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

Percentage Accurate: 100.0% → 100.0%

Time: 14.0s

Alternatives: 9

Speedup: 3.5×

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

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{\color{blue}{\left(3 + 2\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. metadata-eval 100.0%

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

   3. pow-prod-up 100.0%

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

   4. pow3 100.0%

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

   5. metadata-eval 100.0%

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

   6. pow2 100.0%

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

   7. *-un-lft-identity 100.0%

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

   8. *-commutative 100.0%

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

5. Applied egg-rr 100.0%

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

   1. distribute-lft-in 100.0%

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

   2. *-rgt-identity 100.0%

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

   3. *-rgt-identity 100.0%

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

7. Applied egg-rr 100.0%

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

   1. distribute-lft-out 100.0%

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

   2. +-commutative 100.0%

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

   3. associate-/l/ 100.0%

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

9. Simplified 100.0%

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

10. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{\color{blue}{\left(3 + 2\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. metadata-eval 100.0%

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

   3. pow-prod-up 100.0%

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

   4. pow3 100.0%

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

   5. metadata-eval 100.0%

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

   6. pow2 100.0%

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

   7. *-un-lft-identity 100.0%

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

   8. *-commutative 100.0%

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

5. Applied egg-rr 100.0%

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

   1. distribute-lft-in 100.0%

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

   2. *-rgt-identity 100.0%

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

   3. *-rgt-identity 100.0%

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

7. Applied egg-rr 100.0%

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

   1. distribute-lft-out 100.0%

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

   2. +-commutative 100.0%

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

   3. associate-/l/ 100.0%

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

9. Simplified 100.0%

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

    1. pow-exp 99.1%

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

11. Applied egg-rr 99.9%

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

12. Final simplification 99.9%

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

Alternative 3: 99.7% accurate, 4.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (exp (* x x)) (sqrt PI))
  (+ (/ (+ 1.0 (/ 0.5 (* x x))) (fabs x)) (/ 0.75 (pow x 5.0)))))

C

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

Java

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

Python

def code(x):
	return (math.exp((x * x)) / math.sqrt(math.pi)) * (((1.0 + (0.5 / (x * x))) / math.fabs(x)) + (0.75 / math.pow(x, 5.0)))

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * x)) / sqrt(pi)) * Float64(Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x)) + Float64(0.75 / (x ^ 5.0))))
end

MATLAB

function tmp = code(x)
	tmp = (exp((x * x)) / sqrt(pi)) * (((1.0 + (0.5 / (x * x))) / abs(x)) + (0.75 / (x ^ 5.0)));
end

Wolfram

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

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

   1. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{\color{blue}{\left(3 + 2\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. metadata-eval 100.0%

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

   3. pow-prod-up 100.0%

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

   4. pow3 100.0%

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

   5. metadata-eval 100.0%

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

   6. pow2 100.0%

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

   7. *-un-lft-identity 100.0%

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

   8. *-commutative 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in x around inf 99.1%

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

   1. pow-exp 99.1%

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

8. Applied egg-rr 99.1%

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

9. Final simplification 99.1%

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

Alternative 4: 52.0% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{\color{blue}{\left(3 + 2\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. metadata-eval 100.0%

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

   3. pow-prod-up 100.0%

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

   4. pow3 100.0%

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

   5. metadata-eval 100.0%

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

   6. pow2 100.0%

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

   7. *-un-lft-identity 100.0%

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

   8. *-commutative 100.0%

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

5. Applied egg-rr 100.0%

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

   1. distribute-lft-in 100.0%

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

   2. *-rgt-identity 100.0%

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

   3. *-rgt-identity 100.0%

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

7. Applied egg-rr 100.0%

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

   1. distribute-lft-out 100.0%

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

   2. +-commutative 100.0%

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

   3. associate-/l/ 100.0%

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

9. Simplified 100.0%

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

10. Taylor expanded in x around 0 50.5%

    \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {x}^{-5} \cdot \left(\frac{1.875}{x \cdot x} + 0.75\right)\right) \cdot \frac{\color{blue}{1 + {x}^{2}}}{\sqrt{\pi}} \]
11. Step-by-step derivation

    1. unpow2 50.5%

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

12. Simplified 50.5%

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

13. Final simplification 50.5%

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

Alternative 5: 52.0% accurate, 5.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+ (/ (+ 1.0 (/ 0.5 (* x x))) (fabs x)) (/ 0.75 (pow x 5.0)))
  (/ (+ 1.0 (* x x)) (sqrt PI))))

C

double code(double x) {
	return (((1.0 + (0.5 / (x * x))) / fabs(x)) + (0.75 / pow(x, 5.0))) * ((1.0 + (x * x)) / sqrt(((double) M_PI)));
}

Java

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

Python

def code(x):
	return (((1.0 + (0.5 / (x * x))) / math.fabs(x)) + (0.75 / math.pow(x, 5.0))) * ((1.0 + (x * x)) / math.sqrt(math.pi))

Julia

function code(x)
	return Float64(Float64(Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x)) + Float64(0.75 / (x ^ 5.0))) * Float64(Float64(1.0 + Float64(x * x)) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = (((1.0 + (0.5 / (x * x))) / abs(x)) + (0.75 / (x ^ 5.0))) * ((1.0 + (x * x)) / sqrt(pi));
end

Wolfram

code[x_] := N[(N[(N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(0.75 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(x * 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%

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

   1. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{\color{blue}{\left(3 + 2\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. metadata-eval 100.0%

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

   3. pow-prod-up 100.0%

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

   4. pow3 100.0%

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

   5. metadata-eval 100.0%

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

   6. pow2 100.0%

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

   7. *-un-lft-identity 100.0%

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

   8. *-commutative 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in x around inf 99.1%

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

7. Taylor expanded in x around 0 50.5%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{0.75}{{x}^{5}}\right) \cdot \frac{\color{blue}{1 + {x}^{2}}}{\sqrt{\pi}} \]
8. Step-by-step derivation

   1. unpow2 50.5%

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

9. Simplified 50.5%

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

10. Final simplification 50.5%

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

Alternative 6: 52.0% accurate, 5.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (+ 1.0 (* x x)) (sqrt PI))
  (+ (/ (+ 1.0 (/ 0.5 (* x x))) (fabs x)) (/ 1.875 (pow x 7.0)))))

C

double code(double x) {
	return ((1.0 + (x * x)) / sqrt(((double) M_PI))) * (((1.0 + (0.5 / (x * x))) / fabs(x)) + (1.875 / pow(x, 7.0)));
}

Java

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

Python

def code(x):
	return ((1.0 + (x * x)) / math.sqrt(math.pi)) * (((1.0 + (0.5 / (x * x))) / math.fabs(x)) + (1.875 / math.pow(x, 7.0)))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 + Float64(x * x)) / sqrt(pi)) * Float64(Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x)) + Float64(1.875 / (x ^ 7.0))))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 + (x * x)) / sqrt(pi)) * (((1.0 + (0.5 / (x * x))) / abs(x)) + (1.875 / (x ^ 7.0)));
end

Wolfram

code[x_] := N[(N[(N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 7.0], $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%

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

   1. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{\color{blue}{\left(3 + 2\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. metadata-eval 100.0%

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

   3. pow-prod-up 100.0%

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

   4. pow3 100.0%

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

   5. metadata-eval 100.0%

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

   6. pow2 100.0%

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

   7. *-un-lft-identity 100.0%

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

   8. *-commutative 100.0%

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

5. Applied egg-rr 100.0%

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

   1. distribute-lft-in 100.0%

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

   2. *-rgt-identity 100.0%

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

   3. *-rgt-identity 100.0%

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

7. Applied egg-rr 100.0%

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

   1. distribute-lft-out 100.0%

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

   2. +-commutative 100.0%

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

   3. associate-/l/ 100.0%

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

9. Simplified 100.0%

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

10. Taylor expanded in x around 0 50.5%

    \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {x}^{-5} \cdot \left(\frac{1.875}{x \cdot x} + 0.75\right)\right) \cdot \frac{\color{blue}{1 + {x}^{2}}}{\sqrt{\pi}} \]
11. Step-by-step derivation

    1. unpow2 50.5%

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

12. Simplified 50.5%

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

13. Taylor expanded in x around 0 50.5%

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

14. Final simplification 50.5%

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

Alternative 7: 52.0% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{\color{blue}{\left(3 + 2\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. metadata-eval 100.0%

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

   3. pow-prod-up 100.0%

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

   4. pow3 100.0%

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

   5. metadata-eval 100.0%

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

   6. pow2 100.0%

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

   7. *-un-lft-identity 100.0%

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

   8. *-commutative 100.0%

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

5. Applied egg-rr 100.0%

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

   1. distribute-lft-in 100.0%

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

   2. *-rgt-identity 100.0%

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

   3. *-rgt-identity 100.0%

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

7. Applied egg-rr 100.0%

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

   1. distribute-lft-out 100.0%

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

   2. +-commutative 100.0%

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

   3. associate-/l/ 100.0%

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

9. Simplified 100.0%

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

10. Taylor expanded in x around 0 50.5%

    \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {x}^{-5} \cdot \left(\frac{1.875}{x \cdot x} + 0.75\right)\right) \cdot \frac{\color{blue}{1 + {x}^{2}}}{\sqrt{\pi}} \]
11. Step-by-step derivation

    1. unpow2 50.5%

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

12. Simplified 50.5%

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

13. Taylor expanded in x around inf 50.5%

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

    1. *-commutative 50.5%

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

    2. unpow2 50.5%

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

15. Simplified 50.5%

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

16. Final simplification 50.5%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{x \cdot x}{\left|x\right|} \]

Alternative 8: 5.4% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{\color{blue}{\left(3 + 2\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. metadata-eval 100.0%

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

   3. pow-prod-up 100.0%

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

   4. pow3 100.0%

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

   5. metadata-eval 100.0%

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

   6. pow2 100.0%

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

   7. *-un-lft-identity 100.0%

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

   8. *-commutative 100.0%

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

5. Applied egg-rr 100.0%

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

   1. distribute-lft-in 100.0%

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

   2. *-rgt-identity 100.0%

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

   3. *-rgt-identity 100.0%

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

7. Applied egg-rr 100.0%

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

   1. distribute-lft-out 100.0%

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

   2. +-commutative 100.0%

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

   3. associate-/l/ 100.0%

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

9. Simplified 100.0%

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

10. Taylor expanded in x around 0 50.5%

    \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {x}^{-5} \cdot \left(\frac{1.875}{x \cdot x} + 0.75\right)\right) \cdot \frac{\color{blue}{1 + {x}^{2}}}{\sqrt{\pi}} \]
11. Step-by-step derivation

    1. unpow2 50.5%

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

12. Simplified 50.5%

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

13. Taylor expanded in x around inf 50.5%

    \[\leadsto \color{blue}{1.5 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \frac{1}{\left|x\right|}\right) + \frac{{x}^{2}}{\left|x\right|} \cdot \sqrt{\frac{1}{\pi}}} \]
14. Step-by-step derivation

    1. +-commutative 50.5%

       \[\leadsto \color{blue}{\frac{{x}^{2}}{\left|x\right|} \cdot \sqrt{\frac{1}{\pi}} + 1.5 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \frac{1}{\left|x\right|}\right)} \]

    2. *-commutative 50.5%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \frac{{x}^{2}}{\left|x\right|}} + 1.5 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \frac{1}{\left|x\right|}\right) \]

    3. *-commutative 50.5%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{{x}^{2}}{\left|x\right|} + \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \frac{1}{\left|x\right|}\right) \cdot 1.5} \]

    4. associate-*l* 50.5%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{{x}^{2}}{\left|x\right|} + \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{\left|x\right|} \cdot 1.5\right)} \]

    5. distribute-lft-out 50.5%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\frac{{x}^{2}}{\left|x\right|} + \frac{1}{\left|x\right|} \cdot 1.5\right)} \]

    6. unpow2 50.5%

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

    7. unpow1 50.5%

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

    8. sqr-pow 50.5%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\frac{x \cdot x}{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} + \frac{1}{\left|x\right|} \cdot 1.5\right) \]

    9. fabs-sqr 50.5%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\frac{x \cdot x}{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}} + \frac{1}{\left|x\right|} \cdot 1.5\right) \]

    10. sqr-pow 50.5%

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

    11. unpow1 50.5%

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

    12. associate-/l* 5.5%

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

    13. *-inverses 5.5%

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

    14. /-rgt-identity 5.5%

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

15. Simplified 5.5%

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

16. Final simplification 5.5%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(x + \frac{1.5}{x}\right) \]

Alternative 9: 5.4% accurate, 10.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(x * N[Sqrt[N[(1.0 / 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%

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

   1. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{\color{blue}{\left(3 + 2\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. metadata-eval 100.0%

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

   3. pow-prod-up 100.0%

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

   4. pow3 100.0%

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

   5. metadata-eval 100.0%

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

   6. pow2 100.0%

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

   7. *-un-lft-identity 100.0%

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

   8. *-commutative 100.0%

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

5. Applied egg-rr 100.0%

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

   1. distribute-lft-in 100.0%

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

   2. *-rgt-identity 100.0%

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

   3. *-rgt-identity 100.0%

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

7. Applied egg-rr 100.0%

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

   1. distribute-lft-out 100.0%

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

   2. +-commutative 100.0%

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

   3. associate-/l/ 100.0%

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

9. Simplified 100.0%

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

10. Taylor expanded in x around 0 50.5%

    \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {x}^{-5} \cdot \left(\frac{1.875}{x \cdot x} + 0.75\right)\right) \cdot \frac{\color{blue}{1 + {x}^{2}}}{\sqrt{\pi}} \]
11. Step-by-step derivation

    1. unpow2 50.5%

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

12. Simplified 50.5%

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

13. Taylor expanded in x around inf 50.5%

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

    1. unpow2 50.5%

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

    2. associate-/l* 5.5%

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

    3. associate-/l* 50.5%

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

    4. unpow1 50.5%

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

    5. sqr-pow 50.5%

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

    6. fabs-sqr 50.5%

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

    7. sqr-pow 50.5%

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

    8. unpow1 50.5%

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

    9. associate-/l* 5.5%

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

    10. *-inverses 5.5%

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

    11. /-rgt-identity 5.5%

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

15. Simplified 5.5%

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

16. Final simplification 5.5%

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

Reproduce

herbie shell --seed 2023292 
(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)))))))