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

Percentage Accurate: 100.0% → 100.0%

Time: 11.6s

Alternatives: 8

Speedup: 4.9×

• 99.3% 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, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right) \cdot e^{{x}^{2}}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (*
   (fma
    0.5
    (pow x -3.0)
    (/ (+ 1.0 (fma 0.75 (pow x -4.0) (* 1.875 (pow x -6.0)))) x))
   (exp (pow x 2.0)))
  (sqrt PI)))

C

double code(double x) {
	return (fma(0.5, pow(x, -3.0), ((1.0 + fma(0.75, pow(x, -4.0), (1.875 * pow(x, -6.0)))) / x)) * exp(pow(x, 2.0))) / sqrt(((double) M_PI));
}

Julia

function code(x)
	return Float64(Float64(fma(0.5, (x ^ -3.0), Float64(Float64(1.0 + fma(0.75, (x ^ -4.0), Float64(1.875 * (x ^ -6.0)))) / x)) * exp((x ^ 2.0))) / sqrt(pi))
end

Wolfram

code[x_] := N[(N[(N[(0.5 * N[Power[x, -3.0], $MachinePrecision] + N[(N[(1.0 + N[(0.75 * N[Power[x, -4.0], $MachinePrecision] + N[(1.875 * N[Power[x, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * N[Exp[N[Power[x, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $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%

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

   1. *-commutative 100.0%

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

   2. associate-*r/ 100.0%

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

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, {x}^{-3}, \frac{1 + \mathsf{fma}\left(0.75, {x}^{-4}, 1.875 \cdot {x}^{-6}\right)}{x}\right) \cdot e^{{x}^{2}}}{\sqrt{\pi}}} \]
7. Add Preprocessing

Alternative 2: 100.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \left(e^{{x}^{2}} \cdot {\pi}^{-0.5}\right) \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* (exp (pow x 2.0)) (pow PI -0.5))
  (/
   (+ 1.0 (+ (/ 0.75 (pow x 4.0)) (+ (/ 0.5 (* x x)) (/ 1.875 (pow x 6.0)))))
   x)))

C

double code(double x) {
	return (exp(pow(x, 2.0)) * pow(((double) M_PI), -0.5)) * ((1.0 + ((0.75 / pow(x, 4.0)) + ((0.5 / (x * x)) + (1.875 / pow(x, 6.0))))) / x);
}

Java

public static double code(double x) {
	return (Math.exp(Math.pow(x, 2.0)) * Math.pow(Math.PI, -0.5)) * ((1.0 + ((0.75 / Math.pow(x, 4.0)) + ((0.5 / (x * x)) + (1.875 / Math.pow(x, 6.0))))) / x);
}

Python

def code(x):
	return (math.exp(math.pow(x, 2.0)) * math.pow(math.pi, -0.5)) * ((1.0 + ((0.75 / math.pow(x, 4.0)) + ((0.5 / (x * x)) + (1.875 / math.pow(x, 6.0))))) / x)

Julia

function code(x)
	return Float64(Float64(exp((x ^ 2.0)) * (pi ^ -0.5)) * Float64(Float64(1.0 + Float64(Float64(0.75 / (x ^ 4.0)) + Float64(Float64(0.5 / Float64(x * x)) + Float64(1.875 / (x ^ 6.0))))) / x))
end

MATLAB

function tmp = code(x)
	tmp = (exp((x ^ 2.0)) * (pi ^ -0.5)) * ((1.0 + ((0.75 / (x ^ 4.0)) + ((0.5 / (x * x)) + (1.875 / (x ^ 6.0))))) / x);
end

Wolfram

code[x_] := N[(N[(N[Exp[N[Power[x, 2.0], $MachinePrecision]], $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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. fma-undefine 100.0%

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

   2. fma-undefine 100.0%

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

   3. associate-+r+ 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(0.5 \cdot \frac{1}{{x}^{2}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)\right)}{x}} \]
8. Step-by-step derivation

   1. associate-*r/ 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)\right)}{x} \]

   2. metadata-eval 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{\color{blue}{0.5}}{{x}^{2}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)\right)}{x} \]

   3. associate-*r/ 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \color{blue}{\frac{1.875 \cdot 1}{{x}^{6}}}\right)\right)}{x} \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{\color{blue}{1.875}}{{x}^{6}}\right)\right)}{x} \]

9. Simplified 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x}} \]
10. Step-by-step derivation

    1. div-inv 100.0%

       \[\leadsto \color{blue}{\left(e^{x \cdot x} \cdot \frac{1}{\sqrt{\pi}}\right)} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    2. pow2 100.0%

       \[\leadsto \left(e^{\color{blue}{{x}^{2}}} \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    3. pow1/2 100.0%

       \[\leadsto \left(e^{{x}^{2}} \cdot \frac{1}{\color{blue}{{\pi}^{0.5}}}\right) \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    4. pow-flip 100.0%

       \[\leadsto \left(e^{{x}^{2}} \cdot \color{blue}{{\pi}^{\left(-0.5\right)}}\right) \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

    5. metadata-eval 100.0%

       \[\leadsto \left(e^{{x}^{2}} \cdot {\pi}^{\color{blue}{-0.5}}\right) \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

11. Applied egg-rr 100.0%

    \[\leadsto \color{blue}{\left(e^{{x}^{2}} \cdot {\pi}^{-0.5}\right)} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]
12. Step-by-step derivation

    1. pow2 100.0%

       \[\leadsto \left(e^{{x}^{2}} \cdot {\pi}^{-0.5}\right) \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{\color{blue}{x \cdot x}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

13. Applied egg-rr 100.0%

    \[\leadsto \left(e^{{x}^{2}} \cdot {\pi}^{-0.5}\right) \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{\color{blue}{x \cdot x}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]
14. Add Preprocessing

Alternative 3: 100.0% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return ((1.0 + ((0.75 / Math.pow(x, 4.0)) + ((0.5 / (x * x)) + (1.875 / Math.pow(x, 6.0))))) / x) * (Math.exp((x * x)) / Math.sqrt(Math.PI));
}

Python

def code(x):
	return ((1.0 + ((0.75 / math.pow(x, 4.0)) + ((0.5 / (x * x)) + (1.875 / math.pow(x, 6.0))))) / x) * (math.exp((x * x)) / math.sqrt(math.pi))

Julia

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

MATLAB

function tmp = code(x)
	tmp = ((1.0 + ((0.75 / (x ^ 4.0)) + ((0.5 / (x * x)) + (1.875 / (x ^ 6.0))))) / x) * (exp((x * x)) / sqrt(pi));
end

Wolfram

code[x_] := N[(N[(N[(1.0 + N[(N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[(N[Exp[N[(x * x), $MachinePrecision]], $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. Step-by-step derivation

   1. fma-undefine 100.0%

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

   2. fma-undefine 100.0%

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

   3. associate-+r+ 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(0.5 \cdot \frac{1}{{x}^{2}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)\right)}{x}} \]
8. Step-by-step derivation

   1. associate-*r/ 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)\right)}{x} \]

   2. metadata-eval 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{\color{blue}{0.5}}{{x}^{2}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)\right)}{x} \]

   3. associate-*r/ 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \color{blue}{\frac{1.875 \cdot 1}{{x}^{6}}}\right)\right)}{x} \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{\color{blue}{1.875}}{{x}^{6}}\right)\right)}{x} \]

9. Simplified 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x}} \]
10. Step-by-step derivation

    1. pow2 100.0%

       \[\leadsto \left(e^{{x}^{2}} \cdot {\pi}^{-0.5}\right) \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{\color{blue}{x \cdot x}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

11. Applied egg-rr 100.0%

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

12. Final simplification 100.0%

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

Alternative 4: 99.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{\frac{0.5}{{x}^{2}} + \left(1 + \frac{0.75}{{x}^{4}}\right)}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (exp (* x x)) (sqrt PI))
  (/ (+ (/ 0.5 (pow x 2.0)) (+ 1.0 (/ 0.75 (pow x 4.0)))) x)))

C

double code(double x) {
	return (exp((x * x)) / sqrt(((double) M_PI))) * (((0.5 / pow(x, 2.0)) + (1.0 + (0.75 / pow(x, 4.0)))) / x);
}

Java

public static double code(double x) {
	return (Math.exp((x * x)) / Math.sqrt(Math.PI)) * (((0.5 / Math.pow(x, 2.0)) + (1.0 + (0.75 / Math.pow(x, 4.0)))) / x);
}

Python

def code(x):
	return (math.exp((x * x)) / math.sqrt(math.pi)) * (((0.5 / math.pow(x, 2.0)) + (1.0 + (0.75 / math.pow(x, 4.0)))) / x)

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * x)) / sqrt(pi)) * Float64(Float64(Float64(0.5 / (x ^ 2.0)) + Float64(1.0 + Float64(0.75 / (x ^ 4.0)))) / x))
end

MATLAB

function tmp = code(x)
	tmp = (exp((x * x)) / sqrt(pi)) * (((0.5 / (x ^ 2.0)) + (1.0 + (0.75 / (x ^ 4.0)))) / x);
end

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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. fma-undefine 100.0%

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

   2. fma-undefine 100.0%

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

   3. associate-+r+ 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(0.5 \cdot \frac{1}{{x}^{2}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)\right)}{x}} \]
8. Step-by-step derivation

   1. associate-*r/ 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)\right)}{x} \]

   2. metadata-eval 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{\color{blue}{0.5}}{{x}^{2}} + 1.875 \cdot \frac{1}{{x}^{6}}\right)\right)}{x} \]

   3. associate-*r/ 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \color{blue}{\frac{1.875 \cdot 1}{{x}^{6}}}\right)\right)}{x} \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{\color{blue}{1.875}}{{x}^{6}}\right)\right)}{x} \]

9. Simplified 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{{x}^{2}} + \frac{1.875}{{x}^{6}}\right)\right)}{x}} \]

10. Taylor expanded in x around inf 99.4%

    \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1 + \left(\frac{0.75}{{x}^{4}} + 0.5 \cdot \frac{1}{{x}^{2}}\right)}{x}} \]
11. Step-by-step derivation

    1. associate-+r+ 99.4%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{\color{blue}{\left(1 + \frac{0.75}{{x}^{4}}\right) + 0.5 \cdot \frac{1}{{x}^{2}}}}{x} \]

    2. +-commutative 99.4%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{\color{blue}{0.5 \cdot \frac{1}{{x}^{2}} + \left(1 + \frac{0.75}{{x}^{4}}\right)}}{x} \]

    3. associate-*r/ 99.4%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} + \left(1 + \frac{0.75}{{x}^{4}}\right)}{x} \]

    4. metadata-eval 99.4%

       \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{\frac{\color{blue}{0.5}}{{x}^{2}} + \left(1 + \frac{0.75}{{x}^{4}}\right)}{x} \]

12. Simplified 99.4%

    \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{\frac{0.5}{{x}^{2}} + \left(1 + \frac{0.75}{{x}^{4}}\right)}{x}} \]
13. Add Preprocessing

Alternative 5: 99.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \frac{0.5}{{x}^{2}}\right)}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (exp (* x x)) (sqrt PI))
  (/ (+ 1.0 (+ (/ 0.75 (pow x 4.0)) (/ 0.5 (pow x 2.0)))) x)))

C

double code(double x) {
	return (exp((x * x)) / sqrt(((double) M_PI))) * ((1.0 + ((0.75 / pow(x, 4.0)) + (0.5 / pow(x, 2.0)))) / x);
}

Java

public static double code(double x) {
	return (Math.exp((x * x)) / Math.sqrt(Math.PI)) * ((1.0 + ((0.75 / Math.pow(x, 4.0)) + (0.5 / Math.pow(x, 2.0)))) / x);
}

Python

def code(x):
	return (math.exp((x * x)) / math.sqrt(math.pi)) * ((1.0 + ((0.75 / math.pow(x, 4.0)) + (0.5 / math.pow(x, 2.0)))) / x)

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * x)) / sqrt(pi)) * Float64(Float64(1.0 + Float64(Float64(0.75 / (x ^ 4.0)) + Float64(0.5 / (x ^ 2.0)))) / x))
end

MATLAB

function tmp = code(x)
	tmp = (exp((x * x)) / sqrt(pi)) * ((1.0 + ((0.75 / (x ^ 4.0)) + (0.5 / (x ^ 2.0)))) / x);
end

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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. fma-undefine 100.0%

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

   2. fma-undefine 100.0%

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

   3. associate-+r+ 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf 99.4%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1 + \left(\frac{0.75}{{x}^{4}} + 0.5 \cdot \frac{1}{{x}^{2}}\right)}{x}} \]
8. Step-by-step derivation

   1. associate-*r/ 99.4%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right)}{x} \]

   2. metadata-eval 99.4%

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \frac{\color{blue}{0.5}}{{x}^{2}}\right)}{x} \]

9. Simplified 99.4%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1 + \left(\frac{0.75}{{x}^{4}} + \frac{0.5}{{x}^{2}}\right)}{x}} \]
10. Add Preprocessing

Alternative 6: 99.6% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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. fma-undefine 100.0%

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

   2. fma-undefine 100.0%

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

   3. associate-+r+ 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf 99.4%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1 + 0.5 \cdot \frac{1}{{x}^{2}}}{x}} \]
8. Step-by-step derivation

   1. associate-*r/ 99.4%

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

   2. metadata-eval 99.4%

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

9. Simplified 99.4%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1 + \frac{0.5}{{x}^{2}}}{x}} \]
10. Step-by-step derivation

    1. pow2 100.0%

       \[\leadsto \left(e^{{x}^{2}} \cdot {\pi}^{-0.5}\right) \cdot \frac{1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{\color{blue}{x \cdot x}} + \frac{1.875}{{x}^{6}}\right)\right)}{x} \]

11. Applied egg-rr 99.4%

    \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{1 + \frac{0.5}{\color{blue}{x \cdot x}}}{x} \]
12. Add Preprocessing

Alternative 7: 99.5% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $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. fma-undefine 100.0%

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

   2. fma-undefine 100.0%

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

   3. associate-+r+ 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf 99.4%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\frac{1}{x}} \]
8. Add Preprocessing

Alternative 8: 1.8% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left({x}^{-3} \cdot {\pi}^{-0.5}\right) \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(0.5 * N[(N[Power[x, -3.0], $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 0 31.4%

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

   1. unpow2 31.4%

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

   2. sqr-abs 31.4%

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

   3. unpow3 31.4%

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

7. Simplified 31.4%

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

8. Taylor expanded in x around 0 1.9%

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

   1. associate-*r* 1.9%

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

   2. *-commutative 1.9%

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

   3. associate-*l* 1.9%

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

   4. unpow3 1.9%

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

   5. sqr-abs 1.9%

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

   6. unpow2 1.9%

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

   7. unpow2 1.9%

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

   8. rem-square-sqrt 1.9%

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

   9. fabs-sqr 1.9%

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

   10. rem-square-sqrt 1.9%

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

   11. unpow3 1.9%

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

   12. exp-to-pow 1.9%

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

   13. *-commutative 1.9%

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

   14. exp-neg 1.9%

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

   15. distribute-lft-neg-in 1.9%

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

   16. metadata-eval 1.9%

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

   17. *-commutative 1.9%

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

   18. exp-to-pow 1.9%

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

10. Simplified 1.9%

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

11. Taylor expanded in x around 0 1.9%

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

    1. associate-*r* 1.9%

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

    2. associate-*r/ 1.9%

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

    3. metadata-eval 1.9%

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

    4. unpow-1 1.9%

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

    5. metadata-eval 1.9%

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

    6. pow-sqr 1.9%

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

    7. rem-sqrt-square 1.9%

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

    8. rem-square-sqrt 1.9%

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

    9. fabs-sqr 1.9%

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

    10. rem-square-sqrt 1.9%

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

    11. *-commutative 1.9%

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

13. Simplified 1.9%

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

    1. pow1 1.9%

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

    2. div-inv 1.9%

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

    3. pow-flip 1.9%

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

    4. metadata-eval 1.9%

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

15. Applied egg-rr 1.9%

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

    1. unpow1 1.9%

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

    2. associate-*r* 1.9%

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

    3. *-commutative 1.9%

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

    4. associate-*l* 1.9%

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

17. Simplified 1.9%

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

18. Final simplification 1.9%

    \[\leadsto 0.5 \cdot \left({x}^{-3} \cdot {\pi}^{-0.5}\right) \]
19. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(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)))))))