          \[\leadsto {x}^{7} \cdot \frac{\color{blue}{\frac{0.2}{{x}^{2}} + 0.047619047619047616}}{\sqrt{\pi}} \]

       6. div-inv 0.2%

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

       7. fma-define 0.2%

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

       8. pow-flip 0.2%

          \[\leadsto {x}^{7} \cdot \frac{\mathsf{fma}\left(0.2, \color{blue}{{x}^{\left(-2\right)}}, 0.047619047619047616\right)}{\sqrt{\pi}} \]

       9. metadata-eval 0.2%

          \[\leadsto {x}^{7} \cdot \frac{\mathsf{fma}\left(0.2, {x}^{\color{blue}{-2}}, 0.047619047619047616\right)}{\sqrt{\pi}} \]

   14. Applied egg-rr 0.2%

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

4. Final simplification 33.6%

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

Alternative 4: 99.0% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 99.2%

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

   1. associate-*r* 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*l* 99.2%

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

   4. fma-define 99.2%

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

   5. metadata-eval 99.2%

      \[\leadsto \left|\left|x\right| \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{\color{blue}{\left(2 \cdot 3\right)}}, 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right)\right)\right| \]

   6. pow-sqr 99.2%

      \[\leadsto \left|\left|x\right| \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, \color{blue}{{\left(\left|x\right|\right)}^{3} \cdot {\left(\left|x\right|\right)}^{3}}, 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right)\right)\right| \]

   7. cube-prod 99.2%

      \[\leadsto \left|\left|x\right| \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, \color{blue}{{\left(\left|x\right| \cdot \left|x\right|\right)}^{3}}, 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right)\right)\right| \]

   8. sqr-abs 99.2%

      \[\leadsto \left|\left|x\right| \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {\color{blue}{\left(x \cdot x\right)}}^{3}, 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right)\right)\right| \]

   9. cube-prod 99.2%

      \[\leadsto \left|\left|x\right| \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, \color{blue}{{x}^{3} \cdot {x}^{3}}, 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right)\right)\right| \]

   10. pow-sqr 99.2%

       \[\leadsto \left|\left|x\right| \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, \color{blue}{{x}^{\left(2 \cdot 3\right)}}, 0.2 \cdot {\left(\left|x\right|\right)}^{4}\right)\right)\right)\right| \]

   11. metadata-eval 99.2%

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

   12. metadata-eval 99.2%

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

   13. pow-sqr 99.2%

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

   14. unpow2 99.2%

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

   15. sqr-abs 99.2%

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

   16. unpow2 99.2%

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

   17. unpow2 99.2%

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

   18. sqr-abs 99.2%

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

   19. unpow2 99.2%

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

7. Simplified 99.2%

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

   1. add-sqr-sqrt 32.1%

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

   2. fabs-sqr 32.1%

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

   3. add-sqr-sqrt 99.2%

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

   4. distribute-rgt-in 99.2%

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

   5. *-commutative 99.2%

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

   6. distribute-lft-in 99.2%

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

   7. *-commutative 99.2%

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

   8. sqrt-div 99.2%

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

   9. metadata-eval 99.2%

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

   10. un-div-inv 99.2%

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

   11. sqrt-div 99.2%

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

   12. metadata-eval 99.2%

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

9. Applied egg-rr 99.2%

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

    1. associate-*r/ 99.2%

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

    2. associate-*l/ 99.2%

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

    3. associate-*r/ 98.8%

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

    4. associate-*l/ 98.8%

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

    5. distribute-lft-in 98.8%

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

    6. associate-*l/ 98.8%

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

    7. associate-*r/ 99.2%

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

    8. fma-define 99.2%

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

    9. +-commutative 99.2%

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

    10. fma-define 99.2%

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

11. Simplified 99.2%

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

12. Final simplification 99.2%

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

Alternative 5: 34.6% accurate, 4.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (if (<= (fabs x) 1.0)
     (* x (* t_0 (+ 2.0 (* 0.6666666666666666 (pow x 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 (fabs(x) <= 1.0) {
		tmp = x * (t_0 * (2.0 + (0.6666666666666666 * pow(x, 2.0))));
	} else {
		tmp = pow(x, 7.0) * (t_0 * (0.047619047619047616 + (0.2 / pow(x, 2.0))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	double tmp;
	if (Math.abs(x) <= 1.0) {
		tmp = x * (t_0 * (2.0 + (0.6666666666666666 * Math.pow(x, 2.0))));
	} else {
		tmp = Math.pow(x, 7.0) * (t_0 * (0.047619047619047616 + (0.2 / Math.pow(x, 2.0))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / math.pi))
	tmp = 0
	if math.fabs(x) <= 1.0:
		tmp = x * (t_0 * (2.0 + (0.6666666666666666 * math.pow(x, 2.0))))
	else:
		tmp = math.pow(x, 7.0) * (t_0 * (0.047619047619047616 + (0.2 / math.pow(x, 2.0))))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 / pi));
	tmp = 0.0;
	if (abs(x) <= 1.0)
		tmp = x * (t_0 * (2.0 + (0.6666666666666666 * (x ^ 2.0))));
	else
		tmp = (x ^ 7.0) * (t_0 * (0.047619047619047616 + (0.2 / (x ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 1.0], N[(x * N[(t$95$0 * N[(2.0 + N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $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 (fabs.f64 x) < 1

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. expm1-log1p-u 99.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)\right)} \]

      2. expm1-undefine 9.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)} - 1} \]

   7. Applied egg-rr 6.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} - 1} \]
   8. Step-by-step derivation

      1. sub-neg 6.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} + \left(-1\right)} \]

      2. +-commutative 6.3%

         \[\leadsto \color{blue}{\left(-1\right) + e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)}} \]

      3. log1p-undefine 6.3%

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

      4. rem-exp-log 6.3%

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

      5. associate-+r+ 52.3%

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

      6. metadata-eval 52.3%

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

      7. metadata-eval 52.3%

         \[\leadsto \color{blue}{0} + x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \]

      8. *-commutative 52.3%

         \[\leadsto 0 + \color{blue}{\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \cdot x} \]

      9. cancel-sign-sub 52.3%

         \[\leadsto \color{blue}{0 - \left(-\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right) \cdot x} \]

   9. Simplified 52.3%

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

   10. Taylor expanded in x around 0 52.3%

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

       1. associate-*r* 52.3%

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

       2. *-commutative 52.3%

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

       3. distribute-rgt-out 52.3%

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

       4. *-commutative 52.3%

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

   12. Simplified 52.3%

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

   if 1 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. expm1-log1p-u 97.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)\right)} \]

      2. expm1-undefine 97.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)} - 1} \]

   7. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} - 1} \]
   8. Step-by-step derivation

      1. sub-neg 0.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} + \left(-1\right)} \]

      2. +-commutative 0.0%

         \[\leadsto \color{blue}{\left(-1\right) + e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)}} \]

      3. log1p-undefine 0.0%

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

      4. rem-exp-log 0.2%

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

      5. associate-+r+ 0.2%

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

      6. metadata-eval 0.2%

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

      7. metadata-eval 0.2%

         \[\leadsto \color{blue}{0} + x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \]

      8. *-commutative 0.2%

         \[\leadsto 0 + \color{blue}{\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \cdot x} \]

      9. cancel-sign-sub 0.2%

         \[\leadsto \color{blue}{0 - \left(-\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right) \cdot x} \]

   9. Simplified 0.2%

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

   10. Taylor expanded in x around inf 0.2%

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

       1. associate-*r* 0.2%

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

       2. distribute-rgt-out 0.2%

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

       3. associate-*r/ 0.2%

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

       4. metadata-eval 0.2%

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

   12. Simplified 0.2%

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

4. Final simplification 33.6%

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

Alternative 6: 34.5% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(x * N[(N[(2.0 + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision] + N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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 99.8%

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

5. Taylor expanded in x around 0 99.8%

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

   1. expm1-log1p-u 99.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)\right)} \]

   2. expm1-undefine 41.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)} - 1} \]

7. Applied egg-rr 4.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} - 1} \]
8. Step-by-step derivation

   1. sub-neg 4.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} + \left(-1\right)} \]

   2. +-commutative 4.1%

      \[\leadsto \color{blue}{\left(-1\right) + e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)}} \]

   3. log1p-undefine 4.1%

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

   4. rem-exp-log 4.1%

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

   5. associate-+r+ 33.6%

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

   6. metadata-eval 33.6%

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

   7. metadata-eval 33.6%

      \[\leadsto \color{blue}{0} + x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \]

   8. *-commutative 33.6%

      \[\leadsto 0 + \color{blue}{\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \cdot x} \]

   9. cancel-sign-sub 33.6%

      \[\leadsto \color{blue}{0 - \left(-\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right) \cdot x} \]

9. Simplified 33.6%

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

10. Taylor expanded in x around inf 33.5%

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

11. Final simplification 33.5%

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

Alternative 7: 67.5% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. expm1-log1p-u 99.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)\right)} \]

      2. expm1-undefine 9.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)} - 1} \]

   7. Applied egg-rr 6.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} - 1} \]
   8. Step-by-step derivation

      1. sub-neg 6.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} + \left(-1\right)} \]

      2. +-commutative 6.3%

         \[\leadsto \color{blue}{\left(-1\right) + e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)}} \]

      3. log1p-undefine 6.3%

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

      4. rem-exp-log 6.3%

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

      5. associate-+r+ 52.3%

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

      6. metadata-eval 52.3%

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

      7. metadata-eval 52.3%

         \[\leadsto \color{blue}{0} + x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \]

      8. *-commutative 52.3%

         \[\leadsto 0 + \color{blue}{\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \cdot x} \]

      9. cancel-sign-sub 52.3%

         \[\leadsto \color{blue}{0 - \left(-\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right) \cdot x} \]

   9. Simplified 52.3%

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

   10. Taylor expanded in x around 0 52.3%

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

       1. associate-*r* 52.3%

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

       2. *-commutative 52.3%

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

       3. distribute-rgt-out 52.3%

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

       4. *-commutative 52.3%

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

   12. Simplified 52.3%

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

   if 1 < (fabs.f64 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 99.7%

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

   5. Taylor expanded in x around inf 98.3%

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

      1. associate-*r* 98.2%

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

      2. sqrt-div 98.2%

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

      3. metadata-eval 98.2%

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

      4. un-div-inv 98.3%

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

      5. *-commutative 98.3%

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

      6. add-sqr-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. add-sqr-sqrt 98.3%

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

   7. Applied egg-rr 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-/l* 98.3%

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

      3. *-commutative 98.3%

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

      4. pow-plus 98.4%

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

      5. metadata-eval 98.4%

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

   9. Simplified 98.4%

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

4. Final simplification 68.9%

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

Alternative 8: 98.3% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 98.8%

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

6. Taylor expanded in x around inf 98.4%

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

7. Final simplification 98.4%

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

Alternative 9: 34.6% accurate, 8.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. expm1-log1p-u 99.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)\right)} \]

      2. expm1-undefine 41.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)} - 1} \]

   7. Applied egg-rr 4.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} - 1} \]
   8. Step-by-step derivation

      1. sub-neg 4.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} + \left(-1\right)} \]

      2. +-commutative 4.1%

         \[\leadsto \color{blue}{\left(-1\right) + e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)}} \]

      3. log1p-undefine 4.1%

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

      4. rem-exp-log 4.1%

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

      5. associate-+r+ 33.6%

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

      6. metadata-eval 33.6%

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

      7. metadata-eval 33.6%

         \[\leadsto \color{blue}{0} + x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \]

      8. *-commutative 33.6%

         \[\leadsto 0 + \color{blue}{\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \cdot x} \]

      9. cancel-sign-sub 33.6%

         \[\leadsto \color{blue}{0 - \left(-\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right) \cdot x} \]

   9. Simplified 33.6%

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

   10. Taylor expanded in x around 0 33.6%

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

       1. associate-*r* 33.6%

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

       2. *-commutative 33.6%

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

       3. distribute-rgt-out 33.6%

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

       4. *-commutative 33.6%

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

   12. Simplified 33.6%

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

   if 2.2000000000000002 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. expm1-log1p-u 99.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)\right)} \]

      2. expm1-undefine 41.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)} - 1} \]

   7. Applied egg-rr 4.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} - 1} \]
   8. Step-by-step derivation

      1. sub-neg 4.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} + \left(-1\right)} \]

      2. +-commutative 4.1%

         \[\leadsto \color{blue}{\left(-1\right) + e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)}} \]

      3. log1p-undefine 4.1%

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

      4. rem-exp-log 4.1%

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

      5. associate-+r+ 33.6%

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

      6. metadata-eval 33.6%

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

      7. metadata-eval 33.6%

         \[\leadsto \color{blue}{0} + x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \]

      8. *-commutative 33.6%

         \[\leadsto 0 + \color{blue}{\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \cdot x} \]

      9. cancel-sign-sub 33.6%

         \[\leadsto \color{blue}{0 - \left(-\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right) \cdot x} \]

   9. Simplified 33.6%

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

   10. Taylor expanded in x around inf 3.5%

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

       1. associate-*r* 3.5%

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

       2. *-commutative 3.5%

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

   12. Simplified 3.5%

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

       1. *-commutative 3.5%

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

       2. sqrt-div 3.5%

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

       3. metadata-eval 3.5%

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

       4. div-inv 3.5%

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

   14. Applied egg-rr 3.5%

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

4. Final simplification 33.6%

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

Alternative 10: 34.6% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{14} \cdot \frac{0.0022675736961451248}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (* x (/ 2.0 (sqrt PI)))
   (sqrt (* (pow x 14.0) (/ 0.0022675736961451248 PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = sqrt((pow(x, 14.0) * (0.0022675736961451248 / ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.sqrt((Math.pow(x, 14.0) * (0.0022675736961451248 / Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.sqrt((math.pow(x, 14.0) * (0.0022675736961451248 / math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = sqrt(Float64((x ^ 14.0) * Float64(0.0022675736961451248 / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = sqrt(((x ^ 14.0) * (0.0022675736961451248 / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[x, 14.0], $MachinePrecision] * N[(0.0022675736961451248 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. expm1-log1p-u 99.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)\right)} \]

      2. expm1-undefine 41.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)} - 1} \]

   7. Applied egg-rr 4.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} - 1} \]
   8. Step-by-step derivation

      1. sub-neg 4.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} + \left(-1\right)} \]

      2. +-commutative 4.1%

         \[\leadsto \color{blue}{\left(-1\right) + e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)}} \]

      3. log1p-undefine 4.1%

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

      4. rem-exp-log 4.1%

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

      5. associate-+r+ 33.6%

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

      6. metadata-eval 33.6%

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

      7. metadata-eval 33.6%

         \[\leadsto \color{blue}{0} + x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \]

      8. *-commutative 33.6%

         \[\leadsto 0 + \color{blue}{\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \cdot x} \]

      9. cancel-sign-sub 33.6%

         \[\leadsto \color{blue}{0 - \left(-\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right) \cdot x} \]

   9. Simplified 33.6%

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

   10. Taylor expanded in x around 0 33.6%

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

       1. associate-*r* 33.6%

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

       2. *-commutative 33.6%

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

   12. Simplified 33.6%

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

       1. sqrt-div 33.6%

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

       2. metadata-eval 33.6%

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

       3. un-div-inv 33.4%

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

   14. Applied egg-rr 33.4%

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

       1. associate-*r/ 33.6%

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

   16. Simplified 33.6%

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

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. expm1-log1p-u 99.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)\right)} \]

      2. expm1-undefine 41.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)} - 1} \]

   7. Applied egg-rr 4.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} - 1} \]
   8. Step-by-step derivation

      1. sub-neg 4.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} + \left(-1\right)} \]

      2. +-commutative 4.1%

         \[\leadsto \color{blue}{\left(-1\right) + e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)}} \]

      3. log1p-undefine 4.1%

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

      4. rem-exp-log 4.1%

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

      5. associate-+r+ 33.6%

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

      6. metadata-eval 33.6%

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

      7. metadata-eval 33.6%

         \[\leadsto \color{blue}{0} + x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \]

      8. *-commutative 33.6%

         \[\leadsto 0 + \color{blue}{\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \cdot x} \]

      9. cancel-sign-sub 33.6%

         \[\leadsto \color{blue}{0 - \left(-\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right) \cdot x} \]

   9. Simplified 33.6%

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

   10. Taylor expanded in x around inf 3.5%

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

       1. associate-*r* 3.5%

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

       2. *-commutative 3.5%

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

   12. Simplified 3.5%

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

       1. add-sqr-sqrt 3.4%

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

       2. sqrt-unprod 35.3%

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

       3. *-commutative 35.3%

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

       4. sqrt-div 35.3%

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

       5. metadata-eval 35.3%

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

       6. div-inv 35.3%

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

       7. *-commutative 35.3%

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

       8. sqrt-div 35.3%

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

       9. metadata-eval 35.3%

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

       10. div-inv 35.4%

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

       11. frac-times 35.4%

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

   14. Applied egg-rr 35.4%

       \[\leadsto \color{blue}{\sqrt{\frac{{x}^{14} \cdot 0.0022675736961451248}{\pi}}} \]
   15. Step-by-step derivation

       1. associate-/l* 35.4%

          \[\leadsto \sqrt{\color{blue}{{x}^{14} \cdot \frac{0.0022675736961451248}{\pi}}} \]

   16. Simplified 35.4%

       \[\leadsto \color{blue}{\sqrt{{x}^{14} \cdot \frac{0.0022675736961451248}{\pi}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{14} \cdot \frac{0.0022675736961451248}{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 34.6% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. expm1-log1p-u 99.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)\right)} \]

      2. expm1-undefine 41.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)} - 1} \]

   7. Applied egg-rr 4.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} - 1} \]
   8. Step-by-step derivation

      1. sub-neg 4.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} + \left(-1\right)} \]

      2. +-commutative 4.1%

         \[\leadsto \color{blue}{\left(-1\right) + e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)}} \]

      3. log1p-undefine 4.1%

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

      4. rem-exp-log 4.1%

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

      5. associate-+r+ 33.6%

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

      6. metadata-eval 33.6%

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

      7. metadata-eval 33.6%

         \[\leadsto \color{blue}{0} + x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \]

      8. *-commutative 33.6%

         \[\leadsto 0 + \color{blue}{\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \cdot x} \]

      9. cancel-sign-sub 33.6%

         \[\leadsto \color{blue}{0 - \left(-\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right) \cdot x} \]

   9. Simplified 33.6%

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

   10. Taylor expanded in x around 0 33.6%

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

       1. associate-*r* 33.6%

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

       2. *-commutative 33.6%

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

   12. Simplified 33.6%

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

       1. sqrt-div 33.6%

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

       2. metadata-eval 33.6%

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

       3. un-div-inv 33.4%

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

   14. Applied egg-rr 33.4%

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

       1. associate-*r/ 33.6%

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

   16. Simplified 33.6%

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

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. expm1-log1p-u 99.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)\right)} \]

      2. expm1-undefine 41.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)} - 1} \]

   7. Applied egg-rr 4.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} - 1} \]
   8. Step-by-step derivation

      1. sub-neg 4.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} + \left(-1\right)} \]

      2. +-commutative 4.1%

         \[\leadsto \color{blue}{\left(-1\right) + e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)}} \]

      3. log1p-undefine 4.1%

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

      4. rem-exp-log 4.1%

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

      5. associate-+r+ 33.6%

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

      6. metadata-eval 33.6%

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

      7. metadata-eval 33.6%

         \[\leadsto \color{blue}{0} + x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \]

      8. *-commutative 33.6%

         \[\leadsto 0 + \color{blue}{\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \cdot x} \]

      9. cancel-sign-sub 33.6%

         \[\leadsto \color{blue}{0 - \left(-\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right) \cdot x} \]

   9. Simplified 33.6%

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

   10. Taylor expanded in x around inf 3.5%

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

       1. associate-*r* 3.5%

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

       2. *-commutative 3.5%

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

   12. Simplified 3.5%

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

       1. sqrt-div 3.5%

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

       2. metadata-eval 3.5%

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

       3. associate-/r/ 3.5%

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

       4. *-un-lft-identity 3.5%

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

       5. times-frac 3.5%

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

       6. metadata-eval 3.5%

          \[\leadsto \frac{1}{\color{blue}{21} \cdot \frac{\sqrt{\pi}}{{x}^{7}}} \]

   14. Applied egg-rr 3.5%

       \[\leadsto \color{blue}{\frac{1}{21 \cdot \frac{\sqrt{\pi}}{{x}^{7}}}} \]
   15. Step-by-step derivation

       1. associate-/r* 3.5%

          \[\leadsto \color{blue}{\frac{\frac{1}{21}}{\frac{\sqrt{\pi}}{{x}^{7}}}} \]

       2. metadata-eval 3.5%

          \[\leadsto \frac{\color{blue}{0.047619047619047616}}{\frac{\sqrt{\pi}}{{x}^{7}}} \]

   16. Simplified 3.5%

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

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616}{\frac{\sqrt{\pi}}{{x}^{7}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 34.6% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. expm1-log1p-u 99.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)\right)} \]

      2. expm1-undefine 41.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)} - 1} \]

   7. Applied egg-rr 4.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} - 1} \]
   8. Step-by-step derivation

      1. sub-neg 4.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} + \left(-1\right)} \]

      2. +-commutative 4.1%

         \[\leadsto \color{blue}{\left(-1\right) + e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)}} \]

      3. log1p-undefine 4.1%

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

      4. rem-exp-log 4.1%

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

      5. associate-+r+ 33.6%

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

      6. metadata-eval 33.6%

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

      7. metadata-eval 33.6%

         \[\leadsto \color{blue}{0} + x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \]

      8. *-commutative 33.6%

         \[\leadsto 0 + \color{blue}{\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \cdot x} \]

      9. cancel-sign-sub 33.6%

         \[\leadsto \color{blue}{0 - \left(-\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right) \cdot x} \]

   9. Simplified 33.6%

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

   10. Taylor expanded in x around 0 33.6%

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

       1. associate-*r* 33.6%

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

       2. *-commutative 33.6%

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

   12. Simplified 33.6%

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

       1. sqrt-div 33.6%

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

       2. metadata-eval 33.6%

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

       3. un-div-inv 33.4%

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

   14. Applied egg-rr 33.4%

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

       1. associate-*r/ 33.6%

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

   16. Simplified 33.6%

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

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. expm1-log1p-u 99.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)\right)} \]

      2. expm1-undefine 41.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)} - 1} \]

   7. Applied egg-rr 4.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} - 1} \]
   8. Step-by-step derivation

      1. sub-neg 4.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} + \left(-1\right)} \]

      2. +-commutative 4.1%

         \[\leadsto \color{blue}{\left(-1\right) + e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)}} \]

      3. log1p-undefine 4.1%

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

      4. rem-exp-log 4.1%

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

      5. associate-+r+ 33.6%

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

      6. metadata-eval 33.6%

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

      7. metadata-eval 33.6%

         \[\leadsto \color{blue}{0} + x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \]

      8. *-commutative 33.6%

         \[\leadsto 0 + \color{blue}{\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \cdot x} \]

      9. cancel-sign-sub 33.6%

         \[\leadsto \color{blue}{0 - \left(-\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right) \cdot x} \]

   9. Simplified 33.6%

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

   10. Taylor expanded in x around inf 3.5%

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

       1. associate-*r* 3.5%

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

       2. *-commutative 3.5%

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

   12. Simplified 3.5%

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

       1. *-commutative 3.5%

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

       2. sqrt-div 3.5%

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

       3. metadata-eval 3.5%

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

       4. div-inv 3.5%

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

   14. Applied egg-rr 3.5%

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

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 34.6% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(x * N[(2.0 / 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 99.8%

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

5. Taylor expanded in x around 0 99.8%

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

   1. expm1-log1p-u 99.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)\right)} \]

   2. expm1-undefine 41.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|\right)} - 1} \]

7. Applied egg-rr 4.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} - 1} \]
8. Step-by-step derivation

   1. sub-neg 4.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)} + \left(-1\right)} \]

   2. +-commutative 4.1%

      \[\leadsto \color{blue}{\left(-1\right) + e^{\mathsf{log1p}\left(x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right)}} \]

   3. log1p-undefine 4.1%

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

   4. rem-exp-log 4.1%

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

   5. associate-+r+ 33.6%

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

   6. metadata-eval 33.6%

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

   7. metadata-eval 33.6%

      \[\leadsto \color{blue}{0} + x \cdot \frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \]

   8. *-commutative 33.6%

      \[\leadsto 0 + \color{blue}{\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \cdot x} \]

   9. cancel-sign-sub 33.6%

      \[\leadsto \color{blue}{0 - \left(-\frac{2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right) \cdot x} \]

9. Simplified 33.6%

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

10. Taylor expanded in x around 0 33.6%

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

    1. associate-*r* 33.6%

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

    2. *-commutative 33.6%

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

12. Simplified 33.6%

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

    1. sqrt-div 33.6%

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

    2. metadata-eval 33.6%

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

    3. un-div-inv 33.4%

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

14. Applied egg-rr 33.4%

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

    1. associate-*r/ 33.6%

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

16. Simplified 33.6%

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

17. Final simplification 33.6%

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

Reproduce

herbie shell --seed 2024051 
(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)))))))