Jmat.Real.erfi, branch x greater than or equal to 5

Percentage Accurate: 100.0% → 100.0%

Time: 11.2s

Alternatives: 8

Speedup: 3.4×

• 99.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot t\_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot t\_1\right) + \frac{3}{4} \cdot t\_2\right) + \frac{15}{8} \cdot \left(\left(t\_2 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x)))
        (t_1 (* (* t_0 t_0) t_0))
        (t_2 (* (* t_1 t_0) t_0)))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
    (+
     (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
     (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((Math.abs(x) * Math.abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	t_1 = (t_0 * t_0) * t_0
	t_2 = (t_1 * t_0) * t_0
	return ((1.0 / math.sqrt(math.pi)) * math.exp((math.fabs(x) * math.fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	t_1 = Float64(Float64(t_0 * t_0) * t_0)
	t_2 = Float64(Float64(t_1 * t_0) * t_0)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(Float64(t_0 + Float64(Float64(1.0 / 2.0) * t_1)) + Float64(Float64(3.0 / 4.0) * t_2)) + Float64(Float64(15.0 / 8.0) * Float64(Float64(t_2 * t_0) * t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	t_1 = (t_0 * t_0) * t_0;
	t_2 = (t_1 * t_0) * t_0;
	tmp = ((1.0 / sqrt(pi)) * exp((abs(x) * abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 + N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(t$95$2 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot t\_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot t\_1\right) + \frac{3}{4} \cdot t\_2\right) + \frac{15}{8} \cdot \left(\left(t\_2 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x)))
        (t_1 (* (* t_0 t_0) t_0))
        (t_2 (* (* t_1 t_0) t_0)))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
    (+
     (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
     (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((Math.abs(x) * Math.abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	t_1 = (t_0 * t_0) * t_0
	t_2 = (t_1 * t_0) * t_0
	return ((1.0 / math.sqrt(math.pi)) * math.exp((math.fabs(x) * math.fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	t_1 = Float64(Float64(t_0 * t_0) * t_0)
	t_2 = Float64(Float64(t_1 * t_0) * t_0)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(Float64(t_0 + Float64(Float64(1.0 / 2.0) * t_1)) + Float64(Float64(3.0 / 4.0) * t_2)) + Float64(Float64(15.0 / 8.0) * Float64(Float64(t_2 * t_0) * t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	t_1 = (t_0 * t_0) * t_0;
	t_2 = (t_1 * t_0) * t_0;
	tmp = ((1.0 / sqrt(pi)) * exp((abs(x) * abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 + N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(t$95$2 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Alternative 1: 100.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\left({x}^{-0.5}\right)}^{14}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (exp (* x x)) (sqrt PI))
  (fma
   0.75
   (pow x -5.0)
   (fma 1.875 (pow (pow x -0.5) 14.0) (/ (+ 1.0 (/ 0.5 (* x x))) x)))))

C

double code(double x) {
	return (exp((x * x)) / sqrt(((double) M_PI))) * fma(0.75, pow(x, -5.0), fma(1.875, pow(pow(x, -0.5), 14.0), ((1.0 + (0.5 / (x * x))) / x)));
}

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * x)) / sqrt(pi)) * fma(0.75, (x ^ -5.0), fma(1.875, ((x ^ -0.5) ^ 14.0), Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / x))))
end

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(0.75 * N[Power[x, -5.0], $MachinePrecision] + N[(1.875 * N[Power[N[Power[x, -0.5], $MachinePrecision], 14.0], $MachinePrecision] + N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-un-lft-identity 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, \color{blue}{1 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{5}}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   2. inv-pow 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   3. pow-pow 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 1 \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   4. add-sqr-sqrt 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. fabs-sqr 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. add-sqr-sqrt 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\color{blue}{x}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. metadata-eval 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {x}^{\color{blue}{-5}}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, \color{blue}{1 \cdot {x}^{-5}}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
7. Step-by-step derivation

   1. *-lft-identity 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, \color{blue}{{x}^{-5}}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

8. Simplified 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, \color{blue}{{x}^{-5}}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
9. Step-by-step derivation

   1. add-sqr-sqrt 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}\right)\right) \]

   2. fabs-sqr 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)\right) \]

   3. add-sqr-sqrt 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\color{blue}{x}}\right)\right) \]

   4. *-un-lft-identity 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\color{blue}{1 \cdot x}}\right)\right) \]

10. Applied egg-rr 100.0%

    \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\color{blue}{1 \cdot x}}\right)\right) \]
11. Step-by-step derivation

    1. *-lft-identity 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\color{blue}{x}}\right)\right) \]

12. Simplified 100.0%

    \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\color{blue}{x}}\right)\right) \]
13. Step-by-step derivation

    1. add-sqr-sqrt 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\color{blue}{\left(\sqrt{\frac{1}{\left|x\right|}} \cdot \sqrt{\frac{1}{\left|x\right|}}\right)}}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \]

    2. unpow-prod-down 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, \color{blue}{{\left(\sqrt{\frac{1}{\left|x\right|}}\right)}^{7} \cdot {\left(\sqrt{\frac{1}{\left|x\right|}}\right)}^{7}}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \]

    3. inv-pow 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\left(\sqrt{\color{blue}{{\left(\left|x\right|\right)}^{-1}}}\right)}^{7} \cdot {\left(\sqrt{\frac{1}{\left|x\right|}}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \]

    4. sqrt-pow1 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\color{blue}{\left({\left(\left|x\right|\right)}^{\left(\frac{-1}{2}\right)}\right)}}^{7} \cdot {\left(\sqrt{\frac{1}{\left|x\right|}}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \]

    5. add-sqr-sqrt 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\left({\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(\frac{-1}{2}\right)}\right)}^{7} \cdot {\left(\sqrt{\frac{1}{\left|x\right|}}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \]

    6. fabs-sqr 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\left({\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(\frac{-1}{2}\right)}\right)}^{7} \cdot {\left(\sqrt{\frac{1}{\left|x\right|}}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \]

    7. add-sqr-sqrt 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\left({\color{blue}{x}}^{\left(\frac{-1}{2}\right)}\right)}^{7} \cdot {\left(\sqrt{\frac{1}{\left|x\right|}}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \]

    8. metadata-eval 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\left({x}^{\color{blue}{-0.5}}\right)}^{7} \cdot {\left(\sqrt{\frac{1}{\left|x\right|}}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \]

    9. inv-pow 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\left({x}^{-0.5}\right)}^{7} \cdot {\left(\sqrt{\color{blue}{{\left(\left|x\right|\right)}^{-1}}}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \]

    10. sqrt-pow1 100.0%

        \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\left({x}^{-0.5}\right)}^{7} \cdot {\color{blue}{\left({\left(\left|x\right|\right)}^{\left(\frac{-1}{2}\right)}\right)}}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \]

    11. add-sqr-sqrt 100.0%

        \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\left({x}^{-0.5}\right)}^{7} \cdot {\left({\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(\frac{-1}{2}\right)}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \]

    12. fabs-sqr 100.0%

        \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\left({x}^{-0.5}\right)}^{7} \cdot {\left({\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(\frac{-1}{2}\right)}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \]

    13. add-sqr-sqrt 100.0%

        \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\left({x}^{-0.5}\right)}^{7} \cdot {\left({\color{blue}{x}}^{\left(\frac{-1}{2}\right)}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \]

    14. metadata-eval 100.0%

        \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\left({x}^{-0.5}\right)}^{7} \cdot {\left({x}^{\color{blue}{-0.5}}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \]

14. Applied egg-rr 100.0%

    \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, \color{blue}{{\left({x}^{-0.5}\right)}^{7} \cdot {\left({x}^{-0.5}\right)}^{7}}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \]
15. Step-by-step derivation

    1. pow-sqr 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, \color{blue}{{\left({x}^{-0.5}\right)}^{\left(2 \cdot 7\right)}}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \]

    2. metadata-eval 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\left({x}^{-0.5}\right)}^{\color{blue}{14}}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \]

16. Simplified 100.0%

    \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, \color{blue}{{\left({x}^{-0.5}\right)}^{14}}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \]
17. Add Preprocessing

Alternative 2: 100.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-7}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (exp (* x x)) (sqrt PI))
  (fma
   0.75
   (pow x -5.0)
   (fma 1.875 (pow x -7.0) (/ (+ 1.0 (/ 0.5 (* x x))) x)))))

C

double code(double x) {
	return (exp((x * x)) / sqrt(((double) M_PI))) * fma(0.75, pow(x, -5.0), fma(1.875, pow(x, -7.0), ((1.0 + (0.5 / (x * x))) / x)));
}

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * x)) / sqrt(pi)) * fma(0.75, (x ^ -5.0), fma(1.875, (x ^ -7.0), Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / x))))
end

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(0.75 * N[Power[x, -5.0], $MachinePrecision] + N[(1.875 * N[Power[x, -7.0], $MachinePrecision] + N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-un-lft-identity 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, \color{blue}{1 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{5}}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   2. inv-pow 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   3. pow-pow 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 1 \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   4. add-sqr-sqrt 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   5. fabs-sqr 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   6. add-sqr-sqrt 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {\color{blue}{x}}^{\left(-1 \cdot 5\right)}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

   7. metadata-eval 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, 1 \cdot {x}^{\color{blue}{-5}}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, \color{blue}{1 \cdot {x}^{-5}}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
7. Step-by-step derivation

   1. *-lft-identity 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, \color{blue}{{x}^{-5}}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

8. Simplified 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, \color{blue}{{x}^{-5}}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]
9. Step-by-step derivation

   1. add-sqr-sqrt 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}\right)\right) \]

   2. fabs-sqr 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)\right) \]

   3. add-sqr-sqrt 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\color{blue}{x}}\right)\right) \]

   4. *-un-lft-identity 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\color{blue}{1 \cdot x}}\right)\right) \]

10. Applied egg-rr 100.0%

    \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\color{blue}{1 \cdot x}}\right)\right) \]
11. Step-by-step derivation

    1. *-lft-identity 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\color{blue}{x}}\right)\right) \]

12. Simplified 100.0%

    \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\color{blue}{x}}\right)\right) \]
13. Step-by-step derivation

    1. *-un-lft-identity 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, \color{blue}{1 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{7}}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \]

    2. inv-pow 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, 1 \cdot {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \]

    3. pow-pow 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, 1 \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 7\right)}}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \]

    4. add-sqr-sqrt 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, 1 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \]

    5. fabs-sqr 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, 1 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \]

    6. add-sqr-sqrt 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, 1 \cdot {\color{blue}{x}}^{\left(-1 \cdot 7\right)}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \]

    7. metadata-eval 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, 1 \cdot {x}^{\color{blue}{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \]

14. Applied egg-rr 100.0%

    \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, \color{blue}{1 \cdot {x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \]
15. Step-by-step derivation

    1. *-lft-identity 100.0%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, \color{blue}{{x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \]

16. Simplified 100.0%

    \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {x}^{-5}, \mathsf{fma}\left(1.875, \color{blue}{{x}^{-7}}, \frac{1 + \frac{0.5}{x \cdot x}}{x}\right)\right) \]
17. Add Preprocessing

Alternative 3: 99.6% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{x} + \left(\frac{0.75}{{x}^{5}} + \frac{1.875}{{x}^{7}}\right)\right) \cdot \left(e^{{x}^{2}} \cdot {\pi}^{-0.5}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+ (/ 1.0 x) (+ (/ 0.75 (pow x 5.0)) (/ 1.875 (pow x 7.0))))
  (* (exp (pow x 2.0)) (pow PI -0.5))))

C

double code(double x) {
	return ((1.0 / x) + ((0.75 / pow(x, 5.0)) + (1.875 / pow(x, 7.0)))) * (exp(pow(x, 2.0)) * pow(((double) M_PI), -0.5));
}

Java

public static double code(double x) {
	return ((1.0 / x) + ((0.75 / Math.pow(x, 5.0)) + (1.875 / Math.pow(x, 7.0)))) * (Math.exp(Math.pow(x, 2.0)) * Math.pow(Math.PI, -0.5));
}

Python

def code(x):
	return ((1.0 / x) + ((0.75 / math.pow(x, 5.0)) + (1.875 / math.pow(x, 7.0)))) * (math.exp(math.pow(x, 2.0)) * math.pow(math.pi, -0.5))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / x) + Float64(Float64(0.75 / (x ^ 5.0)) + Float64(1.875 / (x ^ 7.0)))) * Float64(exp((x ^ 2.0)) * (pi ^ -0.5)))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / x) + ((0.75 / (x ^ 5.0)) + (1.875 / (x ^ 7.0)))) * (exp((x ^ 2.0)) * (pi ^ -0.5));
end

Wolfram

code[x_] := N[(N[(N[(1.0 / x), $MachinePrecision] + N[(N[(0.75 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[Power[x, 2.0], $MachinePrecision]], $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 99.7%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(e^{{x}^{2}} \cdot \left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 99.7%

      \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(\left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right) \cdot e^{{x}^{2}}\right)} \]

   2. +-commutative 99.7%

      \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\left(\left(\frac{1}{\left|x\right|} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right) + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)} \cdot e^{{x}^{2}}\right) \]

   3. +-commutative 99.7%

      \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left(\color{blue}{\left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \frac{1}{\left|x\right|}\right)} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right) \cdot e^{{x}^{2}}\right) \]

   4. associate-+l+ 99.7%

      \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)} \cdot e^{{x}^{2}}\right) \]

   5. associate-*r/ 99.7%

      \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left(\color{blue}{\frac{1.875 \cdot 1}{{\left(\left|x\right|\right)}^{7}}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right) \cdot e^{{x}^{2}}\right) \]

   6. metadata-eval 99.7%

      \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{\color{blue}{1.875}}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right) \cdot e^{{x}^{2}}\right) \]

   7. associate-*r/ 99.7%

      \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \color{blue}{\frac{0.75 \cdot 1}{{\left(\left|x\right|\right)}^{5}}}\right)\right) \cdot e^{{x}^{2}}\right) \]

   8. metadata-eval 99.7%

      \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{\color{blue}{0.75}}{{\left(\left|x\right|\right)}^{5}}\right)\right) \cdot e^{{x}^{2}}\right) \]

7. Simplified 99.7%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right)\right) \cdot e^{{x}^{2}}\right)} \]

8. Taylor expanded in x around inf 99.7%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(e^{{x}^{2}} \cdot \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right)} \]
9. Step-by-step derivation

   1. *-commutative 99.7%

      \[\leadsto \color{blue}{\left(e^{{x}^{2}} \cdot \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \cdot \sqrt{\frac{1}{\pi}}} \]

   2. unpow-1 99.7%

      \[\leadsto \left(e^{{x}^{2}} \cdot \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \cdot \sqrt{\color{blue}{{\pi}^{-1}}} \]

   3. metadata-eval 99.7%

      \[\leadsto \left(e^{{x}^{2}} \cdot \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \cdot \sqrt{{\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

   4. pow-sqr 99.7%

      \[\leadsto \left(e^{{x}^{2}} \cdot \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \cdot \sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}} \]

   5. rem-sqrt-square 99.7%

      \[\leadsto \left(e^{{x}^{2}} \cdot \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \cdot \color{blue}{\left|{\pi}^{-0.5}\right|} \]

   6. rem-square-sqrt 99.7%

      \[\leadsto \left(e^{{x}^{2}} \cdot \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \cdot \left|\color{blue}{\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}}\right| \]

   7. fabs-sqr 99.7%

      \[\leadsto \left(e^{{x}^{2}} \cdot \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \cdot \color{blue}{\left(\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}\right)} \]

   8. rem-square-sqrt 99.7%

      \[\leadsto \left(e^{{x}^{2}} \cdot \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \cdot \color{blue}{{\pi}^{-0.5}} \]

   9. *-commutative 99.7%

      \[\leadsto \color{blue}{\left(\left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right) \cdot e^{{x}^{2}}\right)} \cdot {\pi}^{-0.5} \]

   10. associate-*l* 99.7%

       \[\leadsto \color{blue}{\left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right) \cdot \left(e^{{x}^{2}} \cdot {\pi}^{-0.5}\right)} \]

10. Simplified 99.7%

    \[\leadsto \color{blue}{\left(\frac{1}{x} + \left(\frac{0.75}{{x}^{5}} + \frac{1.875}{{x}^{7}}\right)\right) \cdot \left(e^{{x}^{2}} \cdot {\pi}^{-0.5}\right)} \]
11. Add Preprocessing

Alternative 4: 51.7% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \left(1 + {x}^{2}\right) \cdot \left(\left(\frac{1}{x} + \left(\frac{0.75}{{x}^{5}} + \frac{1.875}{{x}^{7}}\right)\right) \cdot {\pi}^{-0.5}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+ 1.0 (pow x 2.0))
  (*
   (+ (/ 1.0 x) (+ (/ 0.75 (pow x 5.0)) (/ 1.875 (pow x 7.0))))
   (pow PI -0.5))))

C

double code(double x) {
	return (1.0 + pow(x, 2.0)) * (((1.0 / x) + ((0.75 / pow(x, 5.0)) + (1.875 / pow(x, 7.0)))) * pow(((double) M_PI), -0.5));
}

Java

public static double code(double x) {
	return (1.0 + Math.pow(x, 2.0)) * (((1.0 / x) + ((0.75 / Math.pow(x, 5.0)) + (1.875 / Math.pow(x, 7.0)))) * Math.pow(Math.PI, -0.5));
}

Python

def code(x):
	return (1.0 + math.pow(x, 2.0)) * (((1.0 / x) + ((0.75 / math.pow(x, 5.0)) + (1.875 / math.pow(x, 7.0)))) * math.pow(math.pi, -0.5))

Julia

function code(x)
	return Float64(Float64(1.0 + (x ^ 2.0)) * Float64(Float64(Float64(1.0 / x) + Float64(Float64(0.75 / (x ^ 5.0)) + Float64(1.875 / (x ^ 7.0)))) * (pi ^ -0.5)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 + (x ^ 2.0)) * (((1.0 / x) + ((0.75 / (x ^ 5.0)) + (1.875 / (x ^ 7.0)))) * (pi ^ -0.5));
end

Wolfram

code[x_] := N[(N[(1.0 + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / x), $MachinePrecision] + N[(N[(0.75 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 99.7%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(e^{{x}^{2}} \cdot \left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 99.7%

      \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(\left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right) \cdot e^{{x}^{2}}\right)} \]

   2. +-commutative 99.7%

      \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\left(\left(\frac{1}{\left|x\right|} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right) + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)} \cdot e^{{x}^{2}}\right) \]

   3. +-commutative 99.7%

      \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left(\color{blue}{\left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \frac{1}{\left|x\right|}\right)} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right) \cdot e^{{x}^{2}}\right) \]

   4. associate-+l+ 99.7%

      \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)} \cdot e^{{x}^{2}}\right) \]

   5. associate-*r/ 99.7%

      \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left(\color{blue}{\frac{1.875 \cdot 1}{{\left(\left|x\right|\right)}^{7}}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right) \cdot e^{{x}^{2}}\right) \]

   6. metadata-eval 99.7%

      \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{\color{blue}{1.875}}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right) \cdot e^{{x}^{2}}\right) \]

   7. associate-*r/ 99.7%

      \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \color{blue}{\frac{0.75 \cdot 1}{{\left(\left|x\right|\right)}^{5}}}\right)\right) \cdot e^{{x}^{2}}\right) \]

   8. metadata-eval 99.7%

      \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{\color{blue}{0.75}}{{\left(\left|x\right|\right)}^{5}}\right)\right) \cdot e^{{x}^{2}}\right) \]

7. Simplified 99.7%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right)\right) \cdot e^{{x}^{2}}\right)} \]

8. Taylor expanded in x around 0 56.5%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right) + \left({x}^{2} \cdot \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \cdot \sqrt{\frac{1}{\pi}}} \]
9. Step-by-step derivation

   1. associate-*l* 56.5%

      \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right) + \color{blue}{{x}^{2} \cdot \left(\left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)} \]

   2. *-commutative 56.5%

      \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right) + {x}^{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right)} \]

   3. distribute-rgt1-in 56.5%

      \[\leadsto \color{blue}{\left({x}^{2} + 1\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right)} \]

   4. *-commutative 56.5%

      \[\leadsto \left({x}^{2} + 1\right) \cdot \color{blue}{\left(\left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)} \]

10. Simplified 56.5%

    \[\leadsto \color{blue}{\left({x}^{2} + 1\right) \cdot \left(\left(\frac{1}{x} + \left(\frac{0.75}{{x}^{5}} + \frac{1.875}{{x}^{7}}\right)\right) \cdot {\pi}^{-0.5}\right)} \]

11. Final simplification 56.5%

    \[\leadsto \left(1 + {x}^{2}\right) \cdot \left(\left(\frac{1}{x} + \left(\frac{0.75}{{x}^{5}} + \frac{1.875}{{x}^{7}}\right)\right) \cdot {\pi}^{-0.5}\right) \]
12. Add Preprocessing

Alternative 5: 2.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{x} + \left(\frac{0.75}{{x}^{5}} + \frac{1.875}{{x}^{7}}\right)\right) \cdot {\pi}^{-0.5} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+ (/ 1.0 x) (+ (/ 0.75 (pow x 5.0)) (/ 1.875 (pow x 7.0))))
  (pow PI -0.5)))

C

double code(double x) {
	return ((1.0 / x) + ((0.75 / pow(x, 5.0)) + (1.875 / pow(x, 7.0)))) * pow(((double) M_PI), -0.5);
}

Java

public static double code(double x) {
	return ((1.0 / x) + ((0.75 / Math.pow(x, 5.0)) + (1.875 / Math.pow(x, 7.0)))) * Math.pow(Math.PI, -0.5);
}

Python

def code(x):
	return ((1.0 / x) + ((0.75 / math.pow(x, 5.0)) + (1.875 / math.pow(x, 7.0)))) * math.pow(math.pi, -0.5)

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / x) + Float64(Float64(0.75 / (x ^ 5.0)) + Float64(1.875 / (x ^ 7.0)))) * (pi ^ -0.5))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / x) + ((0.75 / (x ^ 5.0)) + (1.875 / (x ^ 7.0)))) * (pi ^ -0.5);
end

Wolfram

code[x_] := N[(N[(N[(1.0 / x), $MachinePrecision] + N[(N[(0.75 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 99.7%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(e^{{x}^{2}} \cdot \left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 99.7%

      \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(\left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right) \cdot e^{{x}^{2}}\right)} \]

   2. +-commutative 99.7%

      \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\left(\left(\frac{1}{\left|x\right|} + 1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right) + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)} \cdot e^{{x}^{2}}\right) \]

   3. +-commutative 99.7%

      \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left(\color{blue}{\left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \frac{1}{\left|x\right|}\right)} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right) \cdot e^{{x}^{2}}\right) \]

   4. associate-+l+ 99.7%

      \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)} \cdot e^{{x}^{2}}\right) \]

   5. associate-*r/ 99.7%

      \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left(\color{blue}{\frac{1.875 \cdot 1}{{\left(\left|x\right|\right)}^{7}}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right) \cdot e^{{x}^{2}}\right) \]

   6. metadata-eval 99.7%

      \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{\color{blue}{1.875}}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right) \cdot e^{{x}^{2}}\right) \]

   7. associate-*r/ 99.7%

      \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \color{blue}{\frac{0.75 \cdot 1}{{\left(\left|x\right|\right)}^{5}}}\right)\right) \cdot e^{{x}^{2}}\right) \]

   8. metadata-eval 99.7%

      \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{\color{blue}{0.75}}{{\left(\left|x\right|\right)}^{5}}\right)\right) \cdot e^{{x}^{2}}\right) \]

7. Simplified 99.7%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{1.875}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{0.75}{{\left(\left|x\right|\right)}^{5}}\right)\right) \cdot e^{{x}^{2}}\right)} \]

8. Taylor expanded in x around 0 2.3%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)} \]
9. Step-by-step derivation

   1. *-commutative 2.3%

      \[\leadsto \color{blue}{\left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right) \cdot \sqrt{\frac{1}{\pi}}} \]

10. Simplified 2.3%

    \[\leadsto \color{blue}{\left(\frac{1}{x} + \left(\frac{0.75}{{x}^{5}} + \frac{1.875}{{x}^{7}}\right)\right) \cdot {\pi}^{-0.5}} \]
11. Add Preprocessing

Alternative 6: 1.8% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{{x}^{-3}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (/ (pow x -3.0) (sqrt PI))))

C

double code(double x) {
	return 0.5 * (pow(x, -3.0) / sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return 0.5 * (Math.pow(x, -3.0) / Math.sqrt(Math.PI));
}

Python

def code(x):
	return 0.5 * (math.pow(x, -3.0) / math.sqrt(math.pi))

Julia

function code(x)
	return Float64(0.5 * Float64((x ^ -3.0) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * ((x ^ -3.0) / sqrt(pi));
end

Wolfram

code[x_] := N[(0.5 * N[(N[Power[x, -3.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 1.8%

   \[\leadsto \color{blue}{0.5 \cdot \left(\frac{1}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
6. Step-by-step derivation

   1. *-commutative 1.8%

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

   2. sqrt-div 1.8%

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

   3. metadata-eval 1.8%

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

   4. clear-num 1.8%

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

   5. frac-times 1.8%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{1 \cdot 1}{\sqrt{\pi} \cdot \frac{{x}^{2} \cdot \left|x\right|}{1}}} \]

   6. metadata-eval 1.8%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{1}}{\sqrt{\pi} \cdot \frac{{x}^{2} \cdot \left|x\right|}{1}} \]

   7. add-sqr-sqrt 1.8%

      \[\leadsto 0.5 \cdot \frac{1}{\sqrt{\pi} \cdot \frac{{x}^{2} \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{1}} \]

   8. fabs-sqr 1.8%

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

   9. add-sqr-sqrt 1.8%

      \[\leadsto 0.5 \cdot \frac{1}{\sqrt{\pi} \cdot \frac{{x}^{2} \cdot \color{blue}{x}}{1}} \]

   10. pow-plus 1.8%

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

   11. metadata-eval 1.8%

       \[\leadsto 0.5 \cdot \frac{1}{\sqrt{\pi} \cdot \frac{{x}^{\color{blue}{3}}}{1}} \]

7. Applied egg-rr 1.8%

   \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\sqrt{\pi} \cdot \frac{{x}^{3}}{1}}} \]
8. Step-by-step derivation

   1. /-rgt-identity 1.8%

      \[\leadsto 0.5 \cdot \frac{1}{\sqrt{\pi} \cdot \color{blue}{{x}^{3}}} \]

   2. *-commutative 1.8%

      \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{{x}^{3} \cdot \sqrt{\pi}}} \]

   3. associate-/r* 1.8%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\frac{1}{{x}^{3}}}{\sqrt{\pi}}} \]

   4. exp-to-pow 1.8%

      \[\leadsto 0.5 \cdot \frac{\frac{1}{\color{blue}{e^{\log x \cdot 3}}}}{\sqrt{\pi}} \]

   5. *-commutative 1.8%

      \[\leadsto 0.5 \cdot \frac{\frac{1}{e^{\color{blue}{3 \cdot \log x}}}}{\sqrt{\pi}} \]

   6. exp-neg 1.8%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{e^{-3 \cdot \log x}}}{\sqrt{\pi}} \]

   7. *-commutative 1.8%

      \[\leadsto 0.5 \cdot \frac{e^{-\color{blue}{\log x \cdot 3}}}{\sqrt{\pi}} \]

   8. distribute-rgt-neg-in 1.8%

      \[\leadsto 0.5 \cdot \frac{e^{\color{blue}{\log x \cdot \left(-3\right)}}}{\sqrt{\pi}} \]

   9. metadata-eval 1.8%

      \[\leadsto 0.5 \cdot \frac{e^{\log x \cdot \color{blue}{-3}}}{\sqrt{\pi}} \]

   10. exp-to-pow 1.8%

       \[\leadsto 0.5 \cdot \frac{\color{blue}{{x}^{-3}}}{\sqrt{\pi}} \]

9. Simplified 1.8%

   \[\leadsto 0.5 \cdot \color{blue}{\frac{{x}^{-3}}{\sqrt{\pi}}} \]
10. Add Preprocessing

Alternative 7: 1.8% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{\frac{0.25}{\pi}}{{x}^{6}}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (/ (/ 0.25 PI) (pow x 6.0))))

C

double code(double x) {
	return sqrt(((0.25 / ((double) M_PI)) / pow(x, 6.0)));
}

Java

public static double code(double x) {
	return Math.sqrt(((0.25 / Math.PI) / Math.pow(x, 6.0)));
}

Python

def code(x):
	return math.sqrt(((0.25 / math.pi) / math.pow(x, 6.0)))

Julia

function code(x)
	return sqrt(Float64(Float64(0.25 / pi) / (x ^ 6.0)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt(((0.25 / pi) / (x ^ 6.0)));
end

Wolfram

code[x_] := N[Sqrt[N[(N[(0.25 / Pi), $MachinePrecision] / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 1.8%

   \[\leadsto \color{blue}{0.5 \cdot \left(\frac{1}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
6. Step-by-step derivation

   1. *-commutative 1.8%

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

   2. sqrt-div 1.8%

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

   3. metadata-eval 1.8%

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

   4. clear-num 1.8%

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

   5. frac-times 1.8%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{1 \cdot 1}{\sqrt{\pi} \cdot \frac{{x}^{2} \cdot \left|x\right|}{1}}} \]

   6. metadata-eval 1.8%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{1}}{\sqrt{\pi} \cdot \frac{{x}^{2} \cdot \left|x\right|}{1}} \]

   7. add-sqr-sqrt 1.8%

      \[\leadsto 0.5 \cdot \frac{1}{\sqrt{\pi} \cdot \frac{{x}^{2} \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{1}} \]

   8. fabs-sqr 1.8%

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

   9. add-sqr-sqrt 1.8%

      \[\leadsto 0.5 \cdot \frac{1}{\sqrt{\pi} \cdot \frac{{x}^{2} \cdot \color{blue}{x}}{1}} \]

   10. pow-plus 1.8%

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

   11. metadata-eval 1.8%

       \[\leadsto 0.5 \cdot \frac{1}{\sqrt{\pi} \cdot \frac{{x}^{\color{blue}{3}}}{1}} \]

7. Applied egg-rr 1.8%

   \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\sqrt{\pi} \cdot \frac{{x}^{3}}{1}}} \]
8. Step-by-step derivation

   1. /-rgt-identity 1.8%

      \[\leadsto 0.5 \cdot \frac{1}{\sqrt{\pi} \cdot \color{blue}{{x}^{3}}} \]

   2. *-commutative 1.8%

      \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{{x}^{3} \cdot \sqrt{\pi}}} \]

   3. associate-/r* 1.8%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\frac{1}{{x}^{3}}}{\sqrt{\pi}}} \]

   4. exp-to-pow 1.8%

      \[\leadsto 0.5 \cdot \frac{\frac{1}{\color{blue}{e^{\log x \cdot 3}}}}{\sqrt{\pi}} \]

   5. *-commutative 1.8%

      \[\leadsto 0.5 \cdot \frac{\frac{1}{e^{\color{blue}{3 \cdot \log x}}}}{\sqrt{\pi}} \]

   6. exp-neg 1.8%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{e^{-3 \cdot \log x}}}{\sqrt{\pi}} \]

   7. *-commutative 1.8%

      \[\leadsto 0.5 \cdot \frac{e^{-\color{blue}{\log x \cdot 3}}}{\sqrt{\pi}} \]

   8. distribute-rgt-neg-in 1.8%

      \[\leadsto 0.5 \cdot \frac{e^{\color{blue}{\log x \cdot \left(-3\right)}}}{\sqrt{\pi}} \]

   9. metadata-eval 1.8%

      \[\leadsto 0.5 \cdot \frac{e^{\log x \cdot \color{blue}{-3}}}{\sqrt{\pi}} \]

   10. exp-to-pow 1.8%

       \[\leadsto 0.5 \cdot \frac{\color{blue}{{x}^{-3}}}{\sqrt{\pi}} \]

9. Simplified 1.8%

   \[\leadsto 0.5 \cdot \color{blue}{\frac{{x}^{-3}}{\sqrt{\pi}}} \]
10. Step-by-step derivation

    1. metadata-eval 1.8%

       \[\leadsto 0.5 \cdot \frac{{x}^{\color{blue}{\left(-3\right)}}}{\sqrt{\pi}} \]

    2. pow-flip 1.8%

       \[\leadsto 0.5 \cdot \frac{\color{blue}{\frac{1}{{x}^{3}}}}{\sqrt{\pi}} \]

    3. pow-to-exp 1.8%

       \[\leadsto 0.5 \cdot \frac{\frac{1}{\color{blue}{e^{\log x \cdot 3}}}}{\sqrt{\pi}} \]

    4. *-commutative 1.8%

       \[\leadsto 0.5 \cdot \frac{\frac{1}{e^{\color{blue}{3 \cdot \log x}}}}{\sqrt{\pi}} \]

    5. *-commutative 1.8%

       \[\leadsto 0.5 \cdot \frac{\frac{1}{e^{\color{blue}{\log x \cdot 3}}}}{\sqrt{\pi}} \]

    6. pow-to-exp 1.8%

       \[\leadsto 0.5 \cdot \frac{\frac{1}{\color{blue}{{x}^{3}}}}{\sqrt{\pi}} \]

    7. associate-/r* 1.8%

       \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{{x}^{3} \cdot \sqrt{\pi}}} \]

    8. add-sqr-sqrt 1.8%

       \[\leadsto \color{blue}{\sqrt{0.5 \cdot \frac{1}{{x}^{3} \cdot \sqrt{\pi}}} \cdot \sqrt{0.5 \cdot \frac{1}{{x}^{3} \cdot \sqrt{\pi}}}} \]

    9. sqrt-unprod 1.8%

       \[\leadsto \color{blue}{\sqrt{\left(0.5 \cdot \frac{1}{{x}^{3} \cdot \sqrt{\pi}}\right) \cdot \left(0.5 \cdot \frac{1}{{x}^{3} \cdot \sqrt{\pi}}\right)}} \]

    10. un-div-inv 1.8%

        \[\leadsto \sqrt{\color{blue}{\frac{0.5}{{x}^{3} \cdot \sqrt{\pi}}} \cdot \left(0.5 \cdot \frac{1}{{x}^{3} \cdot \sqrt{\pi}}\right)} \]

    11. un-div-inv 1.8%

        \[\leadsto \sqrt{\frac{0.5}{{x}^{3} \cdot \sqrt{\pi}} \cdot \color{blue}{\frac{0.5}{{x}^{3} \cdot \sqrt{\pi}}}} \]

    12. frac-times 1.8%

        \[\leadsto \sqrt{\color{blue}{\frac{0.5 \cdot 0.5}{\left({x}^{3} \cdot \sqrt{\pi}\right) \cdot \left({x}^{3} \cdot \sqrt{\pi}\right)}}} \]

    13. metadata-eval 1.8%

        \[\leadsto \sqrt{\frac{\color{blue}{0.25}}{\left({x}^{3} \cdot \sqrt{\pi}\right) \cdot \left({x}^{3} \cdot \sqrt{\pi}\right)}} \]

11. Applied egg-rr 1.8%

    \[\leadsto \color{blue}{\sqrt{\frac{0.25}{\pi \cdot {x}^{6}}}} \]
12. Step-by-step derivation

    1. associate-/r* 1.8%

       \[\leadsto \sqrt{\color{blue}{\frac{\frac{0.25}{\pi}}{{x}^{6}}}} \]

13. Simplified 1.8%

    \[\leadsto \color{blue}{\sqrt{\frac{\frac{0.25}{\pi}}{{x}^{6}}}} \]
14. Add Preprocessing

Alternative 8: 1.6% accurate, 2083.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 1.8%

   \[\leadsto \color{blue}{0.5 \cdot \left(\frac{1}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
6. Step-by-step derivation

   1. *-commutative 1.8%

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

   2. sqrt-div 1.8%

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

   3. metadata-eval 1.8%

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

   4. clear-num 1.8%

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

   5. frac-times 1.8%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{1 \cdot 1}{\sqrt{\pi} \cdot \frac{{x}^{2} \cdot \left|x\right|}{1}}} \]

   6. metadata-eval 1.8%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{1}}{\sqrt{\pi} \cdot \frac{{x}^{2} \cdot \left|x\right|}{1}} \]

   7. add-sqr-sqrt 1.8%

      \[\leadsto 0.5 \cdot \frac{1}{\sqrt{\pi} \cdot \frac{{x}^{2} \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{1}} \]

   8. fabs-sqr 1.8%

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

   9. add-sqr-sqrt 1.8%

      \[\leadsto 0.5 \cdot \frac{1}{\sqrt{\pi} \cdot \frac{{x}^{2} \cdot \color{blue}{x}}{1}} \]

   10. pow-plus 1.8%

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

   11. metadata-eval 1.8%

       \[\leadsto 0.5 \cdot \frac{1}{\sqrt{\pi} \cdot \frac{{x}^{\color{blue}{3}}}{1}} \]

7. Applied egg-rr 1.8%

   \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\sqrt{\pi} \cdot \frac{{x}^{3}}{1}}} \]
8. Step-by-step derivation

   1. /-rgt-identity 1.8%

      \[\leadsto 0.5 \cdot \frac{1}{\sqrt{\pi} \cdot \color{blue}{{x}^{3}}} \]

   2. *-commutative 1.8%

      \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{{x}^{3} \cdot \sqrt{\pi}}} \]

   3. associate-/r* 1.8%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\frac{1}{{x}^{3}}}{\sqrt{\pi}}} \]

   4. exp-to-pow 1.8%

      \[\leadsto 0.5 \cdot \frac{\frac{1}{\color{blue}{e^{\log x \cdot 3}}}}{\sqrt{\pi}} \]

   5. *-commutative 1.8%

      \[\leadsto 0.5 \cdot \frac{\frac{1}{e^{\color{blue}{3 \cdot \log x}}}}{\sqrt{\pi}} \]

   6. exp-neg 1.8%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{e^{-3 \cdot \log x}}}{\sqrt{\pi}} \]

   7. *-commutative 1.8%

      \[\leadsto 0.5 \cdot \frac{e^{-\color{blue}{\log x \cdot 3}}}{\sqrt{\pi}} \]

   8. distribute-rgt-neg-in 1.8%

      \[\leadsto 0.5 \cdot \frac{e^{\color{blue}{\log x \cdot \left(-3\right)}}}{\sqrt{\pi}} \]

   9. metadata-eval 1.8%

      \[\leadsto 0.5 \cdot \frac{e^{\log x \cdot \color{blue}{-3}}}{\sqrt{\pi}} \]

   10. exp-to-pow 1.8%

       \[\leadsto 0.5 \cdot \frac{\color{blue}{{x}^{-3}}}{\sqrt{\pi}} \]

9. Simplified 1.8%

   \[\leadsto 0.5 \cdot \color{blue}{\frac{{x}^{-3}}{\sqrt{\pi}}} \]

11. Applied egg-rr 1.6%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot \sqrt{\frac{{x}^{-6}}{\pi}}\right)} - 1} \]
12. Step-by-step derivation

    1. sub-neg 1.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot \sqrt{\frac{{x}^{-6}}{\pi}}\right)} + \left(-1\right)} \]

    2. metadata-eval 1.6%

       \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \sqrt{\frac{{x}^{-6}}{\pi}}\right)} + \color{blue}{-1} \]

    3. +-commutative 1.6%

       \[\leadsto \color{blue}{-1 + e^{\mathsf{log1p}\left(0.5 \cdot \sqrt{\frac{{x}^{-6}}{\pi}}\right)}} \]

    4. log1p-undefine 1.6%

       \[\leadsto -1 + e^{\color{blue}{\log \left(1 + 0.5 \cdot \sqrt{\frac{{x}^{-6}}{\pi}}\right)}} \]

    5. rem-exp-log 1.6%

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

    6. +-commutative 1.6%

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

    7. fma-define 1.6%

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

13. Simplified 1.6%

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

14. Taylor expanded in x around inf 1.6%

    \[\leadsto -1 + \color{blue}{1} \]

15. Final simplification 1.6%

    \[\leadsto 0 \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024131 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x greater than or equal to 5"
  :precision binary64
  :pre (>= x 0.5)
  (* (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x)))) (+ (+ (+ (/ 1.0 (fabs x)) (* (/ 1.0 2.0) (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 3.0 4.0) (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 15.0 8.0) (* (* (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))))