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

Percentage Accurate: 99.8% → 99.9%

Time: 11.8s

Alternatives: 13

Speedup: 2.6×

• 28.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% 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 99.9%

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

Alternative 2: 99.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[Abs[N[(N[(N[Abs[x], $MachinePrecision] * N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] + N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 99.5%

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

   1. metadata-eval 99.5%

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

   2. metadata-eval 99.5%

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

   3. pow-prod-up 99.5%

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

   4. pow-prod-down 99.5%

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

   5. sqr-abs 99.5%

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

   6. pow-prod-down 99.5%

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

   7. pow-prod-up 99.5%

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

   8. metadata-eval 99.5%

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

   9. metadata-eval 99.5%

      \[\leadsto \left|\frac{\left|x\right| \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(\frac{1}{5}, {x}^{4}, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{\color{blue}{\left(3 + 3\right)}}\right)\right)}{\sqrt{\pi}}\right| \]

   10. pow-prod-up 99.4%

       \[\leadsto \left|\frac{\left|x\right| \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(\frac{1}{5}, {x}^{4}, 0.047619047619047616 \cdot \color{blue}{\left({\left(\left|x\right|\right)}^{3} \cdot {\left(\left|x\right|\right)}^{3}\right)}\right)\right)}{\sqrt{\pi}}\right| \]

   11. pow-prod-down 99.4%

       \[\leadsto \left|\frac{\left|x\right| \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(\frac{1}{5}, {x}^{4}, 0.047619047619047616 \cdot \color{blue}{{\left(\left|x\right| \cdot \left|x\right|\right)}^{3}}\right)\right)}{\sqrt{\pi}}\right| \]

   12. sqr-abs 99.4%

       \[\leadsto \left|\frac{\left|x\right| \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(\frac{1}{5}, {x}^{4}, 0.047619047619047616 \cdot {\color{blue}{\left(x \cdot x\right)}}^{3}\right)\right)}{\sqrt{\pi}}\right| \]

   13. pow-prod-down 99.4%

       \[\leadsto \left|\frac{\left|x\right| \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(\frac{1}{5}, {x}^{4}, 0.047619047619047616 \cdot \color{blue}{\left({x}^{3} \cdot {x}^{3}\right)}\right)\right)}{\sqrt{\pi}}\right| \]

   14. pow-prod-up 99.5%

       \[\leadsto \left|\frac{\left|x\right| \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(\frac{1}{5}, {x}^{4}, 0.047619047619047616 \cdot \color{blue}{{x}^{\left(3 + 3\right)}}\right)\right)}{\sqrt{\pi}}\right| \]

   15. metadata-eval 99.5%

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

   16. fma-undefine 99.5%

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

   17. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 3: 99.2% accurate, 3.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	return fabs(x) * fabs((((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(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[(0.047619047619047616 * N[Power[x, 6.0], $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 99.9%

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

5. Taylor expanded in x around inf 98.6%

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

Alternative 4: 67.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2:\\ \;\;\;\;x \cdot \sqrt{\frac{4}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 2.0)
   (* x (sqrt (/ 4.0 PI)))
   (fabs (* (pow PI -0.5) (* 0.047619047619047616 (pow x 7.0))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 2.0) {
		tmp = x * sqrt((4.0 / ((double) M_PI)));
	} else {
		tmp = fabs((pow(((double) M_PI), -0.5) * (0.047619047619047616 * pow(x, 7.0))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 2.0) {
		tmp = x * Math.sqrt((4.0 / Math.PI));
	} else {
		tmp = Math.abs((Math.pow(Math.PI, -0.5) * (0.047619047619047616 * Math.pow(x, 7.0))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 2.0:
		tmp = x * math.sqrt((4.0 / math.pi))
	else:
		tmp = math.fabs((math.pow(math.pi, -0.5) * (0.047619047619047616 * math.pow(x, 7.0))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 2.0)
		tmp = Float64(x * sqrt(Float64(4.0 / pi)));
	else
		tmp = abs(Float64((pi ^ -0.5) * Float64(0.047619047619047616 * (x ^ 7.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 2.0)
		tmp = x * sqrt((4.0 / pi));
	else
		tmp = abs(((pi ^ -0.5) * (0.047619047619047616 * (x ^ 7.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 2.0], N[(x * N[Sqrt[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2

   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%

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

   5. Taylor expanded in x around 0 98.5%

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

      1. associate-*r* 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in x around 0 98.5%

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

      1. *-commutative 98.5%

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

      2. associate-*r* 98.5%

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

      3. rem-exp-log 98.5%

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

      4. rec-exp 98.5%

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

      5. unpow1/2 98.5%

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

      6. exp-prod 98.5%

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

      7. distribute-lft-neg-out 98.5%

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

      8. distribute-rgt-neg-in 98.5%

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

      9. metadata-eval 98.5%

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

      10. exp-to-pow 98.5%

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

      11. *-commutative 98.5%

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

      12. associate-*l* 98.5%

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

      13. fabs-mul 98.5%

          \[\leadsto \color{blue}{\left|\left|x\right|\right| \cdot \left|2 \cdot {\pi}^{-0.5}\right|} \]

      14. fabs-fabs 98.5%

          \[\leadsto \color{blue}{\left|x\right|} \cdot \left|2 \cdot {\pi}^{-0.5}\right| \]

      15. rem-square-sqrt 96.8%

          \[\leadsto \left|x\right| \cdot \left|\color{blue}{\sqrt{2 \cdot {\pi}^{-0.5}} \cdot \sqrt{2 \cdot {\pi}^{-0.5}}}\right| \]

   10. Simplified 98.5%

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

   11. Taylor expanded in x around 0 98.5%

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

       1. associate-*r* 98.5%

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

       2. *-commutative 98.5%

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

       3. rem-square-sqrt 50.4%

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

       4. fabs-sqr 50.4%

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

       5. rem-square-sqrt 52.7%

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

       6. metadata-eval 52.7%

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

       7. unpow-1 52.7%

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

       8. metadata-eval 52.7%

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

       9. pow-sqr 52.7%

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

       10. rem-sqrt-square 52.7%

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

       11. fabs-mul 52.7%

           \[\leadsto x \cdot \color{blue}{\left|2 \cdot {\pi}^{-0.5}\right|} \]

       12. rem-sqrt-square 52.7%

           \[\leadsto x \cdot \color{blue}{\sqrt{\left(2 \cdot {\pi}^{-0.5}\right) \cdot \left(2 \cdot {\pi}^{-0.5}\right)}} \]

       13. swap-sqr 52.7%

           \[\leadsto x \cdot \sqrt{\color{blue}{\left(2 \cdot 2\right) \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)}} \]

       14. metadata-eval 52.7%

           \[\leadsto x \cdot \sqrt{\color{blue}{4} \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)} \]

       15. pow-sqr 52.7%

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

       16. metadata-eval 52.7%

           \[\leadsto x \cdot \sqrt{4 \cdot {\pi}^{\color{blue}{-1}}} \]

       17. unpow-1 52.7%

           \[\leadsto x \cdot \sqrt{4 \cdot \color{blue}{\frac{1}{\pi}}} \]

       18. associate-*r/ 52.7%

           \[\leadsto x \cdot \sqrt{\color{blue}{\frac{4 \cdot 1}{\pi}}} \]

       19. metadata-eval 52.7%

           \[\leadsto x \cdot \sqrt{\frac{\color{blue}{4}}{\pi}} \]

   13. Simplified 52.7%

       \[\leadsto \color{blue}{x \cdot \sqrt{\frac{4}{\pi}}} \]

   if 2 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 98.4%

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

      1. *-commutative 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in x around 0 98.5%

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

      1. +-commutative 98.5%

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

      2. *-commutative 98.5%

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

      3. *-commutative 98.5%

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

      4. *-commutative 98.5%

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

      5. associate-*r* 98.4%

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

      6. associate-*l* 98.4%

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

      7. *-commutative 98.4%

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

      8. associate-*l* 98.5%

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

      9. distribute-lft-out 98.5%

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

   10. Simplified 98.5%

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

   11. Taylor expanded in x around inf 98.5%

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

       1. associate-*r* 98.4%

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

       2. *-commutative 98.4%

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

       3. unpow-1 98.4%

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

       4. metadata-eval 98.4%

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

       5. pow-sqr 98.4%

          \[\leadsto \left|\sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}} \cdot \left(0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

       6. rem-sqrt-square 98.4%

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

       7. metadata-eval 98.4%

          \[\leadsto \left|\left|{\pi}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right| \cdot \left(0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

       8. pow-sqr 98.4%

          \[\leadsto \left|\left|\color{blue}{{\pi}^{-0.25} \cdot {\pi}^{-0.25}}\right| \cdot \left(0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

       9. fabs-sqr 98.4%

          \[\leadsto \left|\color{blue}{\left({\pi}^{-0.25} \cdot {\pi}^{-0.25}\right)} \cdot \left(0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

       10. pow-sqr 98.4%

           \[\leadsto \left|\color{blue}{{\pi}^{\left(2 \cdot -0.25\right)}} \cdot \left(0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

       11. metadata-eval 98.4%

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

       12. rem-square-sqrt 0.0%

           \[\leadsto \left|{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left({x}^{6} \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)\right)\right| \]

       13. fabs-sqr 0.0%

           \[\leadsto \left|{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left({x}^{6} \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right)\right)\right| \]

       14. rem-square-sqrt 98.4%

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

       15. pow-plus 98.4%

           \[\leadsto \left|{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \color{blue}{{x}^{\left(6 + 1\right)}}\right)\right| \]

       16. metadata-eval 98.4%

           \[\leadsto \left|{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{\color{blue}{7}}\right)\right| \]

   13. Simplified 98.4%

       \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.9% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(N[(N[(0.047619047619047616 * N[Power[x, 6.0], $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 99.9%

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

5. Taylor expanded in x around inf 98.6%

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

6. Taylor expanded in x around 0 98.5%

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

Alternative 6: 34.7% accurate, 5.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 2.0)
   (* x (sqrt (/ 4.0 PI)))
   (* (* 0.2 (pow x 4.0)) (/ x (sqrt PI)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2

   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%

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

   5. Taylor expanded in x around 0 98.5%

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

      1. associate-*r* 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in x around 0 98.5%

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

      1. *-commutative 98.5%

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

      2. associate-*r* 98.5%

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

      3. rem-exp-log 98.5%

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

      4. rec-exp 98.5%

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

      5. unpow1/2 98.5%

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

      6. exp-prod 98.5%

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

      7. distribute-lft-neg-out 98.5%

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

      8. distribute-rgt-neg-in 98.5%

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

      9. metadata-eval 98.5%

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

      10. exp-to-pow 98.5%

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

      11. *-commutative 98.5%

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

      12. associate-*l* 98.5%

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

      13. fabs-mul 98.5%

          \[\leadsto \color{blue}{\left|\left|x\right|\right| \cdot \left|2 \cdot {\pi}^{-0.5}\right|} \]

      14. fabs-fabs 98.5%

          \[\leadsto \color{blue}{\left|x\right|} \cdot \left|2 \cdot {\pi}^{-0.5}\right| \]

      15. rem-square-sqrt 96.8%

          \[\leadsto \left|x\right| \cdot \left|\color{blue}{\sqrt{2 \cdot {\pi}^{-0.5}} \cdot \sqrt{2 \cdot {\pi}^{-0.5}}}\right| \]

   10. Simplified 98.5%

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

   11. Taylor expanded in x around 0 98.5%

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

       1. associate-*r* 98.5%

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

       2. *-commutative 98.5%

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

       3. rem-square-sqrt 50.4%

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

       4. fabs-sqr 50.4%

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

       5. rem-square-sqrt 52.7%

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

       6. metadata-eval 52.7%

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

       7. unpow-1 52.7%

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

       8. metadata-eval 52.7%

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

       9. pow-sqr 52.7%

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

       10. rem-sqrt-square 52.7%

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

       11. fabs-mul 52.7%

           \[\leadsto x \cdot \color{blue}{\left|2 \cdot {\pi}^{-0.5}\right|} \]

       12. rem-sqrt-square 52.7%

           \[\leadsto x \cdot \color{blue}{\sqrt{\left(2 \cdot {\pi}^{-0.5}\right) \cdot \left(2 \cdot {\pi}^{-0.5}\right)}} \]

       13. swap-sqr 52.7%

           \[\leadsto x \cdot \sqrt{\color{blue}{\left(2 \cdot 2\right) \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)}} \]

       14. metadata-eval 52.7%

           \[\leadsto x \cdot \sqrt{\color{blue}{4} \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)} \]

       15. pow-sqr 52.7%

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

       16. metadata-eval 52.7%

           \[\leadsto x \cdot \sqrt{4 \cdot {\pi}^{\color{blue}{-1}}} \]

       17. unpow-1 52.7%

           \[\leadsto x \cdot \sqrt{4 \cdot \color{blue}{\frac{1}{\pi}}} \]

       18. associate-*r/ 52.7%

           \[\leadsto x \cdot \sqrt{\color{blue}{\frac{4 \cdot 1}{\pi}}} \]

       19. metadata-eval 52.7%

           \[\leadsto x \cdot \sqrt{\frac{\color{blue}{4}}{\pi}} \]

   13. Simplified 52.7%

       \[\leadsto \color{blue}{x \cdot \sqrt{\frac{4}{\pi}}} \]

   if 2 < (fabs.f64 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 100.0%

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

   5. Taylor expanded in x around 0 87.4%

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

      1. pow1 87.4%

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

      2. mul-fabs 87.4%

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

      3. fma-define 87.4%

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

      4. pow2 87.4%

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

   7. Applied egg-rr 87.4%

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

      1. unpow1 87.4%

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

      2. associate-*r/ 87.4%

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

      3. fabs-div 87.4%

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

      4. fma-undefine 87.4%

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

      5. +-commutative 87.4%

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

      6. fma-undefine 87.4%

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

      7. associate-+r+ 87.4%

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

      8. fma-define 87.4%

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

      9. +-commutative 87.4%

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

      10. fma-define 87.4%

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

      11. rem-sqrt-square 87.4%

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

      12. rem-square-sqrt 87.4%

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

   9. Simplified 87.4%

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

       1. *-un-lft-identity 87.4%

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

       2. add-sqr-sqrt 0.0%

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

       3. fabs-sqr 0.0%

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

       4. add-sqr-sqrt 0.1%

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

   11. Applied egg-rr 0.1%

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

       1. *-lft-identity 0.1%

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

       2. *-commutative 0.1%

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

       3. associate-/l* 0.1%

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

   13. Simplified 0.1%

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

   14. Taylor expanded in x around inf 0.1%

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

Alternative 7: 53.7% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-22}:\\ \;\;\;\;x \cdot \sqrt{\frac{4}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{4}{\pi} \cdot {x}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 2e-22)
   (* x (sqrt (/ 4.0 PI)))
   (sqrt (* (/ 4.0 PI) (pow x 2.0)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 2e-22) {
		tmp = x * sqrt((4.0 / ((double) M_PI)));
	} else {
		tmp = sqrt(((4.0 / ((double) M_PI)) * pow(x, 2.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 2e-22) {
		tmp = x * Math.sqrt((4.0 / Math.PI));
	} else {
		tmp = Math.sqrt(((4.0 / Math.PI) * Math.pow(x, 2.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 2e-22:
		tmp = x * math.sqrt((4.0 / math.pi))
	else:
		tmp = math.sqrt(((4.0 / math.pi) * math.pow(x, 2.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 2e-22)
		tmp = Float64(x * sqrt(Float64(4.0 / pi)));
	else
		tmp = sqrt(Float64(Float64(4.0 / pi) * (x ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 2e-22)
		tmp = x * sqrt((4.0 / pi));
	else
		tmp = sqrt(((4.0 / pi) * (x ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 2e-22], N[(x * N[Sqrt[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(4.0 / Pi), $MachinePrecision] * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2.0000000000000001e-22

   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%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. associate-*r* 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*r* 99.8%

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

      3. rem-exp-log 99.8%

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

      4. rec-exp 99.8%

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

      5. unpow1/2 99.8%

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

      6. exp-prod 99.8%

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

      7. distribute-lft-neg-out 99.8%

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

      8. distribute-rgt-neg-in 99.8%

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

      9. metadata-eval 99.8%

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

      10. exp-to-pow 99.8%

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

      11. *-commutative 99.8%

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

      12. associate-*l* 99.8%

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

      13. fabs-mul 99.8%

          \[\leadsto \color{blue}{\left|\left|x\right|\right| \cdot \left|2 \cdot {\pi}^{-0.5}\right|} \]

      14. fabs-fabs 99.8%

          \[\leadsto \color{blue}{\left|x\right|} \cdot \left|2 \cdot {\pi}^{-0.5}\right| \]

      15. rem-square-sqrt 98.1%

          \[\leadsto \left|x\right| \cdot \left|\color{blue}{\sqrt{2 \cdot {\pi}^{-0.5}} \cdot \sqrt{2 \cdot {\pi}^{-0.5}}}\right| \]

   10. Simplified 99.8%

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

   11. Taylor expanded in x around 0 99.8%

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

       1. associate-*r* 99.8%

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

       2. *-commutative 99.8%

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

       3. rem-square-sqrt 50.9%

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

       4. fabs-sqr 50.9%

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

       5. rem-square-sqrt 53.2%

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

       6. metadata-eval 53.2%

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

       7. unpow-1 53.2%

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

       8. metadata-eval 53.2%

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

       9. pow-sqr 53.2%

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

       10. rem-sqrt-square 53.2%

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

       11. fabs-mul 53.2%

           \[\leadsto x \cdot \color{blue}{\left|2 \cdot {\pi}^{-0.5}\right|} \]

       12. rem-sqrt-square 53.2%

           \[\leadsto x \cdot \color{blue}{\sqrt{\left(2 \cdot {\pi}^{-0.5}\right) \cdot \left(2 \cdot {\pi}^{-0.5}\right)}} \]

       13. swap-sqr 53.2%

           \[\leadsto x \cdot \sqrt{\color{blue}{\left(2 \cdot 2\right) \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)}} \]

       14. metadata-eval 53.2%

           \[\leadsto x \cdot \sqrt{\color{blue}{4} \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)} \]

       15. pow-sqr 53.2%

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

       16. metadata-eval 53.2%

           \[\leadsto x \cdot \sqrt{4 \cdot {\pi}^{\color{blue}{-1}}} \]

       17. unpow-1 53.2%

           \[\leadsto x \cdot \sqrt{4 \cdot \color{blue}{\frac{1}{\pi}}} \]

       18. associate-*r/ 53.2%

           \[\leadsto x \cdot \sqrt{\color{blue}{\frac{4 \cdot 1}{\pi}}} \]

       19. metadata-eval 53.2%

           \[\leadsto x \cdot \sqrt{\frac{\color{blue}{4}}{\pi}} \]

   13. Simplified 53.2%

       \[\leadsto \color{blue}{x \cdot \sqrt{\frac{4}{\pi}}} \]

   if 2.0000000000000001e-22 < (fabs.f64 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 99.8%

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

   5. Taylor expanded in x around 0 12.7%

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

      1. associate-*r* 12.7%

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

   7. Simplified 12.7%

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

   8. Taylor expanded in x around 0 12.7%

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

      1. *-commutative 12.7%

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

      2. associate-*r* 12.7%

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

      3. rem-exp-log 12.7%

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

      4. rec-exp 12.7%

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

      5. unpow1/2 12.7%

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

      6. exp-prod 12.7%

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

      7. distribute-lft-neg-out 12.7%

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

      8. distribute-rgt-neg-in 12.7%

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

      9. metadata-eval 12.7%

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

      10. exp-to-pow 12.7%

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

      11. *-commutative 12.7%

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

      12. associate-*l* 12.7%

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

      13. fabs-mul 12.7%

          \[\leadsto \color{blue}{\left|\left|x\right|\right| \cdot \left|2 \cdot {\pi}^{-0.5}\right|} \]

      14. fabs-fabs 12.7%

          \[\leadsto \color{blue}{\left|x\right|} \cdot \left|2 \cdot {\pi}^{-0.5}\right| \]

      15. rem-square-sqrt 12.6%

          \[\leadsto \left|x\right| \cdot \left|\color{blue}{\sqrt{2 \cdot {\pi}^{-0.5}} \cdot \sqrt{2 \cdot {\pi}^{-0.5}}}\right| \]

   10. Simplified 12.7%

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

   11. Taylor expanded in x around 0 12.7%

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

       1. associate-*r* 12.7%

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

       2. *-commutative 12.7%

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

       3. rem-square-sqrt 4.2%

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

       4. fabs-sqr 4.2%

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

       5. rem-square-sqrt 4.6%

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

       6. metadata-eval 4.6%

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

       7. unpow-1 4.6%

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

       8. metadata-eval 4.6%

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

       9. pow-sqr 4.6%

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

       10. rem-sqrt-square 4.6%

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

       11. fabs-mul 4.6%

           \[\leadsto x \cdot \color{blue}{\left|2 \cdot {\pi}^{-0.5}\right|} \]

       12. rem-sqrt-square 4.6%

           \[\leadsto x \cdot \color{blue}{\sqrt{\left(2 \cdot {\pi}^{-0.5}\right) \cdot \left(2 \cdot {\pi}^{-0.5}\right)}} \]

       13. swap-sqr 4.6%

           \[\leadsto x \cdot \sqrt{\color{blue}{\left(2 \cdot 2\right) \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)}} \]

       14. metadata-eval 4.6%

           \[\leadsto x \cdot \sqrt{\color{blue}{4} \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)} \]

       15. pow-sqr 4.6%

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

       16. metadata-eval 4.6%

           \[\leadsto x \cdot \sqrt{4 \cdot {\pi}^{\color{blue}{-1}}} \]

       17. unpow-1 4.6%

           \[\leadsto x \cdot \sqrt{4 \cdot \color{blue}{\frac{1}{\pi}}} \]

       18. associate-*r/ 4.6%

           \[\leadsto x \cdot \sqrt{\color{blue}{\frac{4 \cdot 1}{\pi}}} \]

       19. metadata-eval 4.6%

           \[\leadsto x \cdot \sqrt{\frac{\color{blue}{4}}{\pi}} \]

   13. Simplified 4.6%

       \[\leadsto \color{blue}{x \cdot \sqrt{\frac{4}{\pi}}} \]
   14. Step-by-step derivation

       1. add-sqr-sqrt 4.3%

          \[\leadsto \color{blue}{\sqrt{x \cdot \sqrt{\frac{4}{\pi}}} \cdot \sqrt{x \cdot \sqrt{\frac{4}{\pi}}}} \]

       2. sqrt-unprod 57.6%

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

       3. *-commutative 57.6%

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

       4. *-commutative 57.6%

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

       5. swap-sqr 57.5%

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

       6. add-sqr-sqrt 57.6%

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

       7. unpow2 57.6%

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

   15. Applied egg-rr 57.6%

       \[\leadsto \color{blue}{\sqrt{\frac{4}{\pi} \cdot {x}^{2}}} \]
   16. Step-by-step derivation

       1. *-commutative 57.6%

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

   17. Simplified 57.6%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-22}:\\ \;\;\;\;x \cdot \sqrt{\frac{4}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{4}{\pi} \cdot {x}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.9% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return Math.abs((((0.047619047619047616 * Math.pow(x, 6.0)) + 2.0) * (x * Math.pow(Math.PI, -0.5))));
}

Python

def code(x):
	return math.fabs((((0.047619047619047616 * math.pow(x, 6.0)) + 2.0) * (x * math.pow(math.pi, -0.5))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = abs((((0.047619047619047616 * (x ^ 6.0)) + 2.0) * (x * (pi ^ -0.5))));
end

Wolfram

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

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

5. Taylor expanded in x around 0 98.5%

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

   1. *-commutative 98.5%

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

7. Simplified 98.5%

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

8. Taylor expanded in x around 0 98.5%

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

   1. +-commutative 98.5%

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

   2. *-commutative 98.5%

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

   3. *-commutative 98.5%

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

   4. *-commutative 98.5%

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

   5. associate-*r* 98.5%

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

   6. associate-*l* 98.5%

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

   7. *-commutative 98.5%

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

   8. associate-*l* 98.5%

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

   9. distribute-lft-out 98.5%

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

10. Simplified 98.5%

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

11. Taylor expanded in x around 0 98.5%

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

    1. *-commutative 98.5%

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

    2. rem-square-sqrt 31.9%

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

    3. fabs-sqr 31.9%

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

    4. rem-square-sqrt 98.5%

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

    5. unpow-1 98.5%

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

    6. metadata-eval 98.5%

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

    7. pow-sqr 98.5%

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

    8. rem-sqrt-square 98.5%

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

    9. metadata-eval 98.5%

       \[\leadsto \left|\left(x \cdot \left|{\pi}^{\color{blue}{\left(2 \cdot -0.25\right)}}\right|\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

    10. pow-sqr 98.5%

        \[\leadsto \left|\left(x \cdot \left|\color{blue}{{\pi}^{-0.25} \cdot {\pi}^{-0.25}}\right|\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

    11. fabs-sqr 98.5%

        \[\leadsto \left|\left(x \cdot \color{blue}{\left({\pi}^{-0.25} \cdot {\pi}^{-0.25}\right)}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

    12. pow-sqr 98.5%

        \[\leadsto \left|\left(x \cdot \color{blue}{{\pi}^{\left(2 \cdot -0.25\right)}}\right) \cdot \left(2 + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

    13. metadata-eval 98.5%

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

13. Simplified 98.5%

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

14. Final simplification 98.5%

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

Alternative 9: 34.5% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around inf 98.6%

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

6. Taylor expanded in x around 0 98.5%

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

   1. *-commutative 98.5%

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

   2. add-sqr-sqrt 97.4%

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

   3. fabs-sqr 97.4%

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

   4. add-sqr-sqrt 98.5%

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

   5. add-sqr-sqrt 31.9%

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

   6. fabs-sqr 31.9%

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

   7. add-sqr-sqrt 33.4%

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

   8. associate-*l/ 33.1%

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

   9. fma-define 33.1%

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

8. Applied egg-rr 33.1%

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

9. Final simplification 33.1%

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

Alternative 10: 34.5% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around inf 98.6%

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

6. Taylor expanded in x around 0 98.5%

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

   1. add-sqr-sqrt 31.9%

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

   2. fabs-sqr 31.9%

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

   3. add-sqr-sqrt 33.4%

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

   4. add-sqr-sqrt 32.8%

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

   5. fabs-sqr 32.8%

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

   6. add-sqr-sqrt 33.4%

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

   7. clear-num 33.4%

      \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\sqrt{\pi}}{0.047619047619047616 \cdot {x}^{6} + 2}}} \]

   8. associate-/r/ 33.4%

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

   9. pow1/2 33.4%

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

   10. pow-flip 33.4%

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

   11. metadata-eval 33.4%

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

   12. +-commutative 33.4%

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

   13. associate-*l* 33.4%

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

   14. distribute-rgt-in 33.4%

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

8. Applied egg-rr 33.1%

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

   1. associate-*r* 33.1%

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

   2. distribute-rgt-in 33.1%

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

   3. +-commutative 33.1%

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

   4. fma-undefine 33.1%

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

10. Simplified 33.1%

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

11. Final simplification 33.1%

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

Alternative 11: 34.7% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\mathsf{expm1}\left({\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log1p (expm1 (* (pow PI -0.5) (* x 2.0)))))

C

double code(double x) {
	return log1p(expm1((pow(((double) M_PI), -0.5) * (x * 2.0))));
}

Java

public static double code(double x) {
	return Math.log1p(Math.expm1((Math.pow(Math.PI, -0.5) * (x * 2.0))));
}

Python

def code(x):
	return math.log1p(math.expm1((math.pow(math.pi, -0.5) * (x * 2.0))))

Julia

function code(x)
	return log1p(expm1(Float64((pi ^ -0.5) * Float64(x * 2.0))))
end

Wolfram

code[x_] := N[Log[1 + N[(Exp[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]] - 1), $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%

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

5. Taylor expanded in x around 0 64.4%

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

   1. associate-*r* 64.4%

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

7. Simplified 64.4%

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

8. Taylor expanded in x around 0 64.4%

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

   1. *-commutative 64.4%

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

   2. associate-*r* 64.4%

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

   3. rem-exp-log 64.4%

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

   4. rec-exp 64.4%

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

   5. unpow1/2 64.4%

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

   6. exp-prod 64.4%

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

   7. distribute-lft-neg-out 64.4%

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

   8. distribute-rgt-neg-in 64.4%

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

   9. metadata-eval 64.4%

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

   10. exp-to-pow 64.4%

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

   11. *-commutative 64.4%

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

   12. associate-*l* 64.4%

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

   13. fabs-mul 64.4%

       \[\leadsto \color{blue}{\left|\left|x\right|\right| \cdot \left|2 \cdot {\pi}^{-0.5}\right|} \]

   14. fabs-fabs 64.4%

       \[\leadsto \color{blue}{\left|x\right|} \cdot \left|2 \cdot {\pi}^{-0.5}\right| \]

   15. rem-square-sqrt 63.4%

       \[\leadsto \left|x\right| \cdot \left|\color{blue}{\sqrt{2 \cdot {\pi}^{-0.5}} \cdot \sqrt{2 \cdot {\pi}^{-0.5}}}\right| \]

10. Simplified 64.4%

    \[\leadsto \color{blue}{\left|x\right| \cdot \left(2 \cdot {\pi}^{-0.5}\right)} \]
11. Step-by-step derivation

    1. *-commutative 64.4%

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

    2. associate-*r* 64.4%

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

    3. add-sqr-sqrt 63.9%

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

    4. fabs-sqr 63.9%

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

    5. add-sqr-sqrt 64.4%

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

    6. log1p-expm1-u 94.7%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left|2 \cdot \left({\pi}^{-0.5} \cdot \left|x\right|\right)\right|\right)\right)} \]

    7. add-sqr-sqrt 94.2%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left|\color{blue}{\sqrt{2 \cdot \left({\pi}^{-0.5} \cdot \left|x\right|\right)} \cdot \sqrt{2 \cdot \left({\pi}^{-0.5} \cdot \left|x\right|\right)}}\right|\right)\right) \]

    8. fabs-sqr 94.2%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sqrt{2 \cdot \left({\pi}^{-0.5} \cdot \left|x\right|\right)} \cdot \sqrt{2 \cdot \left({\pi}^{-0.5} \cdot \left|x\right|\right)}}\right)\right) \]

    9. add-sqr-sqrt 94.7%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{2 \cdot \left({\pi}^{-0.5} \cdot \left|x\right|\right)}\right)\right) \]

    10. *-commutative 94.7%

        \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left({\pi}^{-0.5} \cdot \left|x\right|\right) \cdot 2}\right)\right) \]

    11. add-sqr-sqrt 31.9%

        \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left({\pi}^{-0.5} \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right) \cdot 2\right)\right) \]

    12. fabs-sqr 31.9%

        \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left({\pi}^{-0.5} \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) \cdot 2\right)\right) \]

    13. add-sqr-sqrt 33.4%

        \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left({\pi}^{-0.5} \cdot \color{blue}{x}\right) \cdot 2\right)\right) \]

    14. associate-*l* 33.4%

        \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{{\pi}^{-0.5} \cdot \left(x \cdot 2\right)}\right)\right) \]

12. Applied egg-rr 33.4%

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right)\right)} \]
13. Add Preprocessing

Alternative 12: 34.5% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x \cdot 2}{\sqrt{\pi}}\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log1p (expm1 (/ (* x 2.0) (sqrt PI)))))

C

double code(double x) {
	return log1p(expm1(((x * 2.0) / sqrt(((double) M_PI)))));
}

Java

public static double code(double x) {
	return Math.log1p(Math.expm1(((x * 2.0) / Math.sqrt(Math.PI))));
}

Python

def code(x):
	return math.log1p(math.expm1(((x * 2.0) / math.sqrt(math.pi))))

Julia

function code(x)
	return log1p(expm1(Float64(Float64(x * 2.0) / sqrt(pi))))
end

Wolfram

code[x_] := N[Log[1 + N[(Exp[N[(N[(x * 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]] - 1), $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%

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

5. Taylor expanded in x around 0 64.4%

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

   1. associate-*r* 64.4%

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

7. Simplified 64.4%

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

8. Taylor expanded in x around 0 64.4%

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

   1. *-commutative 64.4%

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

   2. associate-*r* 64.4%

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

   3. rem-exp-log 64.4%

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

   4. rec-exp 64.4%

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

   5. unpow1/2 64.4%

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

   6. exp-prod 64.4%

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

   7. distribute-lft-neg-out 64.4%

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

   8. distribute-rgt-neg-in 64.4%

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

   9. metadata-eval 64.4%

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

   10. exp-to-pow 64.4%

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

   11. *-commutative 64.4%

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

   12. associate-*l* 64.4%

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

   13. fabs-mul 64.4%

       \[\leadsto \color{blue}{\left|\left|x\right|\right| \cdot \left|2 \cdot {\pi}^{-0.5}\right|} \]

   14. fabs-fabs 64.4%

       \[\leadsto \color{blue}{\left|x\right|} \cdot \left|2 \cdot {\pi}^{-0.5}\right| \]

   15. rem-square-sqrt 63.4%

       \[\leadsto \left|x\right| \cdot \left|\color{blue}{\sqrt{2 \cdot {\pi}^{-0.5}} \cdot \sqrt{2 \cdot {\pi}^{-0.5}}}\right| \]

10. Simplified 64.4%

    \[\leadsto \color{blue}{\left|x\right| \cdot \left(2 \cdot {\pi}^{-0.5}\right)} \]
11. Step-by-step derivation

    1. *-commutative 64.4%

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

    2. associate-*r* 64.4%

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

    3. add-sqr-sqrt 63.9%

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

    4. fabs-sqr 63.9%

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

    5. add-sqr-sqrt 64.4%

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

    6. log1p-expm1-u 94.7%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left|2 \cdot \left({\pi}^{-0.5} \cdot \left|x\right|\right)\right|\right)\right)} \]

    7. add-sqr-sqrt 94.2%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left|\color{blue}{\sqrt{2 \cdot \left({\pi}^{-0.5} \cdot \left|x\right|\right)} \cdot \sqrt{2 \cdot \left({\pi}^{-0.5} \cdot \left|x\right|\right)}}\right|\right)\right) \]

    8. fabs-sqr 94.2%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sqrt{2 \cdot \left({\pi}^{-0.5} \cdot \left|x\right|\right)} \cdot \sqrt{2 \cdot \left({\pi}^{-0.5} \cdot \left|x\right|\right)}}\right)\right) \]

    9. add-sqr-sqrt 94.7%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{2 \cdot \left({\pi}^{-0.5} \cdot \left|x\right|\right)}\right)\right) \]

    10. *-commutative 94.7%

        \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left({\pi}^{-0.5} \cdot \left|x\right|\right) \cdot 2}\right)\right) \]

    11. add-sqr-sqrt 31.9%

        \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left({\pi}^{-0.5} \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right) \cdot 2\right)\right) \]

    12. fabs-sqr 31.9%

        \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left({\pi}^{-0.5} \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) \cdot 2\right)\right) \]

    13. add-sqr-sqrt 33.4%

        \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\left({\pi}^{-0.5} \cdot \color{blue}{x}\right) \cdot 2\right)\right) \]

    14. associate-*l* 33.4%

        \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{{\pi}^{-0.5} \cdot \left(x \cdot 2\right)}\right)\right) \]

12. Applied egg-rr 33.4%

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right)\right)} \]
13. Step-by-step derivation

    1. expm1-undefine 3.9%

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

    2. sub-neg 3.9%

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

    3. *-commutative 3.9%

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

    4. *-commutative 3.9%

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

    5. associate-*r* 3.9%

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

    6. *-commutative 3.9%

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

    7. associate-*r* 3.9%

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

    8. metadata-eval 3.9%

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

    9. pow-flip 3.9%

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

    10. pow1/2 3.9%

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

    11. div-inv 3.9%

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

    12. metadata-eval 3.9%

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

    13. sqrt-div 3.9%

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

    14. exp-prod 3.9%

        \[\leadsto \mathsf{log1p}\left(\color{blue}{{\left(e^{\sqrt{\frac{4}{\pi}}}\right)}^{x}} + \left(-1\right)\right) \]

    15. sqrt-div 3.9%

        \[\leadsto \mathsf{log1p}\left({\left(e^{\color{blue}{\frac{\sqrt{4}}{\sqrt{\pi}}}}\right)}^{x} + \left(-1\right)\right) \]

    16. metadata-eval 3.9%

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

    17. metadata-eval 3.9%

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

14. Applied egg-rr 3.9%

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

    1. metadata-eval 3.9%

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

    2. sub-neg 3.9%

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

    3. exp-prod 3.9%

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

    4. associate-*l/ 3.9%

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

    5. associate-*r/ 3.9%

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

    6. expm1-undefine 33.1%

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

    7. associate-*r/ 33.1%

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

16. Simplified 33.1%

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

17. Final simplification 33.1%

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x \cdot 2}{\sqrt{\pi}}\right)\right) \]
18. Add Preprocessing

Alternative 13: 34.8% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \sqrt{\frac{4}{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (sqrt (/ 4.0 PI))))

C

double code(double x) {
	return x * sqrt((4.0 / ((double) M_PI)));
}

Java

public static double code(double x) {
	return x * Math.sqrt((4.0 / Math.PI));
}

Python

def code(x):
	return x * math.sqrt((4.0 / math.pi))

Julia

function code(x)
	return Float64(x * sqrt(Float64(4.0 / pi)))
end

MATLAB

function tmp = code(x)
	tmp = x * sqrt((4.0 / pi));
end

Wolfram

code[x_] := N[(x * N[Sqrt[N[(4.0 / 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%

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

5. Taylor expanded in x around 0 64.4%

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

   1. associate-*r* 64.4%

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

7. Simplified 64.4%

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

8. Taylor expanded in x around 0 64.4%

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

   1. *-commutative 64.4%

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

   2. associate-*r* 64.4%

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

   3. rem-exp-log 64.4%

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

   4. rec-exp 64.4%

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

   5. unpow1/2 64.4%

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

   6. exp-prod 64.4%

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

   7. distribute-lft-neg-out 64.4%

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

   8. distribute-rgt-neg-in 64.4%

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

   9. metadata-eval 64.4%

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

   10. exp-to-pow 64.4%

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

   11. *-commutative 64.4%

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

   12. associate-*l* 64.4%

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

   13. fabs-mul 64.4%

       \[\leadsto \color{blue}{\left|\left|x\right|\right| \cdot \left|2 \cdot {\pi}^{-0.5}\right|} \]

   14. fabs-fabs 64.4%

       \[\leadsto \color{blue}{\left|x\right|} \cdot \left|2 \cdot {\pi}^{-0.5}\right| \]

   15. rem-square-sqrt 63.4%

       \[\leadsto \left|x\right| \cdot \left|\color{blue}{\sqrt{2 \cdot {\pi}^{-0.5}} \cdot \sqrt{2 \cdot {\pi}^{-0.5}}}\right| \]

10. Simplified 64.4%

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

11. Taylor expanded in x around 0 64.4%

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

    1. associate-*r* 64.4%

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

    2. *-commutative 64.4%

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

    3. rem-square-sqrt 31.9%

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

    4. fabs-sqr 31.9%

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

    5. rem-square-sqrt 33.5%

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

    6. metadata-eval 33.5%

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

    7. unpow-1 33.5%

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

    8. metadata-eval 33.5%

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

    9. pow-sqr 33.5%

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

    10. rem-sqrt-square 33.5%

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

    11. fabs-mul 33.5%

        \[\leadsto x \cdot \color{blue}{\left|2 \cdot {\pi}^{-0.5}\right|} \]

    12. rem-sqrt-square 33.5%

        \[\leadsto x \cdot \color{blue}{\sqrt{\left(2 \cdot {\pi}^{-0.5}\right) \cdot \left(2 \cdot {\pi}^{-0.5}\right)}} \]

    13. swap-sqr 33.5%

        \[\leadsto x \cdot \sqrt{\color{blue}{\left(2 \cdot 2\right) \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)}} \]

    14. metadata-eval 33.5%

        \[\leadsto x \cdot \sqrt{\color{blue}{4} \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)} \]

    15. pow-sqr 33.5%

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

    16. metadata-eval 33.5%

        \[\leadsto x \cdot \sqrt{4 \cdot {\pi}^{\color{blue}{-1}}} \]

    17. unpow-1 33.5%

        \[\leadsto x \cdot \sqrt{4 \cdot \color{blue}{\frac{1}{\pi}}} \]

    18. associate-*r/ 33.5%

        \[\leadsto x \cdot \sqrt{\color{blue}{\frac{4 \cdot 1}{\pi}}} \]

    19. metadata-eval 33.5%

        \[\leadsto x \cdot \sqrt{\frac{\color{blue}{4}}{\pi}} \]

13. Simplified 33.5%

    \[\leadsto \color{blue}{x \cdot \sqrt{\frac{4}{\pi}}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024166 
(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)))))))