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

Percentage Accurate: 100.0% → 99.7%

Time: 7.0s

Alternatives: 3

Speedup: N/A×

• 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: 99.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{0.3333333333333333 \cdot \left(3 \cdot \left(x \cdot \sqrt{\pi}\right)\right)} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (* 0.3333333333333333 (* 3.0 (* x (sqrt PI)))))
  (+
   1.0
   (+ (/ 0.75 (pow (fabs x) 4.0)) (+ (/ 0.5 (* x x)) (/ 1.875 (pow x 6.0)))))))

C

double code(double x) {
	return (pow(exp(x), x) / (0.3333333333333333 * (3.0 * (x * sqrt(((double) M_PI)))))) * (1.0 + ((0.75 / pow(fabs(x), 4.0)) + ((0.5 / (x * x)) + (1.875 / pow(x, 6.0)))));
}

Java

public static double code(double x) {
	return (Math.pow(Math.exp(x), x) / (0.3333333333333333 * (3.0 * (x * Math.sqrt(Math.PI))))) * (1.0 + ((0.75 / Math.pow(Math.abs(x), 4.0)) + ((0.5 / (x * x)) + (1.875 / Math.pow(x, 6.0)))));
}

Python

def code(x):
	return (math.pow(math.exp(x), x) / (0.3333333333333333 * (3.0 * (x * math.sqrt(math.pi))))) * (1.0 + ((0.75 / math.pow(math.fabs(x), 4.0)) + ((0.5 / (x * x)) + (1.875 / math.pow(x, 6.0)))))

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / Float64(0.3333333333333333 * Float64(3.0 * Float64(x * sqrt(pi))))) * Float64(1.0 + Float64(Float64(0.75 / (abs(x) ^ 4.0)) + Float64(Float64(0.5 / Float64(x * x)) + Float64(1.875 / (x ^ 6.0))))))
end

MATLAB

function tmp = code(x)
	tmp = ((exp(x) ^ x) / (0.3333333333333333 * (3.0 * (x * sqrt(pi))))) * (1.0 + ((0.75 / (abs(x) ^ 4.0)) + ((0.5 / (x * x)) + (1.875 / (x ^ 6.0)))));
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[(0.3333333333333333 * N[(3.0 * N[(x * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(0.75 / N[Power[N[Abs[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. *-commutative 100.0%

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

   2. add-sqr-sqrt 100.0%

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

   3. fabs-sqr 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. add-log-exp 1.6%

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

   6. log-pow 1.6%

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

5. Applied egg-rr 1.6%

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

   1. add-cbrt-cube 1.6%

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

   2. pow1/3 1.6%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\log \color{blue}{\left({\left(\left({\left(e^{x}\right)}^{\left(\sqrt{\pi}\right)} \cdot {\left(e^{x}\right)}^{\left(\sqrt{\pi}\right)}\right) \cdot {\left(e^{x}\right)}^{\left(\sqrt{\pi}\right)}\right)}^{0.3333333333333333}\right)}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   3. log-pow 1.6%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{0.3333333333333333 \cdot \log \left(\left({\left(e^{x}\right)}^{\left(\sqrt{\pi}\right)} \cdot {\left(e^{x}\right)}^{\left(\sqrt{\pi}\right)}\right) \cdot {\left(e^{x}\right)}^{\left(\sqrt{\pi}\right)}\right)}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   4. pow3 1.6%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{0.3333333333333333 \cdot \log \color{blue}{\left({\left({\left(e^{x}\right)}^{\left(\sqrt{\pi}\right)}\right)}^{3}\right)}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   5. metadata-eval 1.6%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{0.3333333333333333 \cdot \log \left({\left({\left(e^{x}\right)}^{\left(\sqrt{\pi}\right)}\right)}^{\color{blue}{\left(2 + 1\right)}}\right)} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   6. log-pow 1.6%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{0.3333333333333333 \cdot \color{blue}{\left(\left(2 + 1\right) \cdot \log \left({\left(e^{x}\right)}^{\left(\sqrt{\pi}\right)}\right)\right)}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   7. metadata-eval 1.6%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{0.3333333333333333 \cdot \left(\color{blue}{3} \cdot \log \left({\left(e^{x}\right)}^{\left(\sqrt{\pi}\right)}\right)\right)} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   8. pow-exp 1.6%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{0.3333333333333333 \cdot \left(3 \cdot \log \color{blue}{\left(e^{x \cdot \sqrt{\pi}}\right)}\right)} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   9. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\left(x \cdot \sqrt{\pi}\right)}\right)} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

7. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \left(x \cdot \sqrt{\pi}\right)\right)}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

8. Final simplification 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{0.3333333333333333 \cdot \left(3 \cdot \left(x \cdot \sqrt{\pi}\right)\right)} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

Alternative 2: 99.8% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. clear-num 100.0%

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

   2. associate-/r/ 100.0%

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

   3. pow-flip 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. fabs-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

6. Taylor expanded in x around 0 100.0%

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

   1. *-commutative 100.0%

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

   2. unpow1 100.0%

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

   3. sqr-pow 100.0%

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

   4. fabs-sqr 100.0%

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

   5. sqr-pow 100.0%

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

   6. unpow1 100.0%

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

8. Simplified 100.0%

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

9. Final simplification 100.0%

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

Alternative 3: 99.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{0} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+
   1.0
   (+ (/ 0.75 (pow (fabs x) 4.0)) (+ (/ 0.5 (* x x)) (/ 1.875 (pow x 6.0)))))
  (/ (pow (exp x) x) 0.0)))

C

double code(double x) {
	return (1.0 + ((0.75 / pow(fabs(x), 4.0)) + ((0.5 / (x * x)) + (1.875 / pow(x, 6.0))))) * (pow(exp(x), x) / 0.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 + ((0.75d0 / (abs(x) ** 4.0d0)) + ((0.5d0 / (x * x)) + (1.875d0 / (x ** 6.0d0))))) * ((exp(x) ** x) / 0.0d0)
end function

Java

public static double code(double x) {
	return (1.0 + ((0.75 / Math.pow(Math.abs(x), 4.0)) + ((0.5 / (x * x)) + (1.875 / Math.pow(x, 6.0))))) * (Math.pow(Math.exp(x), x) / 0.0);
}

Python

def code(x):
	return (1.0 + ((0.75 / math.pow(math.fabs(x), 4.0)) + ((0.5 / (x * x)) + (1.875 / math.pow(x, 6.0))))) * (math.pow(math.exp(x), x) / 0.0)

Julia

function code(x)
	return Float64(Float64(1.0 + Float64(Float64(0.75 / (abs(x) ^ 4.0)) + Float64(Float64(0.5 / Float64(x * x)) + Float64(1.875 / (x ^ 6.0))))) * Float64((exp(x) ^ x) / 0.0))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 + ((0.75 / (abs(x) ^ 4.0)) + ((0.5 / (x * x)) + (1.875 / (x ^ 6.0))))) * ((exp(x) ^ x) / 0.0);
end

Wolfram

code[x_] := N[(N[(1.0 + N[(N[(0.75 / N[Power[N[Abs[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / 0.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. *-commutative 100.0%

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

   2. add-sqr-sqrt 100.0%

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

   3. fabs-sqr 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. add-log-exp 1.6%

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

   6. log-pow 1.6%

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

5. Applied egg-rr 1.6%

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

6. Taylor expanded in x around 0 99.2%

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

7. Final simplification 99.2%

   \[\leadsto \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{0} \]

Reproduce

herbie shell --seed 2023306 
(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)))))))