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

Percentage Accurate: 99.8% → 99.9%

Time: 10.7s

Alternatives: 8

Speedup: 3.6×

• 29.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

   1. div-inv 99.9%

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

   2. pow1/2 99.9%

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

   3. pow-flip 99.9%

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

   4. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

   1. expm1-log1p-u 99.7%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot {\pi}^{-0.5}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   2. expm1-udef 36.7%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\left|x\right| \cdot {\pi}^{-0.5}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   3. add-sqr-sqrt 2.7%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   4. fabs-sqr 2.7%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   5. add-sqr-sqrt 6.5%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{x} \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

8. Applied egg-rr 6.5%

   \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]
9. Step-by-step derivation

   1. expm1-def 69.4%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   2. expm1-log1p 99.9%

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

10. Simplified 99.9%

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

11. Taylor expanded in x around 0 99.9%

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

    1. fma-def 99.9%

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

13. Simplified 99.9%

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

14. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

   1. div-inv 99.9%

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

   2. pow1/2 99.9%

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

   3. pow-flip 99.9%

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

   4. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

   1. expm1-log1p-u 99.7%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot {\pi}^{-0.5}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   2. expm1-udef 36.7%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\left|x\right| \cdot {\pi}^{-0.5}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   3. add-sqr-sqrt 2.7%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   4. fabs-sqr 2.7%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   5. add-sqr-sqrt 6.5%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{x} \cdot {\pi}^{-0.5}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

8. Applied egg-rr 6.5%

   \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]
9. Step-by-step derivation

   1. expm1-def 69.4%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)\right)} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

   2. expm1-log1p 99.9%

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

10. Simplified 99.9%

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

    1. fma-udef 36.2%

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

12. Applied egg-rr 99.9%

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

13. Final simplification 99.9%

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

Alternative 3: 33.7% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 98.5%

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

   1. *-un-lft-identity 98.5%

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

   2. add-sqr-sqrt 98.7%

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

   3. fabs-sqr 98.7%

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

   4. add-sqr-sqrt 98.5%

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

   5. div-inv 98.5%

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

   6. times-frac 98.9%

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

   7. pow1/2 98.9%

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

   8. pow-flip 98.9%

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

   9. metadata-eval 98.9%

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

7. Applied egg-rr 36.2%

   \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \frac{x}{\frac{1}{2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}}} \]
8. Step-by-step derivation

   1. associate-/r/ 36.2%

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

   2. /-rgt-identity 36.2%

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

   3. associate-*r* 36.2%

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

   4. *-commutative 36.2%

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

   5. fma-def 36.2%

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

   6. +-commutative 36.2%

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

   7. fma-def 36.2%

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

9. Simplified 36.2%

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

10. Final simplification 36.2%

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

Alternative 4: 33.7% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 98.5%

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

   1. div-inv 98.9%

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

   2. add-sqr-sqrt 34.7%

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

   3. fabs-sqr 34.7%

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

   4. add-sqr-sqrt 36.2%

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

   5. add-sqr-sqrt 36.2%

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

   6. fabs-sqr 36.2%

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

   7. add-sqr-sqrt 36.2%

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

   8. clear-num 36.2%

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

   9. +-commutative 36.2%

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

7. Applied egg-rr 36.2%

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

   1. fma-udef 36.2%

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

9. Applied egg-rr 36.2%

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

10. Final simplification 36.2%

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

Alternative 5: 33.8% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	double tmp;
	if (x <= 1.85) {
		tmp = 2.0 * (x * t_0);
	} else {
		tmp = 0.047619047619047616 * (t_0 * Math.pow(x, 7.0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / math.pi))
	tmp = 0
	if x <= 1.85:
		tmp = 2.0 * (x * t_0)
	else:
		tmp = 0.047619047619047616 * (t_0 * math.pow(x, 7.0))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 / pi));
	tmp = 0.0;
	if (x <= 1.85)
		tmp = 2.0 * (x * t_0);
	else
		tmp = 0.047619047619047616 * (t_0 * (x ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 98.5%

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

      1. div-inv 98.9%

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

      2. add-sqr-sqrt 34.7%

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

      3. fabs-sqr 34.7%

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

      4. add-sqr-sqrt 36.2%

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

      5. add-sqr-sqrt 36.2%

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

      6. fabs-sqr 36.2%

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

      7. add-sqr-sqrt 36.2%

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

      8. clear-num 36.2%

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

      9. +-commutative 36.2%

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

   7. Applied egg-rr 36.2%

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

   8. Taylor expanded in x around 0 36.3%

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

      1. *-commutative 36.3%

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

   10. Simplified 36.3%

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

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 98.5%

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

      1. div-inv 98.9%

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

      2. add-sqr-sqrt 34.7%

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

      3. fabs-sqr 34.7%

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

      4. add-sqr-sqrt 36.2%

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

      5. add-sqr-sqrt 36.2%

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

      6. fabs-sqr 36.2%

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

      7. add-sqr-sqrt 36.2%

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

      8. clear-num 36.2%

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

      9. +-commutative 36.2%

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

   7. Applied egg-rr 36.2%

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

   8. Taylor expanded in x around inf 3.8%

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

4. Final simplification 36.3%

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

Alternative 6: 33.7% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 98.5%

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

   1. div-inv 98.9%

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

   2. add-sqr-sqrt 34.7%

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

   3. fabs-sqr 34.7%

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

   4. add-sqr-sqrt 36.2%

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

   5. add-sqr-sqrt 36.2%

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

   6. fabs-sqr 36.2%

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

   7. add-sqr-sqrt 36.2%

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

   8. clear-num 36.2%

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

   9. +-commutative 36.2%

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

7. Applied egg-rr 36.2%

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

8. Taylor expanded in x around inf 36.2%

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

9. Final simplification 36.2%

   \[\leadsto x \cdot \frac{2 + 0.047619047619047616 \cdot {x}^{6}}{\sqrt{\pi}} \]
10. Add Preprocessing

Alternative 7: 33.8% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 98.5%

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

   1. div-inv 98.9%

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

   2. add-sqr-sqrt 34.7%

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

   3. fabs-sqr 34.7%

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

   4. add-sqr-sqrt 36.2%

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

   5. add-sqr-sqrt 36.2%

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

   6. fabs-sqr 36.2%

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

   7. add-sqr-sqrt 36.2%

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

   8. clear-num 36.2%

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

   9. +-commutative 36.2%

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

7. Applied egg-rr 36.2%

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

8. Taylor expanded in x around 0 36.3%

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

   1. *-commutative 36.3%

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

10. Simplified 36.3%

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

11. Final simplification 36.3%

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

Alternative 8: 33.6% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 98.5%

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

   1. div-inv 98.9%

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

   2. add-sqr-sqrt 34.7%

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

   3. fabs-sqr 34.7%

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

   4. add-sqr-sqrt 36.2%

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

   5. add-sqr-sqrt 36.2%

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

   6. fabs-sqr 36.2%

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

   7. add-sqr-sqrt 36.2%

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

   8. clear-num 36.2%

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

   9. +-commutative 36.2%

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

7. Applied egg-rr 36.2%

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

8. Taylor expanded in x around 0 36.3%

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

   1. *-commutative 36.3%

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

10. Simplified 36.3%

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

    1. expm1-log1p-u 36.2%

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

    2. expm1-udef 4.4%

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

    3. sqrt-div 4.4%

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

    4. metadata-eval 4.4%

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

    5. un-div-inv 4.4%

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

12. Applied egg-rr 4.4%

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

    1. expm1-def 36.0%

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

    2. expm1-log1p 36.1%

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

14. Simplified 36.1%

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

15. Final simplification 36.1%

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

Reproduce

herbie shell --seed 2024024 
(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)))))))