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

Percentage Accurate: 100.0% → 100.0%

Time: 14.5s

Alternatives: 9

Speedup: 2.9×

• 99.4% 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

\[\begin{array}{l} \\ {\left(e^{x}\right)}^{x} \cdot \left(\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \left(\frac{0.75}{{x}^{4}} + \frac{1.875}{{x}^{6}}\right)}{x}\right) \cdot {\pi}^{-0.5}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (pow (exp x) x)
  (*
   (fma
    0.5
    (pow x -3.0)
    (/ (+ 1.0 (+ (/ 0.75 (pow x 4.0)) (/ 1.875 (pow x 6.0)))) x))
   (pow PI -0.5))))

C

double code(double x) {
	return pow(exp(x), x) * (fma(0.5, pow(x, -3.0), ((1.0 + ((0.75 / pow(x, 4.0)) + (1.875 / pow(x, 6.0)))) / x)) * pow(((double) M_PI), -0.5));
}

Julia

function code(x)
	return Float64((exp(x) ^ x) * Float64(fma(0.5, (x ^ -3.0), Float64(Float64(1.0 + Float64(Float64(0.75 / (x ^ 4.0)) + Float64(1.875 / (x ^ 6.0)))) / x)) * (pi ^ -0.5)))
end

Wolfram

code[x_] := N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] * N[(N[(0.5 * N[Power[x, -3.0], $MachinePrecision] + N[(N[(1.0 + N[(N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{{\left(e^{x}\right)}^{x} \cdot \frac{\mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)}{\sqrt{\pi}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. div-inv 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \color{blue}{\left(\mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right)} \]

6. Applied egg-rr 100.0%

   \[\leadsto {\left(e^{x}\right)}^{x} \cdot \color{blue}{\left(\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right) \cdot {\pi}^{-0.5}\right)} \]

7. Taylor expanded in x around 0 100.0%

   \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \color{blue}{\left(0.75 \cdot \frac{1}{{x}^{4}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)}}{x}\right) \cdot {\pi}^{-0.5}\right) \]
8. Step-by-step derivation

   1. associate-*r/ 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \left(\color{blue}{\frac{0.75 \cdot 1}{{x}^{4}}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)}{x}\right) \cdot {\pi}^{-0.5}\right) \]

   2. metadata-eval 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \left(\frac{\color{blue}{0.75}}{{x}^{4}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)}{x}\right) \cdot {\pi}^{-0.5}\right) \]

   3. associate-*r/ 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \left(\frac{0.75}{{x}^{4}} + \color{blue}{\frac{1.875 \cdot 1}{{x}^{6}}}\right)}{x}\right) \cdot {\pi}^{-0.5}\right) \]

   4. metadata-eval 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \left(\frac{0.75}{{x}^{4}} + \frac{\color{blue}{1.875}}{{x}^{6}}\right)}{x}\right) \cdot {\pi}^{-0.5}\right) \]

9. Simplified 100.0%

   \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \color{blue}{\left(\frac{0.75}{{x}^{4}} + \frac{1.875}{{x}^{6}}\right)}}{x}\right) \cdot {\pi}^{-0.5}\right) \]

10. Final simplification 100.0%

    \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \left(\frac{0.75}{{x}^{4}} + \frac{1.875}{{x}^{6}}\right)}{x}\right) \cdot {\pi}^{-0.5}\right) \]
11. Add Preprocessing

Alternative 2: 100.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ {\left(e^{x}\right)}^{x} \cdot \frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \left(\frac{0.75}{{x}^{4}} + \frac{1.875}{{x}^{6}}\right)}{x}\right)}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (pow (exp x) x)
  (/
   (fma
    0.5
    (pow x -3.0)
    (/ (+ 1.0 (+ (/ 0.75 (pow x 4.0)) (/ 1.875 (pow x 6.0)))) x))
   (sqrt PI))))

C

double code(double x) {
	return pow(exp(x), x) * (fma(0.5, pow(x, -3.0), ((1.0 + ((0.75 / pow(x, 4.0)) + (1.875 / pow(x, 6.0)))) / x)) / sqrt(((double) M_PI)));
}

Julia

function code(x)
	return Float64((exp(x) ^ x) * Float64(fma(0.5, (x ^ -3.0), Float64(Float64(1.0 + Float64(Float64(0.75 / (x ^ 4.0)) + Float64(1.875 / (x ^ 6.0)))) / x)) / sqrt(pi)))
end

Wolfram

code[x_] := N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] * N[(N[(0.5 * N[Power[x, -3.0], $MachinePrecision] + N[(N[(1.0 + N[(N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{{\left(e^{x}\right)}^{x} \cdot \frac{\mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)}{\sqrt{\pi}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-un-lft-identity 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \color{blue}{\left(1 \cdot \frac{\mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)}{\sqrt{\pi}}\right)} \]

6. Applied egg-rr 100.0%

   \[\leadsto {\left(e^{x}\right)}^{x} \cdot \color{blue}{\left(1 \cdot \frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right)}{\sqrt{\pi}}\right)} \]
7. Step-by-step derivation

   1. *-lft-identity 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \color{blue}{\frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right)}{\sqrt{\pi}}} \]

8. Simplified 100.0%

   \[\leadsto {\left(e^{x}\right)}^{x} \cdot \color{blue}{\frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right)}{\sqrt{\pi}}} \]

9. Taylor expanded in x around 0 100.0%

   \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \color{blue}{\left(0.75 \cdot \frac{1}{{x}^{4}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)}}{x}\right)}{\sqrt{\pi}} \]
10. Step-by-step derivation

    1. associate-*r/ 100.0%

       \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \left(\color{blue}{\frac{0.75 \cdot 1}{{x}^{4}}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)}{x}\right) \cdot {\pi}^{-0.5}\right) \]

    2. metadata-eval 100.0%

       \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \left(\frac{\color{blue}{0.75}}{{x}^{4}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)}{x}\right) \cdot {\pi}^{-0.5}\right) \]

    3. associate-*r/ 100.0%

       \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \left(\frac{0.75}{{x}^{4}} + \color{blue}{\frac{1.875 \cdot 1}{{x}^{6}}}\right)}{x}\right) \cdot {\pi}^{-0.5}\right) \]

    4. metadata-eval 100.0%

       \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \left(\frac{0.75}{{x}^{4}} + \frac{\color{blue}{1.875}}{{x}^{6}}\right)}{x}\right) \cdot {\pi}^{-0.5}\right) \]

11. Simplified 100.0%

    \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \color{blue}{\left(\frac{0.75}{{x}^{4}} + \frac{1.875}{{x}^{6}}\right)}}{x}\right)}{\sqrt{\pi}} \]

12. Final simplification 100.0%

    \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \left(\frac{0.75}{{x}^{4}} + \frac{1.875}{{x}^{6}}\right)}{x}\right)}{\sqrt{\pi}} \]
13. Add Preprocessing

Alternative 3: 99.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ {\left(e^{x}\right)}^{x} \cdot \frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \frac{0.75}{{x}^{4}}}{x}\right)}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (pow (exp x) x)
  (/ (fma 0.5 (pow x -3.0) (/ (+ 1.0 (/ 0.75 (pow x 4.0))) x)) (sqrt PI))))

C

double code(double x) {
	return pow(exp(x), x) * (fma(0.5, pow(x, -3.0), ((1.0 + (0.75 / pow(x, 4.0))) / x)) / sqrt(((double) M_PI)));
}

Julia

function code(x)
	return Float64((exp(x) ^ x) * Float64(fma(0.5, (x ^ -3.0), Float64(Float64(1.0 + Float64(0.75 / (x ^ 4.0))) / x)) / sqrt(pi)))
end

Wolfram

code[x_] := N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] * N[(N[(0.5 * N[Power[x, -3.0], $MachinePrecision] + N[(N[(1.0 + N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{{\left(e^{x}\right)}^{x} \cdot \frac{\mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)}{\sqrt{\pi}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-un-lft-identity 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \color{blue}{\left(1 \cdot \frac{\mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)}{\sqrt{\pi}}\right)} \]

6. Applied egg-rr 100.0%

   \[\leadsto {\left(e^{x}\right)}^{x} \cdot \color{blue}{\left(1 \cdot \frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right)}{\sqrt{\pi}}\right)} \]
7. Step-by-step derivation

   1. *-lft-identity 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \color{blue}{\frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right)}{\sqrt{\pi}}} \]

8. Simplified 100.0%

   \[\leadsto {\left(e^{x}\right)}^{x} \cdot \color{blue}{\frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right)}{\sqrt{\pi}}} \]

9. Taylor expanded in x around inf 98.9%

   \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \color{blue}{\frac{0.75}{{x}^{4}}}}{x}\right)}{\sqrt{\pi}} \]

10. Final simplification 98.9%

    \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \frac{0.75}{{x}^{4}}}{x}\right)}{\sqrt{\pi}} \]
11. Add Preprocessing

Alternative 4: 99.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ e^{{x}^{2} + \log \left({\pi}^{-0.5} \cdot \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{x}\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (exp
  (+ (pow x 2.0) (log (* (pow PI -0.5) (fma 0.5 (pow x -3.0) (/ 1.0 x)))))))

C

double code(double x) {
	return exp((pow(x, 2.0) + log((pow(((double) M_PI), -0.5) * fma(0.5, pow(x, -3.0), (1.0 / x))))));
}

Julia

function code(x)
	return exp(Float64((x ^ 2.0) + log(Float64((pi ^ -0.5) * fma(0.5, (x ^ -3.0), Float64(1.0 / x))))))
end

Wolfram

code[x_] := N[Exp[N[(N[Power[x, 2.0], $MachinePrecision] + N[Log[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(0.5 * N[Power[x, -3.0], $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{{\left(e^{x}\right)}^{x} \cdot \frac{\mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)}{\sqrt{\pi}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. div-inv 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \color{blue}{\left(\mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right)} \]

6. Applied egg-rr 100.0%

   \[\leadsto {\left(e^{x}\right)}^{x} \cdot \color{blue}{\left(\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right) \cdot {\pi}^{-0.5}\right)} \]

7. Taylor expanded in x around inf 98.7%

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

   1. associate-*r* 98.7%

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

   2. distribute-rgt-out 98.7%

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

   3. associate-*r/ 98.7%

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

   4. metadata-eval 98.7%

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

9. Simplified 98.7%

   \[\leadsto {\left(e^{x}\right)}^{x} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1}{x}\right)\right)} \]
10. Step-by-step derivation

    1. add-exp-log 98.7%

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

    2. log-prod 98.7%

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

    3. pow-exp 98.7%

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

    4. rem-log-exp 98.7%

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

    5. pow2 98.7%

       \[\leadsto e^{\color{blue}{{x}^{2}} + \log \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1}{x}\right)\right)} \]

    6. pow1/2 98.7%

       \[\leadsto e^{{x}^{2} + \log \left(\color{blue}{{\left(\frac{1}{\pi}\right)}^{0.5}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1}{x}\right)\right)} \]

    7. inv-pow 98.7%

       \[\leadsto e^{{x}^{2} + \log \left({\color{blue}{\left({\pi}^{-1}\right)}}^{0.5} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1}{x}\right)\right)} \]

    8. pow-pow 98.7%

       \[\leadsto e^{{x}^{2} + \log \left(\color{blue}{{\pi}^{\left(-1 \cdot 0.5\right)}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1}{x}\right)\right)} \]

    9. metadata-eval 98.7%

       \[\leadsto e^{{x}^{2} + \log \left({\pi}^{\color{blue}{-0.5}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1}{x}\right)\right)} \]

    10. div-inv 98.7%

        \[\leadsto e^{{x}^{2} + \log \left({\pi}^{-0.5} \cdot \left(\color{blue}{0.5 \cdot \frac{1}{{x}^{3}}} + \frac{1}{x}\right)\right)} \]

    11. fma-define 98.7%

        \[\leadsto e^{{x}^{2} + \log \left({\pi}^{-0.5} \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{1}{{x}^{3}}, \frac{1}{x}\right)}\right)} \]

    12. pow-flip 98.7%

        \[\leadsto e^{{x}^{2} + \log \left({\pi}^{-0.5} \cdot \mathsf{fma}\left(0.5, \color{blue}{{x}^{\left(-3\right)}}, \frac{1}{x}\right)\right)} \]

    13. metadata-eval 98.7%

        \[\leadsto e^{{x}^{2} + \log \left({\pi}^{-0.5} \cdot \mathsf{fma}\left(0.5, {x}^{\color{blue}{-3}}, \frac{1}{x}\right)\right)} \]

11. Applied egg-rr 98.7%

    \[\leadsto \color{blue}{e^{{x}^{2} + \log \left({\pi}^{-0.5} \cdot \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{x}\right)\right)}} \]

12. Final simplification 98.7%

    \[\leadsto e^{{x}^{2} + \log \left({\pi}^{-0.5} \cdot \mathsf{fma}\left(0.5, {x}^{-3}, \frac{1}{x}\right)\right)} \]
13. Add Preprocessing

Alternative 5: 99.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ e^{{x}^{2}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (exp (pow x 2.0)) (* (sqrt (/ 1.0 PI)) (+ (/ 1.0 x) (/ 0.5 (pow x 3.0))))))

C

double code(double x) {
	return exp(pow(x, 2.0)) * (sqrt((1.0 / ((double) M_PI))) * ((1.0 / x) + (0.5 / pow(x, 3.0))));
}

Java

public static double code(double x) {
	return Math.exp(Math.pow(x, 2.0)) * (Math.sqrt((1.0 / Math.PI)) * ((1.0 / x) + (0.5 / Math.pow(x, 3.0))));
}

Python

def code(x):
	return math.exp(math.pow(x, 2.0)) * (math.sqrt((1.0 / math.pi)) * ((1.0 / x) + (0.5 / math.pow(x, 3.0))))

Julia

function code(x)
	return Float64(exp((x ^ 2.0)) * Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(1.0 / x) + Float64(0.5 / (x ^ 3.0)))))
end

MATLAB

function tmp = code(x)
	tmp = exp((x ^ 2.0)) * (sqrt((1.0 / pi)) * ((1.0 / x) + (0.5 / (x ^ 3.0))));
end

Wolfram

code[x_] := N[(N[Exp[N[Power[x, 2.0], $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] + N[(0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{{\left(e^{x}\right)}^{x} \cdot \frac{\mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)}{\sqrt{\pi}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. div-inv 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \color{blue}{\left(\mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right)} \]

6. Applied egg-rr 100.0%

   \[\leadsto {\left(e^{x}\right)}^{x} \cdot \color{blue}{\left(\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right) \cdot {\pi}^{-0.5}\right)} \]

7. Taylor expanded in x around inf 98.7%

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

   1. associate-*r* 98.7%

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

   2. distribute-rgt-out 98.7%

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

   3. associate-*r/ 98.7%

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

   4. metadata-eval 98.7%

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

9. Simplified 98.7%

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

10. Taylor expanded in x around inf 98.7%

    \[\leadsto \color{blue}{e^{{x}^{2}}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1}{x}\right)\right) \]

11. Final simplification 98.7%

    \[\leadsto e^{{x}^{2}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)\right) \]
12. Add Preprocessing

Alternative 6: 99.6% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\pi}} \cdot \frac{e^{{x}^{2}}}{x} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

   1. *-un-lft-identity 99.9%

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

6. Applied egg-rr 99.9%

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

   1. *-lft-identity 99.9%

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

8. Simplified 99.9%

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

9. Taylor expanded in x around inf 98.6%

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

10. Final simplification 98.6%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{e^{{x}^{2}}}{x} \]
11. Add Preprocessing

Alternative 7: 2.3% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{0.5}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (sqrt (/ 1.0 PI)) (+ (/ 0.5 (pow x 3.0)) (/ 0.5 x))))

C

double code(double x) {
	return sqrt((1.0 / ((double) M_PI))) * ((0.5 / pow(x, 3.0)) + (0.5 / x));
}

Java

public static double code(double x) {
	return Math.sqrt((1.0 / Math.PI)) * ((0.5 / Math.pow(x, 3.0)) + (0.5 / x));
}

Python

def code(x):
	return math.sqrt((1.0 / math.pi)) * ((0.5 / math.pow(x, 3.0)) + (0.5 / x))

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(0.5 / (x ^ 3.0)) + Float64(0.5 / x)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 / pi)) * ((0.5 / (x ^ 3.0)) + (0.5 / x));
end

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 36.2%

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

   1. associate-*l/ 36.2%

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

   2. *-lft-identity 36.2%

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

   3. unpow2 36.2%

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

   4. sqr-abs 36.2%

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

   5. unpow3 36.2%

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

7. Simplified 36.2%

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

   1. *-un-lft-identity 36.2%

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

   2. div-inv 36.2%

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

   3. pow1/2 36.2%

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

   4. inv-pow 36.2%

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

   5. pow-pow 36.2%

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

   6. metadata-eval 36.2%

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

   7. pow-flip 37.0%

      \[\leadsto e^{x \cdot x} \cdot \left(0.5 \cdot \left(1 \cdot \left({\pi}^{-0.5} \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(-3\right)}}\right)\right)\right) \]

   8. metadata-eval 37.0%

      \[\leadsto e^{x \cdot x} \cdot \left(0.5 \cdot \left(1 \cdot \left({\pi}^{-0.5} \cdot {\left(\left|x\right|\right)}^{\color{blue}{-3}}\right)\right)\right) \]

9. Applied egg-rr 37.0%

   \[\leadsto e^{x \cdot x} \cdot \left(0.5 \cdot \color{blue}{\left(1 \cdot \left({\pi}^{-0.5} \cdot {\left(\left|x\right|\right)}^{-3}\right)\right)}\right) \]
10. Step-by-step derivation

    1. *-lft-identity 37.0%

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

    2. sqr-pow 37.0%

       \[\leadsto e^{x \cdot x} \cdot \left(0.5 \cdot \left({\pi}^{-0.5} \cdot \color{blue}{\left({\left(\left|x\right|\right)}^{\left(\frac{-3}{2}\right)} \cdot {\left(\left|x\right|\right)}^{\left(\frac{-3}{2}\right)}\right)}\right)\right) \]

    3. sqr-pow 37.0%

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

    4. unpow1 37.0%

       \[\leadsto e^{x \cdot x} \cdot \left(0.5 \cdot \left({\pi}^{-0.5} \cdot {\left(\left|\color{blue}{{x}^{1}}\right|\right)}^{-3}\right)\right) \]

    5. sqr-pow 37.0%

       \[\leadsto e^{x \cdot x} \cdot \left(0.5 \cdot \left({\pi}^{-0.5} \cdot {\left(\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|\right)}^{-3}\right)\right) \]

    6. fabs-sqr 37.0%

       \[\leadsto e^{x \cdot x} \cdot \left(0.5 \cdot \left({\pi}^{-0.5} \cdot {\color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}}^{-3}\right)\right) \]

    7. sqr-pow 37.0%

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

    8. unpow1 37.0%

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

11. Simplified 37.0%

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

12. Taylor expanded in x around 0 2.4%

    \[\leadsto \color{blue}{0.5 \cdot \left(\frac{1}{x} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.5 \cdot \left(\frac{1}{{x}^{3}} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
13. Step-by-step derivation

    1. +-commutative 2.4%

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

    2. associate-*r* 2.4%

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

    3. associate-*r* 2.4%

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

    4. distribute-rgt-out 2.4%

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

    5. associate-*r/ 2.4%

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

    6. metadata-eval 2.4%

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

    7. associate-*r/ 2.4%

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

    8. metadata-eval 2.4%

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

14. Simplified 2.4%

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

15. Final simplification 2.4%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{0.5}{x}\right) \]
16. Add Preprocessing

Alternative 8: 2.3% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1.5}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (sqrt (/ 1.0 PI)) (+ (/ 0.5 (pow x 3.0)) (/ 1.5 x))))

C

double code(double x) {
	return sqrt((1.0 / ((double) M_PI))) * ((0.5 / pow(x, 3.0)) + (1.5 / x));
}

Java

public static double code(double x) {
	return Math.sqrt((1.0 / Math.PI)) * ((0.5 / Math.pow(x, 3.0)) + (1.5 / x));
}

Python

def code(x):
	return math.sqrt((1.0 / math.pi)) * ((0.5 / math.pow(x, 3.0)) + (1.5 / x))

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(0.5 / (x ^ 3.0)) + Float64(1.5 / x)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 / pi)) * ((0.5 / (x ^ 3.0)) + (1.5 / x));
end

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(1.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{{\left(e^{x}\right)}^{x} \cdot \frac{\mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right)}{\sqrt{\pi}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. div-inv 100.0%

      \[\leadsto {\left(e^{x}\right)}^{x} \cdot \color{blue}{\left(\mathsf{fma}\left(0.5, \frac{1}{{\left(\left|x\right|\right)}^{3}}, \frac{1}{\left|x\right|} \cdot \left(1 + \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{4}, 1.875 \cdot {\left(\frac{1}{\left|x\right|}\right)}^{6}\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right)} \]

6. Applied egg-rr 100.0%

   \[\leadsto {\left(e^{x}\right)}^{x} \cdot \color{blue}{\left(\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right) \cdot {\pi}^{-0.5}\right)} \]

7. Taylor expanded in x around inf 98.7%

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

   1. associate-*r* 98.7%

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

   2. distribute-rgt-out 98.7%

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

   3. associate-*r/ 98.7%

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

   4. metadata-eval 98.7%

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

9. Simplified 98.7%

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

10. Taylor expanded in x around 0 2.4%

    \[\leadsto \color{blue}{0.5 \cdot \left(\frac{1}{{x}^{3}} \cdot \sqrt{\frac{1}{\pi}}\right) + 1.5 \cdot \left(\frac{1}{x} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
11. Step-by-step derivation

    1. associate-*r* 2.4%

       \[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{{x}^{3}}\right) \cdot \sqrt{\frac{1}{\pi}}} + 1.5 \cdot \left(\frac{1}{x} \cdot \sqrt{\frac{1}{\pi}}\right) \]

    2. associate-*r* 2.4%

       \[\leadsto \left(0.5 \cdot \frac{1}{{x}^{3}}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(1.5 \cdot \frac{1}{x}\right) \cdot \sqrt{\frac{1}{\pi}}} \]

    3. distribute-rgt-out 2.4%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.5 \cdot \frac{1}{{x}^{3}} + 1.5 \cdot \frac{1}{x}\right)} \]

    4. associate-*r/ 2.4%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{3}}} + 1.5 \cdot \frac{1}{x}\right) \]

    5. metadata-eval 2.4%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\frac{\color{blue}{0.5}}{{x}^{3}} + 1.5 \cdot \frac{1}{x}\right) \]

    6. associate-*r/ 2.4%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \color{blue}{\frac{1.5 \cdot 1}{x}}\right) \]

    7. metadata-eval 2.4%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{\color{blue}{1.5}}{x}\right) \]

12. Simplified 2.4%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1.5}{x}\right)} \]

13. Final simplification 2.4%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1.5}{x}\right) \]
14. Add Preprocessing

Alternative 9: 2.3% accurate, 19.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\pi}} \cdot \frac{0.5}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (sqrt (/ 1.0 PI)) (/ 0.5 x)))

C

double code(double x) {
	return sqrt((1.0 / ((double) M_PI))) * (0.5 / x);
}

Java

public static double code(double x) {
	return Math.sqrt((1.0 / Math.PI)) * (0.5 / x);
}

Python

def code(x):
	return math.sqrt((1.0 / math.pi)) * (0.5 / x)

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 / pi)) * Float64(0.5 / x))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 / pi)) * (0.5 / x);
end

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(0.5 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 36.2%

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

   1. associate-*l/ 36.2%

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

   2. *-lft-identity 36.2%

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

   3. unpow2 36.2%

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

   4. sqr-abs 36.2%

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

   5. unpow3 36.2%

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

7. Simplified 36.2%

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

   1. *-un-lft-identity 36.2%

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

   2. div-inv 36.2%

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

   3. pow1/2 36.2%

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

   4. inv-pow 36.2%

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

   5. pow-pow 36.2%

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

   6. metadata-eval 36.2%

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

   7. pow-flip 37.0%

      \[\leadsto e^{x \cdot x} \cdot \left(0.5 \cdot \left(1 \cdot \left({\pi}^{-0.5} \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(-3\right)}}\right)\right)\right) \]

   8. metadata-eval 37.0%

      \[\leadsto e^{x \cdot x} \cdot \left(0.5 \cdot \left(1 \cdot \left({\pi}^{-0.5} \cdot {\left(\left|x\right|\right)}^{\color{blue}{-3}}\right)\right)\right) \]

9. Applied egg-rr 37.0%

   \[\leadsto e^{x \cdot x} \cdot \left(0.5 \cdot \color{blue}{\left(1 \cdot \left({\pi}^{-0.5} \cdot {\left(\left|x\right|\right)}^{-3}\right)\right)}\right) \]
10. Step-by-step derivation

    1. *-lft-identity 37.0%

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

    2. sqr-pow 37.0%

       \[\leadsto e^{x \cdot x} \cdot \left(0.5 \cdot \left({\pi}^{-0.5} \cdot \color{blue}{\left({\left(\left|x\right|\right)}^{\left(\frac{-3}{2}\right)} \cdot {\left(\left|x\right|\right)}^{\left(\frac{-3}{2}\right)}\right)}\right)\right) \]

    3. sqr-pow 37.0%

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

    4. unpow1 37.0%

       \[\leadsto e^{x \cdot x} \cdot \left(0.5 \cdot \left({\pi}^{-0.5} \cdot {\left(\left|\color{blue}{{x}^{1}}\right|\right)}^{-3}\right)\right) \]

    5. sqr-pow 37.0%

       \[\leadsto e^{x \cdot x} \cdot \left(0.5 \cdot \left({\pi}^{-0.5} \cdot {\left(\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|\right)}^{-3}\right)\right) \]

    6. fabs-sqr 37.0%

       \[\leadsto e^{x \cdot x} \cdot \left(0.5 \cdot \left({\pi}^{-0.5} \cdot {\color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}}^{-3}\right)\right) \]

    7. sqr-pow 37.0%

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

    8. unpow1 37.0%

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

11. Simplified 37.0%

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

12. Taylor expanded in x around 0 2.4%

    \[\leadsto \color{blue}{0.5 \cdot \left(\frac{1}{x} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.5 \cdot \left(\frac{1}{{x}^{3}} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
13. Step-by-step derivation

    1. +-commutative 2.4%

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

    2. associate-*r* 2.4%

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

    3. associate-*r* 2.4%

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

    4. distribute-rgt-out 2.4%

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

    5. associate-*r/ 2.4%

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

    6. metadata-eval 2.4%

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

    7. associate-*r/ 2.4%

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

    8. metadata-eval 2.4%

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

14. Simplified 2.4%

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

15. Taylor expanded in x around inf 2.4%

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

    1. associate-*r* 2.4%

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

    2. associate-*r/ 2.4%

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

    3. metadata-eval 2.4%

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

17. Simplified 2.4%

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

18. Final simplification 2.4%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{0.5}{x} \]
19. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(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)))))))