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

Percentage Accurate: 100.0% → 99.7%

Time: 7.0s

Alternatives: 4

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, 7.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around inf 100.0%

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

   1. associate-*r/ 100.0%

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

   2. metadata-eval 100.0%

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

   3. unpow1 100.0%

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

   4. sqr-pow 100.0%

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

   5. fabs-sqr 100.0%

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

   6. sqr-pow 100.0%

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

   7. unpow1 100.0%

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

   8. +-commutative 100.0%

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

   9. associate-*r/ 100.0%

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

   10. metadata-eval 100.0%

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

   11. unpow1 100.0%

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

   12. sqr-pow 100.0%

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

   13. fabs-sqr 100.0%

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

   14. sqr-pow 100.0%

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

   15. unpow1 100.0%

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

   16. unpow1 100.0%

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

   17. sqr-pow 100.0%

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

   18. fabs-sqr 100.0%

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

   19. sqr-pow 100.0%

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

   20. unpow1 100.0%

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

6. Simplified 100.0%

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

7. Taylor expanded in x around inf 100.0%

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

   1. associate-*l/ 100.0%

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

   2. *-lft-identity 100.0%

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

9. Simplified 100.0%

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

    1. add-log-exp 3.5%

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

    2. *-un-lft-identity 3.5%

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

    3. log-prod 3.5%

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

    4. metadata-eval 3.5%

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

    5. add-log-exp 100.0%

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

    6. inv-pow 100.0%

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

    7. sqrt-pow1 100.0%

       \[\leadsto e^{x \cdot x} \cdot \left(0 + \frac{\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}}{x}\right) \]

    8. metadata-eval 100.0%

       \[\leadsto e^{x \cdot x} \cdot \left(0 + \frac{{\pi}^{\color{blue}{-0.5}}}{x}\right) \]

11. Applied egg-rr 100.0%

    \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(0 + \frac{{\pi}^{-0.5}}{x}\right)} \]
12. Step-by-step derivation

    1. +-lft-identity 100.0%

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

13. Simplified 100.0%

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

14. Final simplification 100.0%

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

Alternative 2: 1.8% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{0.25}{\pi} \cdot {x}^{-6}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (* (/ 0.25 PI) (pow x -6.0))))

C

double code(double x) {
	return sqrt(((0.25 / ((double) M_PI)) * pow(x, -6.0)));
}

Java

public static double code(double x) {
	return Math.sqrt(((0.25 / Math.PI) * Math.pow(x, -6.0)));
}

Python

def code(x):
	return math.sqrt(((0.25 / math.pi) * math.pow(x, -6.0)))

Julia

function code(x)
	return sqrt(Float64(Float64(0.25 / pi) * (x ^ -6.0)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt(((0.25 / pi) * (x ^ -6.0)));
end

Wolfram

code[x_] := N[Sqrt[N[(N[(0.25 / Pi), $MachinePrecision] * N[Power[x, -6.0], $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%

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

4. Taylor expanded in x around 0 37.5%

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

   1. associate-*r* 37.5%

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

   2. *-commutative 37.5%

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

   3. associate-/r* 38.7%

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

   4. associate-*r/ 38.7%

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

   5. unpow1 38.7%

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

   6. sqr-pow 38.7%

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

   7. fabs-sqr 38.7%

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

   8. sqr-pow 38.7%

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

   9. unpow1 38.7%

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

   10. associate-*r/ 38.7%

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

   11. metadata-eval 38.7%

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

   12. associate-/r* 37.5%

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

   13. pow-plus 37.5%

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

   14. metadata-eval 37.5%

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

6. Simplified 37.5%

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

7. Taylor expanded in x around 0 1.8%

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

   1. associate-*r* 1.8%

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

   2. associate-*r/ 1.8%

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

   3. metadata-eval 1.8%

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

   4. *-commutative 1.8%

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

9. Simplified 1.8%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \frac{0.5}{{x}^{3}}} \]
10. Step-by-step derivation

    1. *-commutative 1.8%

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

    2. sqrt-div 1.8%

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

    3. metadata-eval 1.8%

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

    4. un-div-inv 1.8%

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

    5. div-inv 1.8%

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

    6. pow-flip 1.8%

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

    7. metadata-eval 1.8%

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

11. Applied egg-rr 1.8%

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

    1. add-sqr-sqrt 1.8%

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

    2. sqrt-unprod 1.8%

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

    3. frac-times 1.8%

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

    4. *-commutative 1.8%

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

    5. *-commutative 1.8%

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

    6. swap-sqr 1.8%

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

    7. pow-prod-up 1.8%

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

    8. metadata-eval 1.8%

       \[\leadsto \sqrt{\frac{{x}^{\color{blue}{-6}} \cdot \left(0.5 \cdot 0.5\right)}{\sqrt{\pi} \cdot \sqrt{\pi}}} \]

    9. metadata-eval 1.8%

       \[\leadsto \sqrt{\frac{{x}^{-6} \cdot \color{blue}{0.25}}{\sqrt{\pi} \cdot \sqrt{\pi}}} \]

    10. add-sqr-sqrt 1.8%

        \[\leadsto \sqrt{\frac{{x}^{-6} \cdot 0.25}{\color{blue}{\pi}}} \]

13. Applied egg-rr 1.8%

    \[\leadsto \color{blue}{\sqrt{\frac{{x}^{-6} \cdot 0.25}{\pi}}} \]
14. Step-by-step derivation

    1. *-commutative 1.8%

       \[\leadsto \sqrt{\frac{\color{blue}{0.25 \cdot {x}^{-6}}}{\pi}} \]

    2. associate-/l* 1.7%

       \[\leadsto \sqrt{\color{blue}{\frac{0.25}{\frac{\pi}{{x}^{-6}}}}} \]

    3. associate-/r/ 1.8%

       \[\leadsto \sqrt{\color{blue}{\frac{0.25}{\pi} \cdot {x}^{-6}}} \]

15. Simplified 1.8%

    \[\leadsto \color{blue}{\sqrt{\frac{0.25}{\pi} \cdot {x}^{-6}}} \]

16. Final simplification 1.8%

    \[\leadsto \sqrt{\frac{0.25}{\pi} \cdot {x}^{-6}} \]

Alternative 3: 1.8% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\sqrt{\pi} \cdot {x}^{3}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(0.5 / N[(N[Sqrt[Pi], $MachinePrecision] * N[Power[x, 3.0], $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%

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

4. Taylor expanded in x around 0 37.5%

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

   1. associate-*r* 37.5%

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

   2. *-commutative 37.5%

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

   3. associate-/r* 38.7%

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

   4. associate-*r/ 38.7%

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

   5. unpow1 38.7%

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

   6. sqr-pow 38.7%

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

   7. fabs-sqr 38.7%

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

   8. sqr-pow 38.7%

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

   9. unpow1 38.7%

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

   10. associate-*r/ 38.7%

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

   11. metadata-eval 38.7%

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

   12. associate-/r* 37.5%

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

   13. pow-plus 37.5%

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

   14. metadata-eval 37.5%

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

6. Simplified 37.5%

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

7. Taylor expanded in x around 0 1.8%

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

   1. associate-*r* 1.8%

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

   2. associate-*r/ 1.8%

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

   3. metadata-eval 1.8%

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

   4. *-commutative 1.8%

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

9. Simplified 1.8%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \frac{0.5}{{x}^{3}}} \]
10. Step-by-step derivation

    1. expm1-log1p-u 1.8%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \frac{0.5}{{x}^{3}}\right)\right)} \]

    2. expm1-udef 1.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \frac{0.5}{{x}^{3}}\right)} - 1} \]

    3. sqrt-div 1.6%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}} \cdot \frac{0.5}{{x}^{3}}\right)} - 1 \]

    4. metadata-eval 1.6%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\sqrt{\pi}} \cdot \frac{0.5}{{x}^{3}}\right)} - 1 \]

    5. frac-times 1.6%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot 0.5}{\sqrt{\pi} \cdot {x}^{3}}}\right)} - 1 \]

    6. metadata-eval 1.6%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{0.5}}{\sqrt{\pi} \cdot {x}^{3}}\right)} - 1 \]

11. Applied egg-rr 1.6%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.5}{\sqrt{\pi} \cdot {x}^{3}}\right)} - 1} \]
12. Step-by-step derivation

    1. expm1-def 1.8%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5}{\sqrt{\pi} \cdot {x}^{3}}\right)\right)} \]

    2. expm1-log1p 1.8%

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

    3. *-commutative 1.8%

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

13. Simplified 1.8%

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

14. Final simplification 1.8%

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

Alternative 4: 1.8% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot {x}^{-3}}{\sqrt{\pi}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(0.5 * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 37.5%

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

   1. associate-*r* 37.5%

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

   2. *-commutative 37.5%

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

   3. associate-/r* 38.7%

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

   4. associate-*r/ 38.7%

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

   5. unpow1 38.7%

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

   6. sqr-pow 38.7%

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

   7. fabs-sqr 38.7%

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

   8. sqr-pow 38.7%

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

   9. unpow1 38.7%

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

   10. associate-*r/ 38.7%

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

   11. metadata-eval 38.7%

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

   12. associate-/r* 37.5%

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

   13. pow-plus 37.5%

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

   14. metadata-eval 37.5%

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

6. Simplified 37.5%

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

7. Taylor expanded in x around 0 1.8%

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

   1. associate-*r* 1.8%

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

   2. associate-*r/ 1.8%

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

   3. metadata-eval 1.8%

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

   4. *-commutative 1.8%

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

9. Simplified 1.8%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \frac{0.5}{{x}^{3}}} \]
10. Step-by-step derivation

    1. *-commutative 1.8%

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

    2. sqrt-div 1.8%

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

    3. metadata-eval 1.8%

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

    4. un-div-inv 1.8%

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

    5. div-inv 1.8%

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

    6. pow-flip 1.8%

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

    7. metadata-eval 1.8%

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

11. Applied egg-rr 1.8%

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

12. Final simplification 1.8%

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

Reproduce

herbie shell --seed 2023331 
(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)))))))