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

Percentage Accurate: 100.0% → 100.0%

Time: 5.4s

Alternatives: 10

Speedup: 2.7×

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

Specification

\[x \geq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot t\_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot t\_1\right) + \frac{3}{4} \cdot t\_2\right) + \frac{15}{8} \cdot \left(\left(t\_2 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot t\_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot t\_1\right) + \frac{3}{4} \cdot t\_2\right) + \frac{15}{8} \cdot \left(\left(t\_2 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({x}^{-5}, 0.75, \mathsf{fma}\left({x}^{-7}, 1.875, \frac{\frac{0.5}{x \cdot x} + 1}{x}\right)\right) \cdot \frac{{\left(e^{\left(-x\right) \cdot 2}\right)}^{\left(\frac{x}{-2}\right)}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma
   (pow x -5.0)
   0.75
   (fma (pow x -7.0) 1.875 (/ (+ (/ 0.5 (* x x)) 1.0) x)))
  (/ (pow (exp (* (- x) 2.0)) (/ x -2.0)) (sqrt PI))))

C

double code(double x) {
	return fma(pow(x, -5.0), 0.75, fma(pow(x, -7.0), 1.875, (((0.5 / (x * x)) + 1.0) / x))) * (pow(exp((-x * 2.0)), (x / -2.0)) / sqrt(((double) M_PI)));
}

Julia

function code(x)
	return Float64(fma((x ^ -5.0), 0.75, fma((x ^ -7.0), 1.875, Float64(Float64(Float64(0.5 / Float64(x * x)) + 1.0) / x))) * Float64((exp(Float64(Float64(-x) * 2.0)) ^ Float64(x / -2.0)) / sqrt(pi)))
end

Wolfram

code[x_] := N[(N[(N[Power[x, -5.0], $MachinePrecision] * 0.75 + N[(N[Power[x, -7.0], $MachinePrecision] * 1.875 + N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[N[((-x) * 2.0), $MachinePrecision]], $MachinePrecision], N[(x / -2.0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)\right)} \]

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   2. lift-exp.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\color{blue}{\pi}}} \]

   3. sqr-abs-rev N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \]

   4. sqr-neg-rev N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{e^{\left(\mathsf{neg}\left(\left|x\right|\right)\right) \cdot \left(\mathsf{neg}\left(\left|x\right|\right)\right)}}{\sqrt{\pi}} \]

   5. lift-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{e^{\left(-\left|x\right|\right) \cdot \left(\mathsf{neg}\left(\left|x\right|\right)\right)}}{\sqrt{\pi}} \]

   6. lift-fabs.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{e^{\left(-\left|x\right|\right) \cdot \left(\mathsf{neg}\left(\left|x\right|\right)\right)}}{\sqrt{\pi}} \]

   7. lift-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{e^{\left(-\left|x\right|\right) \cdot \left(-\left|x\right|\right)}}{\sqrt{\pi}} \]

   8. lift-fabs.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{e^{\left(-\left|x\right|\right) \cdot \left(-\left|x\right|\right)}}{\sqrt{\pi}} \]

   9. pow-exp N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{{\left(e^{-\left|x\right|}\right)}^{\left(-\left|x\right|\right)}}{\sqrt{\color{blue}{\pi}}} \]

   10. lift-fabs.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{{\left(e^{-\left|x\right|}\right)}^{\left(-\left|x\right|\right)}}{\sqrt{\pi}} \]

   11. lift-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{{\left(e^{\mathsf{neg}\left(\left|x\right|\right)}\right)}^{\left(-\left|x\right|\right)}}{\sqrt{\pi}} \]

   12. sqr-pow N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{{\left(e^{\mathsf{neg}\left(\left|x\right|\right)}\right)}^{\left(\frac{-\left|x\right|}{2}\right)} \cdot {\left(e^{\mathsf{neg}\left(\left|x\right|\right)}\right)}^{\left(\frac{-\left|x\right|}{2}\right)}}{\sqrt{\color{blue}{\pi}}} \]

   13. pow-prod-down N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{{\left(e^{\mathsf{neg}\left(\left|x\right|\right)} \cdot e^{\mathsf{neg}\left(\left|x\right|\right)}\right)}^{\left(\frac{-\left|x\right|}{2}\right)}}{\sqrt{\color{blue}{\pi}}} \]

   14. lower-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{{\left(e^{\mathsf{neg}\left(\left|x\right|\right)} \cdot e^{\mathsf{neg}\left(\left|x\right|\right)}\right)}^{\left(\frac{-\left|x\right|}{2}\right)}}{\sqrt{\color{blue}{\pi}}} \]

6. Applied rewrites 100.0%

   \[\leadsto \mathsf{fma}\left(\frac{0.5}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, 1.875, {\left(\left|x\right|\right)}^{-5} \cdot 0.75\right)\right) \cdot \frac{{\left(e^{-x} \cdot e^{-x}\right)}^{\left(\frac{-x}{2}\right)}}{\sqrt{\color{blue}{\pi}}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{{\left(e^{-x} \cdot e^{-x}\right)}^{\left(\frac{-x}{2}\right)}}{\sqrt{\pi}} \]

   2. lift-exp.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{{\left(e^{-x} \cdot e^{-x}\right)}^{\left(\frac{-x}{2}\right)}}{\sqrt{\pi}} \]

   3. lift-exp.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{{\left(e^{-x} \cdot e^{-x}\right)}^{\left(\frac{-x}{2}\right)}}{\sqrt{\pi}} \]

   4. exp-lft-sqr-rev N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{{\left(e^{\left(-x\right) \cdot 2}\right)}^{\left(\frac{-x}{2}\right)}}{\sqrt{\pi}} \]

   5. rem-log-exp N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{{\left(e^{\log \left(e^{\left(-x\right) \cdot 2}\right)}\right)}^{\left(\frac{-x}{2}\right)}}{\sqrt{\pi}} \]

   6. lift-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{{\left(e^{\log \left(e^{\left(\mathsf{neg}\left(x\right)\right) \cdot 2}\right)}\right)}^{\left(\frac{-x}{2}\right)}}{\sqrt{\pi}} \]

   7. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{{\left(e^{\log \left(e^{\left(-1 \cdot x\right) \cdot 2}\right)}\right)}^{\left(\frac{-x}{2}\right)}}{\sqrt{\pi}} \]

   8. pow-exp N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{{\left(e^{\log \left({\left(e^{-1 \cdot x}\right)}^{2}\right)}\right)}^{\left(\frac{-x}{2}\right)}}{\sqrt{\pi}} \]

   9. lower-exp.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{{\left(e^{\log \left({\left(e^{-1 \cdot x}\right)}^{2}\right)}\right)}^{\left(\frac{-x}{2}\right)}}{\sqrt{\pi}} \]

   10. pow-exp N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{{\left(e^{\log \left(e^{\left(-1 \cdot x\right) \cdot 2}\right)}\right)}^{\left(\frac{-x}{2}\right)}}{\sqrt{\pi}} \]

   11. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{{\left(e^{\log \left(e^{\left(\mathsf{neg}\left(x\right)\right) \cdot 2}\right)}\right)}^{\left(\frac{-x}{2}\right)}}{\sqrt{\pi}} \]

   12. lift-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{{\left(e^{\log \left(e^{\left(-x\right) \cdot 2}\right)}\right)}^{\left(\frac{-x}{2}\right)}}{\sqrt{\pi}} \]

   13. rem-log-exp N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{{\left(e^{\left(-x\right) \cdot 2}\right)}^{\left(\frac{-x}{2}\right)}}{\sqrt{\pi}} \]

   14. lower-*.f64 100.0

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

   15. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{{\left(e^{\left(-x\right) \cdot 2}\right)}^{\left(\frac{-x}{2}\right)}}{\sqrt{\pi}} \]

   16. lift-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{{\left(e^{\left(-x\right) \cdot 2}\right)}^{\left(\frac{\mathsf{neg}\left(x\right)}{2}\right)}}{\sqrt{\pi}} \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{{\left(e^{\left(-x\right) \cdot 2}\right)}^{\left(\frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(-2\right)}\right)}}{\sqrt{\pi}} \]

   18. frac-2neg-rev N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, \frac{15}{8}, {\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4}\right)\right) \cdot \frac{{\left(e^{\left(-x\right) \cdot 2}\right)}^{\left(\frac{x}{-2}\right)}}{\sqrt{\pi}} \]

   19. lower-/.f64 100.0

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

8. Applied rewrites 100.0%

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

9. Taylor expanded in x around inf

   \[\leadsto \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right)\right)\right) \cdot \frac{\color{blue}{{\left(e^{\left(-x\right) \cdot 2}\right)}^{\left(\frac{x}{-2}\right)}}}{\sqrt{\pi}} \]

11. Applied rewrites 100.0%

    \[\leadsto \mathsf{fma}\left({x}^{-5}, 0.75, \mathsf{fma}\left({x}^{-7}, 1.875, \frac{\frac{0.5}{x \cdot x} + 1}{x}\right)\right) \cdot \frac{\color{blue}{{\left(e^{\left(-x\right) \cdot 2}\right)}^{\left(\frac{x}{-2}\right)}}}{\sqrt{\pi}} \]
12. Add Preprocessing

Alternative 2: 100.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({x}^{-5}, 0.75, \mathsf{fma}\left({x}^{-7}, 1.875, \frac{\frac{0.5}{x \cdot x} + 1}{x}\right)\right) \cdot \frac{{\left(e^{-x}\right)}^{\left(-x\right)}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma
   (pow x -5.0)
   0.75
   (fma (pow x -7.0) 1.875 (/ (+ (/ 0.5 (* x x)) 1.0) x)))
  (/ (pow (exp (- x)) (- x)) (sqrt PI))))

C

double code(double x) {
	return fma(pow(x, -5.0), 0.75, fma(pow(x, -7.0), 1.875, (((0.5 / (x * x)) + 1.0) / x))) * (pow(exp(-x), -x) / sqrt(((double) M_PI)));
}

Julia

function code(x)
	return Float64(fma((x ^ -5.0), 0.75, fma((x ^ -7.0), 1.875, Float64(Float64(Float64(0.5 / Float64(x * x)) + 1.0) / x))) * Float64((exp(Float64(-x)) ^ Float64(-x)) / sqrt(pi)))
end

Wolfram

code[x_] := N[(N[(N[Power[x, -5.0], $MachinePrecision] * 0.75 + N[(N[Power[x, -7.0], $MachinePrecision] * 1.875 + N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[(-x)], $MachinePrecision], (-x)], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)\right)} \]

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

   \[\leadsto \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right)\right)\right) \cdot \frac{\color{blue}{e^{x \cdot x}}}{\sqrt{\pi}} \]

7. Applied rewrites 100.0%

   \[\leadsto \mathsf{fma}\left({x}^{-5}, 0.75, \mathsf{fma}\left({x}^{-7}, 1.875, \frac{\frac{0.5}{x \cdot x} + 1}{x}\right)\right) \cdot \frac{\color{blue}{e^{x \cdot x}}}{\sqrt{\pi}} \]
8. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \mathsf{fma}\left({x}^{-5}, \frac{3}{4}, \mathsf{fma}\left({x}^{-7}, \frac{15}{8}, \frac{\frac{\frac{1}{2}}{x \cdot x} + 1}{x}\right)\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\color{blue}{\pi}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left({x}^{-5}, \frac{3}{4}, \mathsf{fma}\left({x}^{-7}, \frac{15}{8}, \frac{\frac{\frac{1}{2}}{x \cdot x} + 1}{x}\right)\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   3. sqr-neg-rev N/A

      \[\leadsto \mathsf{fma}\left({x}^{-5}, \frac{3}{4}, \mathsf{fma}\left({x}^{-7}, \frac{15}{8}, \frac{\frac{\frac{1}{2}}{x \cdot x} + 1}{x}\right)\right) \cdot \frac{e^{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)}}{\sqrt{\pi}} \]

   4. lift-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left({x}^{-5}, \frac{3}{4}, \mathsf{fma}\left({x}^{-7}, \frac{15}{8}, \frac{\frac{\frac{1}{2}}{x \cdot x} + 1}{x}\right)\right) \cdot \frac{e^{\left(-x\right) \cdot \left(\mathsf{neg}\left(x\right)\right)}}{\sqrt{\pi}} \]

   5. lift-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left({x}^{-5}, \frac{3}{4}, \mathsf{fma}\left({x}^{-7}, \frac{15}{8}, \frac{\frac{\frac{1}{2}}{x \cdot x} + 1}{x}\right)\right) \cdot \frac{e^{\left(-x\right) \cdot \left(-x\right)}}{\sqrt{\pi}} \]

   6. exp-prod N/A

      \[\leadsto \mathsf{fma}\left({x}^{-5}, \frac{3}{4}, \mathsf{fma}\left({x}^{-7}, \frac{15}{8}, \frac{\frac{\frac{1}{2}}{x \cdot x} + 1}{x}\right)\right) \cdot \frac{{\left(e^{-x}\right)}^{\left(-x\right)}}{\sqrt{\color{blue}{\pi}}} \]

   7. lift-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left({x}^{-5}, \frac{3}{4}, \mathsf{fma}\left({x}^{-7}, \frac{15}{8}, \frac{\frac{\frac{1}{2}}{x \cdot x} + 1}{x}\right)\right) \cdot \frac{{\left(e^{\mathsf{neg}\left(x\right)}\right)}^{\left(-x\right)}}{\sqrt{\pi}} \]

   8. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left({x}^{-5}, \frac{3}{4}, \mathsf{fma}\left({x}^{-7}, \frac{15}{8}, \frac{\frac{\frac{1}{2}}{x \cdot x} + 1}{x}\right)\right) \cdot \frac{{\left(e^{\mathsf{neg}\left(x\right)}\right)}^{\left(-x\right)}}{\sqrt{\color{blue}{\pi}}} \]

   9. lift-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left({x}^{-5}, \frac{3}{4}, \mathsf{fma}\left({x}^{-7}, \frac{15}{8}, \frac{\frac{\frac{1}{2}}{x \cdot x} + 1}{x}\right)\right) \cdot \frac{{\left(e^{-x}\right)}^{\left(-x\right)}}{\sqrt{\pi}} \]

   10. lift-exp.f64 100.0

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

9. Applied rewrites 100.0%

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

Alternative 3: 100.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. sqr-abs N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 100.0

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{x}\right)}^{x}\right) \cdot \color{blue}{\left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)} \]

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around -inf

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

9. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{x}\right)}^{x}\right) \cdot \left(-\frac{\left(\left(-\frac{\frac{1.875}{x \cdot x} + 0.75}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\right) - 1\right) - \frac{0.5}{x \cdot x}}{x}\right) \]
10. Add Preprocessing

Alternative 4: 100.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)\right)} \]

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

   \[\leadsto \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right)\right)\right) \cdot \frac{\color{blue}{e^{x \cdot x}}}{\sqrt{\pi}} \]

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around -inf

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

10. Applied rewrites 100.0%

    \[\leadsto \left(-\frac{\left(\left(-\frac{\frac{1.875}{x \cdot x} + 0.75}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\right) - 1\right) - \frac{0.5}{x \cdot x}}{x}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]
11. Add Preprocessing

Alternative 5: 99.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)\right)} \]

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

   \[\leadsto \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right)\right)\right) \cdot \frac{\color{blue}{e^{x \cdot x}}}{\sqrt{\pi}} \]

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left({x}^{-5}, \frac{3}{4}, \frac{1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}}{x}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]
9. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({x}^{-5}, \frac{3}{4}, \frac{1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}}{x}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   2. associate-*r/ N/A

      \[\leadsto \mathsf{fma}\left({x}^{-5}, \frac{3}{4}, \frac{1 + \frac{\frac{1}{2} \cdot 1}{{x}^{2}}}{x}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({x}^{-5}, \frac{3}{4}, \frac{1 + \frac{\frac{1}{2} \cdot 1}{{x}^{2}}}{x}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({x}^{-5}, \frac{3}{4}, \frac{1 + \frac{\frac{1}{2}}{{x}^{2}}}{x}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({x}^{-5}, \frac{3}{4}, \frac{1 + \frac{\frac{1}{2}}{{x}^{2}}}{x}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   6. pow2 N/A

      \[\leadsto \mathsf{fma}\left({x}^{-5}, \frac{3}{4}, \frac{1 + \frac{\frac{1}{2}}{x \cdot x}}{x}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   7. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left({x}^{-5}, \frac{3}{4}, \frac{1 + \frac{\frac{1}{2}}{x \cdot x}}{x}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   8. lift-/.f64 N/A

      \[\leadsto \mathsf{fma}\left({x}^{-5}, \frac{3}{4}, \frac{1 + \frac{\frac{1}{2}}{x \cdot x}}{x}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left({x}^{-5}, \frac{3}{4}, \frac{\frac{\frac{1}{2}}{x \cdot x} + 1}{x}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   10. lift-+.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{-5}, \frac{3}{4}, \frac{\frac{\frac{1}{2}}{x \cdot x} + 1}{x}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   11. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{-5}, \frac{3}{4}, \frac{\frac{\frac{1}{2}}{x \cdot x} + 1}{x}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   12. metadata-eval 99.6

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

10. Applied rewrites 99.6%

    \[\leadsto \mathsf{fma}\left({x}^{-5}, 0.75, \frac{\frac{0.5}{x \cdot x} + 1}{x}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]
11. Add Preprocessing

Alternative 6: 99.6% accurate, 3.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	return -((-(((0.75 / (x * x)) + 0.5) / (x * x)) - 1.0) / x) * (exp((x * x)) / sqrt(((double) M_PI)));
}

Java

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

Python

def code(x):
	return -((-(((0.75 / (x * x)) + 0.5) / (x * x)) - 1.0) / x) * (math.exp((x * x)) / math.sqrt(math.pi))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)\right)} \]

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

   \[\leadsto \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right)\right)\right) \cdot \frac{\color{blue}{e^{x \cdot x}}}{\sqrt{\pi}} \]

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around -inf

   \[\leadsto \left(-1 \cdot \frac{-1 \cdot \frac{\frac{1}{2} + \frac{3}{4} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - 1}{x}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]
9. Step-by-step derivation

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{1}{2} + \frac{3}{4} \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - 1}{x}\right)\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

10. Applied rewrites 99.6%

    \[\leadsto \left(-\frac{\left(-\frac{\frac{0.75}{x \cdot x} + 0.5}{x \cdot x}\right) - 1}{x}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]
11. Add Preprocessing

Alternative 7: 99.6% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{x \cdot x} + 1}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (/ (+ (/ 0.5 (* x x)) 1.0) x) (/ (exp (* x x)) (sqrt PI))))

C

double code(double x) {
	return (((0.5 / (x * x)) + 1.0) / x) * (exp((x * x)) / sqrt(((double) M_PI)));
}

Java

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

Python

def code(x):
	return (((0.5 / (x * x)) + 1.0) / x) * (math.exp((x * x)) / math.sqrt(math.pi))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)\right)} \]

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

   \[\leadsto \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right)\right)\right) \cdot \frac{\color{blue}{e^{x \cdot x}}}{\sqrt{\pi}} \]

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-+.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. metadata-eval 99.6

       \[\leadsto \frac{\frac{0.5}{x \cdot x} + 1}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

10. Applied rewrites 99.6%

    \[\leadsto \frac{\frac{0.5}{x \cdot x} + 1}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]
11. Add Preprocessing

Alternative 8: 99.6% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)\right)} \]

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

   \[\leadsto \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right)\right)\right) \cdot \frac{\color{blue}{e^{x \cdot x}}}{\sqrt{\pi}} \]

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around inf

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

   1. lower-/.f64 99.6

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

10. Applied rewrites 99.6%

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

Alternative 9: 2.3% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{x \cdot x} + 1}{x} \cdot \frac{1}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (/ (+ (/ 0.5 (* x x)) 1.0) x) (/ 1.0 (sqrt PI))))

C

double code(double x) {
	return (((0.5 / (x * x)) + 1.0) / x) * (1.0 / sqrt(((double) M_PI)));
}

Java

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

Python

def code(x):
	return (((0.5 / (x * x)) + 1.0) / x) * (1.0 / math.sqrt(math.pi))

Julia

function code(x)
	return Float64(Float64(Float64(Float64(0.5 / Float64(x * x)) + 1.0) / x) * Float64(1.0 / sqrt(pi)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. sqr-abs N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 100.0

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)} \]

6. Applied rewrites 2.3%

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

7. Taylor expanded in x around inf

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

   1. metadata-eval N/A

      \[\leadsto \frac{1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}}{x} \cdot \frac{1}{\sqrt{\pi}} \]

   2. associate-*r/ N/A

      \[\leadsto \frac{1 + \frac{\frac{1}{2} \cdot 1}{{x}^{2}}}{x} \cdot \frac{1}{\sqrt{\pi}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{1 + \frac{\frac{1}{2} \cdot 1}{{x}^{2}}}{x} \cdot \frac{1}{\sqrt{\pi}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{1 + \frac{\frac{1}{2}}{{x}^{2}}}{x} \cdot \frac{1}{\sqrt{\pi}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{1 + \frac{\frac{1}{2}}{{x}^{2}}}{x} \cdot \frac{1}{\sqrt{\pi}} \]

   6. pow2 N/A

      \[\leadsto \frac{1 + \frac{\frac{1}{2}}{x \cdot x}}{x} \cdot \frac{1}{\sqrt{\pi}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{1 + \frac{\frac{1}{2}}{x \cdot x}}{x} \cdot \frac{1}{\sqrt{\pi}} \]

   8. lift-/.f64 N/A

      \[\leadsto \frac{1 + \frac{\frac{1}{2}}{x \cdot x}}{x} \cdot \frac{1}{\sqrt{\pi}} \]

   9. +-commutative N/A

      \[\leadsto \frac{\frac{\frac{1}{2}}{x \cdot x} + 1}{x} \cdot \frac{1}{\sqrt{\pi}} \]

   10. lift-+.f64 N/A

       \[\leadsto \frac{\frac{\frac{1}{2}}{x \cdot x} + 1}{x} \cdot \frac{1}{\sqrt{\pi}} \]

   11. lift-/.f64 N/A

       \[\leadsto \frac{\frac{\frac{1}{2}}{x \cdot x} + 1}{x} \cdot \frac{1}{\sqrt{\pi}} \]

   12. metadata-eval 2.3

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

9. Applied rewrites 2.3%

   \[\leadsto \frac{\frac{0.5}{x \cdot x} + 1}{x} \cdot \frac{\color{blue}{1}}{\sqrt{\pi}} \]
10. Add Preprocessing

Alternative 10: 2.3% accurate, 17.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. sqr-abs N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 100.0

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)} \]

6. Applied rewrites 2.3%

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

7. Taylor expanded in x around inf

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

   1. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{1}{\color{blue}{x}} \]

   2. sqrt-div N/A

      \[\leadsto \frac{\sqrt{1}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{1}{x} \]

   3. metadata-eval N/A

      \[\leadsto \frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{1}{x} \]

   4. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{1}{x} \]

   5. lift-PI.f64 N/A

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

   6. frac-times N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 2.3

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

9. Applied rewrites 2.3%

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

Reproduce

herbie shell --seed 2025116 
(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)))))))