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

Percentage Accurate: 100.0% → 100.0%

Time: 8.6s

Alternatives: 7

Speedup: 3.5×

• 99.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	return (((1.0 + (0.5 / (x * x))) / fabs(x)) + (log(exp(pow(x, -5.0))) * (0.75 + ((1.875 / x) / x)))) * (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.log(Math.exp(Math.pow(x, -5.0))) * (0.75 + ((1.875 / x) / x)))) * (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.log(math.exp(math.pow(x, -5.0))) * (0.75 + ((1.875 / x) / x)))) * (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(log(exp((x ^ -5.0))) * Float64(0.75 + Float64(Float64(1.875 / x) / x)))) * Float64((exp(x) ^ x) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = (((1.0 + (0.5 / (x * x))) / abs(x)) + (log(exp((x ^ -5.0))) * (0.75 + ((1.875 / x) / x)))) * ((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[Log[N[Exp[N[Power[x, -5.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(0.75 + N[(N[(1.875 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x], $MachinePrecision], x], $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%

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

   1. add-log-exp 100.0%

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

   2. inv-pow 100.0%

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

   3. pow-pow 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. fabs-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right)\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. add-log-exp 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. inv-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. pow-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{x}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {x}^{\color{blue}{-5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left(0 + {x}^{-5}\right)} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. +-lft-identity 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 3: 99.7% accurate, 3.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (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 (pow(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.pow(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.pow(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(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[Power[N[Exp[x], $MachinePrecision], x], $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 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%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right)\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. add-log-exp 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. inv-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. pow-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{x}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {x}^{\color{blue}{-5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left(0 + {x}^{-5}\right)} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. +-lft-identity 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.5%

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

9. Final simplification 99.5%

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

Alternative 4: 99.6% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right)\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. add-log-exp 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. inv-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. pow-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{x}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {x}^{\color{blue}{-5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left(0 + {x}^{-5}\right)} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. +-lft-identity 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.5%

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

9. Taylor expanded in x around inf 99.4%

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

    1. unpow1 99.4%

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

    2. metadata-eval 99.4%

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

    3. pow-sqr 99.4%

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

    4. unpow1/2 99.4%

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

    5. unpow1/2 99.4%

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

    6. fabs-sqr 99.4%

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

    7. unpow1/2 99.4%

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

    8. unpow1/2 99.4%

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

    9. pow-sqr 99.4%

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

    10. metadata-eval 99.4%

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

    11. unpow1 99.4%

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

11. Simplified 99.4%

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

    1. pow-exp 99.4%

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

13. Applied egg-rr 99.4%

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

14. Final simplification 99.4%

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

Alternative 5: 99.6% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(x)
	return Float64(Float64((exp(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[Power[N[Exp[x], $MachinePrecision], x], $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%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right)\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. add-log-exp 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. inv-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. pow-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{x}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {x}^{\color{blue}{-5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left(0 + {x}^{-5}\right)} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. +-lft-identity 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.5%

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

9. Taylor expanded in x around inf 99.4%

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

    1. unpow1 99.4%

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

    2. metadata-eval 99.4%

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

    3. pow-sqr 99.4%

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

    4. unpow1/2 99.4%

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

    5. unpow1/2 99.4%

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

    6. fabs-sqr 99.4%

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

    7. unpow1/2 99.4%

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

    8. unpow1/2 99.4%

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

    9. pow-sqr 99.4%

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

    10. metadata-eval 99.4%

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

    11. unpow1 99.4%

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

11. Simplified 99.4%

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

12. Taylor expanded in x around inf 99.4%

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

13. Final simplification 99.4%

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

Alternative 6: 51.6% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right)\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. add-log-exp 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. inv-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. pow-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{x}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {x}^{\color{blue}{-5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left(0 + {x}^{-5}\right)} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. +-lft-identity 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.5%

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

9. Taylor expanded in x around inf 99.4%

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

    1. unpow1 99.4%

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

    2. metadata-eval 99.4%

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

    3. pow-sqr 99.4%

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

    4. unpow1/2 99.4%

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

    5. unpow1/2 99.4%

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

    6. fabs-sqr 99.4%

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

    7. unpow1/2 99.4%

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

    8. unpow1/2 99.4%

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

    9. pow-sqr 99.4%

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

    10. metadata-eval 99.4%

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

    11. unpow1 99.4%

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

11. Simplified 99.4%

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

12. Taylor expanded in x around 0 56.1%

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

    1. unpow2 56.1%

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

14. Simplified 56.1%

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

15. Final simplification 56.1%

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

Alternative 7: 2.3% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right)\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. add-log-exp 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. inv-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. pow-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{x}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {x}^{\color{blue}{-5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left(0 + {x}^{-5}\right)} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. +-lft-identity 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.5%

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

9. Taylor expanded in x around inf 99.4%

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

    1. unpow1 99.4%

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

    2. metadata-eval 99.4%

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

    3. pow-sqr 99.4%

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

    4. unpow1/2 99.4%

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

    5. unpow1/2 99.4%

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

    6. fabs-sqr 99.4%

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

    7. unpow1/2 99.4%

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

    8. unpow1/2 99.4%

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

    9. pow-sqr 99.4%

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

    10. metadata-eval 99.4%

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

    11. unpow1 99.4%

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

11. Simplified 99.4%

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

12. Taylor expanded in x around 0 2.3%

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

13. Final simplification 2.3%

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

Reproduce

herbie shell --seed 2023266 
(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)))))))