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

Percentage Accurate: 100.0% → 99.7%

Time: 35.5s

Alternatives: 5

Speedup: N/A×

• 99.6% 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, 10.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around inf 100.0%

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

   1. *-commutative 100.0%

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

   2. +-commutative 100.0%

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

   3. +-commutative 100.0%

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

   4. associate-+l+ 100.0%

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

   5. associate-*r/ 100.0%

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

   6. metadata-eval 100.0%

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

   7. associate-*r/ 100.0%

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

   8. metadata-eval 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 100.0%

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

   1. *-commutative 100.0%

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

   2. *-commutative 100.0%

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

   3. unpow-1 100.0%

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

   4. metadata-eval 100.0%

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

   5. pow-sqr 100.0%

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

   6. rem-sqrt-square 100.0%

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

   7. rem-square-sqrt 100.0%

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

   8. fabs-sqr 100.0%

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

   9. rem-square-sqrt 100.0%

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

   10. associate-*l* 100.0%

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

10. Simplified 100.0%

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

11. Taylor expanded in x around inf 100.0%

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

    1. unpow2 100.0%

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

13. Applied egg-rr 100.0%

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

Alternative 2: 84.0% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \sqrt{\frac{{x}^{6}}{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (sqrt (/ (pow x 6.0) PI))))

C

double code(double x) {
	return 0.5 * sqrt((pow(x, 6.0) / ((double) M_PI)));
}

Java

public static double code(double x) {
	return 0.5 * Math.sqrt((Math.pow(x, 6.0) / Math.PI));
}

Python

def code(x):
	return 0.5 * math.sqrt((math.pow(x, 6.0) / math.pi))

Julia

function code(x)
	return Float64(0.5 * sqrt(Float64((x ^ 6.0) / pi)))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * sqrt(((x ^ 6.0) / pi));
end

Wolfram

code[x_] := N[(0.5 * N[Sqrt[N[(N[Power[x, 6.0], $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 1.8%

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

   1. sqrt-div 1.8%

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

   2. metadata-eval 1.8%

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

   3. un-div-inv 1.8%

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

   4. associate-/r* 1.8%

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

   5. div-inv 1.8%

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

   6. pow-flip 1.8%

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

   7. add-sqr-sqrt 1.8%

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

   8. fabs-sqr 1.8%

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

   9. add-sqr-sqrt 1.8%

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

   10. metadata-eval 1.8%

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

   11. metadata-eval 1.8%

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

   12. pow-sqr 1.8%

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

   13. inv-pow 1.8%

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

   14. inv-pow 1.8%

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

   15. pow3 1.8%

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

   16. inv-pow 1.8%

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

   17. pow-pow 1.8%

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

   18. add-sqr-sqrt 1.8%

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

   19. fabs-sqr 1.8%

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

   20. add-sqr-sqrt 1.8%

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

   21. metadata-eval 1.8%

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

7. Applied egg-rr 1.8%

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

9. Applied egg-rr 88.0%

   \[\leadsto 0.5 \cdot \color{blue}{\sqrt{\frac{{x}^{6}}{\pi}}} \]
10. Add Preprocessing

Alternative 3: 52.5% accurate, 19.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Taylor expanded in x around 0 1.8%

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

   1. *-commutative 1.8%

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

   2. associate-/r* 1.8%

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

   3. sqrt-div 1.8%

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

   4. metadata-eval 1.8%

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

   5. frac-times 1.8%

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

   6. *-un-lft-identity 1.8%

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

   7. pow-flip 1.8%

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

   8. metadata-eval 1.8%

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

   9. add-sqr-sqrt 1.8%

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

   10. fabs-sqr 1.8%

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

   11. add-sqr-sqrt 1.8%

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

7. Applied egg-rr 1.8%

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

   1. add-sqr-sqrt 1.8%

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

   2. pow2 1.8%

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

   3. sqrt-pow1 1.8%

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

   4. metadata-eval 1.8%

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

   5. inv-pow 1.8%

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

   6. add-exp-log 1.8%

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

   7. rec-exp 1.8%

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

   8. add-sqr-sqrt 0.0%

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

   9. sqrt-unprod 52.7%

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

   10. sqr-neg 52.7%

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

   11. sqrt-unprod 52.7%

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

   12. add-sqr-sqrt 52.7%

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

   13. add-exp-log 52.7%

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

   14. unpow2 52.7%

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

9. Applied egg-rr 52.7%

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

10. Final simplification 52.7%

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

Alternative 4: 5.4% accurate, 19.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 1.8%

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

   1. *-commutative 1.8%

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

   2. associate-/r* 1.8%

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

   3. sqrt-div 1.8%

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

   4. metadata-eval 1.8%

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

   5. frac-times 1.8%

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

   6. *-un-lft-identity 1.8%

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

   7. pow-flip 1.8%

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

   8. metadata-eval 1.8%

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

   9. add-sqr-sqrt 1.8%

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

   10. fabs-sqr 1.8%

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

   11. add-sqr-sqrt 1.8%

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

7. Applied egg-rr 1.8%

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

   1. add-sqr-sqrt 1.8%

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

   2. pow2 1.8%

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

   3. sqrt-pow1 1.8%

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

   4. metadata-eval 1.8%

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

   5. inv-pow 1.8%

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

   6. add-exp-log 1.8%

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

   7. rec-exp 1.8%

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

   8. add-sqr-sqrt 0.0%

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

   9. sqrt-unprod 52.7%

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

   10. sqr-neg 52.7%

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

   11. sqrt-unprod 52.7%

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

   12. add-sqr-sqrt 52.7%

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

   13. add-exp-log 52.7%

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

   14. unpow2 52.7%

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

9. Applied egg-rr 52.7%

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

    1. clear-num 52.7%

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

    2. un-div-inv 52.7%

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

    3. clear-num 52.7%

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

    4. *-commutative 52.7%

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

    5. times-frac 5.3%

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

    6. *-inverses 5.3%

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

    7. *-un-lft-identity 5.3%

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

11. Applied egg-rr 5.3%

    \[\leadsto \color{blue}{\frac{0.5}{\frac{1}{\frac{x}{\sqrt{\pi}}}}} \]
12. Step-by-step derivation

    1. associate-/r/ 5.3%

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

    2. metadata-eval 5.3%

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

    3. *-commutative 5.3%

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

    4. associate-*l/ 5.3%

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

13. Simplified 5.3%

    \[\leadsto \color{blue}{\frac{x \cdot 0.5}{\sqrt{\pi}}} \]
14. Add Preprocessing

Alternative 5: 2.3% accurate, 20.0× speedup

Math

\[\begin{array}{l} \\ \frac{{\pi}^{-0.5}}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (pow PI -0.5) x))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Power[Pi, -0.5], $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around inf 100.0%

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

   1. *-commutative 100.0%

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

   2. +-commutative 100.0%

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

   3. +-commutative 100.0%

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

   4. associate-+l+ 100.0%

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

   5. associate-*r/ 100.0%

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

   6. metadata-eval 100.0%

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

   7. associate-*r/ 100.0%

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

   8. metadata-eval 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 100.0%

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

   1. *-commutative 100.0%

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

   2. *-commutative 100.0%

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

   3. unpow-1 100.0%

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

   4. metadata-eval 100.0%

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

   5. pow-sqr 100.0%

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

   6. rem-sqrt-square 100.0%

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

   7. rem-square-sqrt 100.0%

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

   8. fabs-sqr 100.0%

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

   9. rem-square-sqrt 100.0%

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

   10. associate-*l* 100.0%

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

10. Simplified 100.0%

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

11. Taylor expanded in x around inf 100.0%

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

12. Taylor expanded in x around 0 2.2%

    \[\leadsto \color{blue}{\frac{1}{x} \cdot \sqrt{\frac{1}{\pi}}} \]
13. Step-by-step derivation

    1. associate-*l/ 2.2%

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

    2. *-lft-identity 2.2%

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

    3. rem-exp-log 2.2%

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

    4. exp-neg 2.2%

       \[\leadsto \frac{\sqrt{\color{blue}{e^{-\log \pi}}}}{x} \]

    5. unpow1/2 2.2%

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

    6. exp-prod 2.2%

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

    7. distribute-lft-neg-out 2.2%

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

    8. distribute-rgt-neg-in 2.2%

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

    9. metadata-eval 2.2%

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

    10. exp-to-pow 2.2%

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

14. Simplified 2.2%

    \[\leadsto \color{blue}{\frac{{\pi}^{-0.5}}{x}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024116 
(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)))))))