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

Percentage Accurate: 100.0% → 100.0%

Time: 6.9s

Alternatives: 13

Speedup: 2.2×

• 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, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{\sqrt[3]{\sqrt{\left(\pi \cdot \pi\right) \cdot \pi}}} \cdot {\left(e^{-x}\right)}^{\left(-x\right)}\right) \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{-5}, 0.75, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, 1.875, \frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|}\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(1.0 / N[Power[N[Sqrt[N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[(-x)], $MachinePrecision], (-x)], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Abs[x], $MachinePrecision], -5.0], $MachinePrecision] * 0.75 + N[(N[Power[N[Abs[x], $MachinePrecision], -7.0], $MachinePrecision] * 1.875 + N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
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. sqr-neg-rev N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-neg.f64 100.0

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

3. Applied rewrites 100.0%

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

   1. lift-PI.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. add-cbrt-cube N/A

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

   4. sqrt-cbrt N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-PI.f64 100.0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt[3]{\sqrt{\left(\pi \cdot \pi\right) \cdot \pi}}} \cdot {\left(e^{-x}\right)}^{\left(-x\right)}\right) \cdot \color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-5}, 0.75, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, 1.875, \frac{0.5}{\left(x \cdot x\right) \cdot \left|x\right|} + \frac{1}{\left|x\right|}\right)\right)} \]
9. Step-by-step derivation

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

10. Applied rewrites 100.0%

    \[\leadsto \left(\frac{1}{\sqrt[3]{\sqrt{\left(\pi \cdot \pi\right) \cdot \pi}}} \cdot {\left(e^{-x}\right)}^{\left(-x\right)}\right) \cdot \color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{-5}, 0.75, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, 1.875, \frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|}\right)\right)} \]
11. Add Preprocessing

Alternative 2: 100.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * (exp(Float64(-x)) ^ Float64(-x))) * fma(0.75, Float64(1.0 / Float64(Float64(Float64(x * x) * abs(x)) * Float64(x * x))), fma((abs(x) ^ -7.0), 1.875, Float64(Float64(Float64(0.5 / Float64(x * x)) + 1.0) / abs(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[(0.75 * N[(1.0 / N[(N[(N[(x * x), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Abs[x], $MachinePrecision], -7.0], $MachinePrecision] * 1.875 + N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
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. sqr-neg-rev N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-neg.f64 100.0

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 100.0%

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

Alternative 3: 100.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * (exp(x) ^ x)) * fma(0.75, Float64(1.0 / Float64(Float64(Float64(x * x) * abs(x)) * Float64(x * x))), fma((abs(x) ^ -7.0), 1.875, Float64(Float64(Float64(0.5 / Float64(x * x)) + 1.0) / abs(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[(0.75 * N[(1.0 / N[(N[(N[(x * x), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Abs[x], $MachinePrecision], -7.0], $MachinePrecision] * 1.875 + N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

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

Alternative 4: 100.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \frac{1}{\sqrt{\pi}} \cdot \left(e^{x \cdot x} \cdot \left(\mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, 1.875, \frac{1}{\left|x\right|} - \frac{-0.75}{t\_0 \cdot \left(x \cdot x\right)}\right) - \frac{-0.5}{t\_0}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (*
    (/ 1.0 (sqrt PI))
    (*
     (exp (* x x))
     (-
      (fma
       (pow (fabs x) -7.0)
       1.875
       (- (/ 1.0 (fabs x)) (/ -0.75 (* t_0 (* x x)))))
      (/ -0.5 t_0))))))

C

double code(double x) {
	double t_0 = (x * x) * x;
	return (1.0 / sqrt(((double) M_PI))) * (exp((x * x)) * (fma(pow(fabs(x), -7.0), 1.875, ((1.0 / fabs(x)) - (-0.75 / (t_0 * (x * x))))) - (-0.5 / t_0)));
}

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	return Float64(Float64(1.0 / sqrt(pi)) * Float64(exp(Float64(x * x)) * Float64(fma((abs(x) ^ -7.0), 1.875, Float64(Float64(1.0 / abs(x)) - Float64(-0.75 / Float64(t_0 * Float64(x * x))))) - Float64(-0.5 / t_0))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Power[N[Abs[x], $MachinePrecision], -7.0], $MachinePrecision] * 1.875 + N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] - N[(-0.75 / N[(t$95$0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-0.5 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

Alternative 5: 100.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, 1.875, \left(\frac{1}{\left|x\right|} - \frac{-0.75}{t\_0 \cdot \left(x \cdot x\right)}\right) - \frac{-0.5}{t\_0}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (*
    (/ (exp (* x x)) (sqrt PI))
    (fma
     (pow (fabs x) -7.0)
     1.875
     (- (- (/ 1.0 (fabs x)) (/ -0.75 (* t_0 (* x x)))) (/ -0.5 t_0))))))

C

double code(double x) {
	double t_0 = (x * x) * x;
	return (exp((x * x)) / sqrt(((double) M_PI))) * fma(pow(fabs(x), -7.0), 1.875, (((1.0 / fabs(x)) - (-0.75 / (t_0 * (x * x)))) - (-0.5 / t_0)));
}

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	return Float64(Float64(exp(Float64(x * x)) / sqrt(pi)) * fma((abs(x) ^ -7.0), 1.875, Float64(Float64(Float64(1.0 / abs(x)) - Float64(-0.75 / Float64(t_0 * Float64(x * x)))) - Float64(-0.5 / t_0))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Abs[x], $MachinePrecision], -7.0], $MachinePrecision] * 1.875 + N[(N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] - N[(-0.75 / N[(t$95$0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-0.5 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 100.0%

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

Alternative 6: 100.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return (1.0 / Math.sqrt(Math.PI)) * (Math.exp((x * x)) * ((1.0 / x) - ((-((0.75 / (x * x)) - -0.5) / ((x * x) * x)) - (Math.pow(Math.abs(x), -7.0) * 1.875))));
}

Python

def code(x):
	return (1.0 / math.sqrt(math.pi)) * (math.exp((x * x)) * ((1.0 / x) - ((-((0.75 / (x * x)) - -0.5) / ((x * x) * x)) - (math.pow(math.fabs(x), -7.0) * 1.875))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = (1.0 / sqrt(pi)) * (exp((x * x)) * ((1.0 / x) - ((-((0.75 / (x * x)) - -0.5) / ((x * x) * x)) - ((abs(x) ^ -7.0) * 1.875))));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. lower--.f64 N/A

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

6. Applied rewrites 100.0%

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

8. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{\pi}} \cdot \left(e^{x \cdot x} \cdot \left(\frac{1}{x} - \left(\frac{-\left(\frac{0.75}{x \cdot x} - -0.5\right)}{\left(x \cdot x\right) \cdot x} - {\left(\left|x\right|\right)}^{-7} \cdot 1.875\right)\right)\right)} \]
9. Add Preprocessing

Alternative 7: 100.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{e^{0 + x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{1}{x} - \left(\frac{-\left(\frac{0.75}{x \cdot x} - -0.5\right)}{\left(x \cdot x\right) \cdot x} - {\left(\left|x\right|\right)}^{-7} \cdot 1.875\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (exp (+ 0.0 (* x x))) (sqrt PI))
  (-
   (/ 1.0 x)
   (-
    (/ (- (- (/ 0.75 (* x x)) -0.5)) (* (* x x) x))
    (* (pow (fabs x) -7.0) 1.875)))))

C

double code(double x) {
	return (exp((0.0 + (x * x))) / sqrt(((double) M_PI))) * ((1.0 / x) - ((-((0.75 / (x * x)) - -0.5) / ((x * x) * x)) - (pow(fabs(x), -7.0) * 1.875)));
}

Java

public static double code(double x) {
	return (Math.exp((0.0 + (x * x))) / Math.sqrt(Math.PI)) * ((1.0 / x) - ((-((0.75 / (x * x)) - -0.5) / ((x * x) * x)) - (Math.pow(Math.abs(x), -7.0) * 1.875)));
}

Python

def code(x):
	return (math.exp((0.0 + (x * x))) / math.sqrt(math.pi)) * ((1.0 / x) - ((-((0.75 / (x * x)) - -0.5) / ((x * x) * x)) - (math.pow(math.fabs(x), -7.0) * 1.875)))

Julia

function code(x)
	return Float64(Float64(exp(Float64(0.0 + Float64(x * x))) / sqrt(pi)) * Float64(Float64(1.0 / x) - Float64(Float64(Float64(-Float64(Float64(0.75 / Float64(x * x)) - -0.5)) / Float64(Float64(x * x) * x)) - Float64((abs(x) ^ -7.0) * 1.875))))
end

MATLAB

function tmp = code(x)
	tmp = (exp((0.0 + (x * x))) / sqrt(pi)) * ((1.0 / x) - ((-((0.75 / (x * x)) - -0.5) / ((x * x) * x)) - ((abs(x) ^ -7.0) * 1.875)));
end

Wolfram

code[x_] := N[(N[(N[Exp[N[(0.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] - N[(N[((-N[(N[(0.75 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision]) / N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - N[(N[Power[N[Abs[x], $MachinePrecision], -7.0], $MachinePrecision] * 1.875), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. lower--.f64 N/A

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

6. Applied rewrites 100.0%

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

8. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{e^{0 + x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{1}{x} - \left(\frac{-\left(\frac{0.75}{x \cdot x} - -0.5\right)}{\left(x \cdot x\right) \cdot x} - {\left(\left|x\right|\right)}^{-7} \cdot 1.875\right)\right)} \]
9. Add Preprocessing

Alternative 8: 99.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := 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[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] - N[(-0.75 / N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \color{blue}{\frac{\frac{-3}{4}}{{x}^{5}}}\right) \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-prod-up N/A

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

   4. lower-*.f64 N/A

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

   5. pow3 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 99.6

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

6. Applied rewrites 99.6%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\left|x\right|} - \color{blue}{\frac{-0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)}}\right) \]
7. Add Preprocessing

Alternative 9: 99.6% accurate, 3.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (exp (* x x)) (sqrt PI))
  (fma (pow (fabs x) -7.0) 1.875 (/ 1.0 (fabs x)))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{\frac{\frac{3}{4}}{{x}^{5}}}}{\sqrt{\pi}} \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{\color{blue}{{x}^{5}}}}{\sqrt{\pi}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{{x}^{\left(3 + \color{blue}{2}\right)}}}{\sqrt{\pi}} \]

   3. pow-prod-up N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{{x}^{3} \cdot \color{blue}{{x}^{2}}}}{\sqrt{\pi}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{{x}^{3} \cdot \color{blue}{{x}^{2}}}}{\sqrt{\pi}} \]

   5. pow3 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot {\color{blue}{x}}^{2}}}{\sqrt{\pi}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot {\color{blue}{x}}^{2}}}{\sqrt{\pi}} \]

   7. lift-*.f64 N/A

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

   8. pow2 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)}}{\sqrt{\pi}} \]

   9. lift-*.f64 1.7

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

6. Applied rewrites 1.7%

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

7. Taylor expanded in x around inf

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

9. Applied rewrites 99.6%

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

Alternative 10: 99.6% accurate, 3.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/ (* (fma (pow (fabs x) -7.0) 1.875 (/ 1.0 x)) (exp (* x x))) (sqrt PI)))

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(N[Power[N[Abs[x], $MachinePrecision], -7.0], $MachinePrecision] * 1.875 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $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. Applied rewrites 99.6%

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

4. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{\frac{\frac{3}{4}}{{x}^{5}}}}{\sqrt{\pi}} \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{\color{blue}{{x}^{5}}}}{\sqrt{\pi}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{{x}^{\left(3 + \color{blue}{2}\right)}}}{\sqrt{\pi}} \]

   3. pow-prod-up N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{{x}^{3} \cdot \color{blue}{{x}^{2}}}}{\sqrt{\pi}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{{x}^{3} \cdot \color{blue}{{x}^{2}}}}{\sqrt{\pi}} \]

   5. pow3 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot {\color{blue}{x}}^{2}}}{\sqrt{\pi}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot {\color{blue}{x}}^{2}}}{\sqrt{\pi}} \]

   7. lift-*.f64 N/A

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

   8. pow2 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)}}{\sqrt{\pi}} \]

   9. lift-*.f64 1.7

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

6. Applied rewrites 1.7%

   \[\leadsto \frac{\color{blue}{\frac{0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)}}}{\sqrt{\pi}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)}}{\sqrt{\pi}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}}}{\sqrt{\pi}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\color{blue}{x} \cdot x\right)}}{\sqrt{\pi}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)}}{\sqrt{\pi}} \]

   5. pow3 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{{x}^{3} \cdot \left(\color{blue}{x} \cdot x\right)}}{\sqrt{\pi}} \]

   6. associate-*r* N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{\left({x}^{3} \cdot x\right) \cdot \color{blue}{x}}}{\sqrt{\pi}} \]

   7. pow-plus N/A

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

   8. metadata-eval N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{{x}^{4} \cdot x}}{\sqrt{\pi}} \]

   9. metadata-eval N/A

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

   10. pow-prod-up N/A

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

   11. pow2-fabs-rev N/A

       \[\leadsto \frac{\frac{\frac{3}{4}}{\left({\left(\left|x\right|\right)}^{2} \cdot {x}^{2}\right) \cdot x}}{\sqrt{\pi}} \]

   12. pow2-fabs-rev N/A

       \[\leadsto \frac{\frac{\frac{3}{4}}{\left({\left(\left|x\right|\right)}^{2} \cdot {\left(\left|x\right|\right)}^{2}\right) \cdot x}}{\sqrt{\pi}} \]

   13. pow-prod-up N/A

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

   14. metadata-eval N/A

       \[\leadsto \frac{\frac{\frac{3}{4}}{{\left(\left|x\right|\right)}^{4} \cdot x}}{\sqrt{\pi}} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{3}{4}}{{\left(\left|x\right|\right)}^{4} \cdot \color{blue}{x}}}{\sqrt{\pi}} \]

   16. metadata-eval N/A

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

   17. pow-prod-up N/A

       \[\leadsto \frac{\frac{\frac{3}{4}}{\left({\left(\left|x\right|\right)}^{2} \cdot {\left(\left|x\right|\right)}^{2}\right) \cdot x}}{\sqrt{\pi}} \]

   18. pow2-fabs-rev N/A

       \[\leadsto \frac{\frac{\frac{3}{4}}{\left({x}^{2} \cdot {\left(\left|x\right|\right)}^{2}\right) \cdot x}}{\sqrt{\pi}} \]

   19. pow2-fabs-rev N/A

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

   20. pow-prod-up N/A

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

   21. metadata-eval N/A

       \[\leadsto \frac{\frac{\frac{3}{4}}{{x}^{4} \cdot x}}{\sqrt{\pi}} \]

   22. metadata-eval N/A

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

   23. pow-plus N/A

       \[\leadsto \frac{\frac{\frac{3}{4}}{\left({x}^{3} \cdot x\right) \cdot x}}{\sqrt{\pi}} \]

   24. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{3}{4}}{\left({x}^{3} \cdot x\right) \cdot x}}{\sqrt{\pi}} \]

   25. pow3 N/A

       \[\leadsto \frac{\frac{\frac{3}{4}}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}}{\sqrt{\pi}} \]

   26. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{3}{4}}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}}{\sqrt{\pi}} \]

   27. lift-*.f64 1.7

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

8. Applied rewrites 1.7%

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

9. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{e^{{x}^{2}} \cdot \left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)}}{\sqrt{\pi}} \]
10. Step-by-step derivation

    1. metadata-eval N/A

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

    2. *-commutative N/A

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

    3. lower-*.f64 N/A

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

11. Applied rewrites 99.6%

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

Alternative 11: 32.0% accurate, 4.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (* (/ (exp (* x x)) (* (sqrt PI) (* (* x x) x))) (/ 0.75 (* x x))))

C

double code(double x) {
	return (exp((x * x)) / (sqrt(((double) M_PI)) * ((x * x) * x))) * (0.75 / (x * x));
}

Java

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

Python

def code(x):
	return (math.exp((x * x)) / (math.sqrt(math.pi) * ((x * x) * x))) * (0.75 / (x * x))

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * x)) / Float64(sqrt(pi) * Float64(Float64(x * x) * x))) * Float64(0.75 / Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = (exp((x * x)) / (sqrt(pi) * ((x * x) * x))) * (0.75 / (x * x));
end

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.75 / N[(x * x), $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. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 18.5%

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

   1. lift-/.f64 N/A

      \[\leadsto \frac{\frac{e^{0 + x \cdot x}}{\sqrt{\pi}}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)} \cdot \frac{3}{4} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\frac{e^{0 + x \cdot x}}{\sqrt{\pi}}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)} \cdot \frac{3}{4} \]

   3. associate-/l/ N/A

      \[\leadsto \frac{e^{0 + x \cdot x}}{\sqrt{\pi} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot \frac{3}{4} \]

   4. lift-exp.f64 N/A

      \[\leadsto \frac{e^{0 + x \cdot x}}{\sqrt{\pi} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot \frac{3}{4} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{e^{0 + x \cdot x}}{\sqrt{\pi} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot \frac{3}{4} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{e^{0 + x \cdot x}}{\sqrt{\pi} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot \frac{3}{4} \]

   7. pow2 N/A

      \[\leadsto \frac{e^{0 + {x}^{2}}}{\sqrt{\pi} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot \frac{3}{4} \]

   8. +-lft-identity N/A

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

   9. lower-/.f64 N/A

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

   10. pow2 N/A

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

   11. lift-exp.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

8. Applied rewrites 18.5%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot 0.75 \]
9. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot \frac{3}{\color{blue}{4}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot \color{blue}{\frac{3}{4}} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot \frac{\color{blue}{3}}{4} \]

   4. associate-*l/ N/A

      \[\leadsto \frac{e^{x \cdot x} \cdot \frac{3}{4}}{\color{blue}{\sqrt{\pi} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right)}} \]

   5. lift-exp.f64 N/A

      \[\leadsto \frac{e^{x \cdot x} \cdot \frac{3}{4}}{\sqrt{\color{blue}{\pi}} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. lower-*.f64 N/A

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

   12. associate-*r* N/A

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

   13. times-frac N/A

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

10. Applied rewrites 32.0%

    \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi} \cdot \left(\left(x \cdot x\right) \cdot x\right)} \cdot \color{blue}{\frac{0.75}{x \cdot x}} \]
11. Add Preprocessing

Alternative 12: 18.5% accurate, 4.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (* (/ (exp (* x x)) (* (sqrt PI) (* (* (* x x) x) (* x x)))) 0.75))

C

double code(double x) {
	return (exp((x * x)) / (sqrt(((double) M_PI)) * (((x * x) * x) * (x * x)))) * 0.75;
}

Java

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

Python

def code(x):
	return (math.exp((x * x)) / (math.sqrt(math.pi) * (((x * x) * x) * (x * x)))) * 0.75

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * x)) / Float64(sqrt(pi) * Float64(Float64(Float64(x * x) * x) * Float64(x * x)))) * 0.75)
end

MATLAB

function tmp = code(x)
	tmp = (exp((x * x)) / (sqrt(pi) * (((x * x) * x) * (x * x)))) * 0.75;
end

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.75), $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. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 18.5%

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

   1. lift-/.f64 N/A

      \[\leadsto \frac{\frac{e^{0 + x \cdot x}}{\sqrt{\pi}}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)} \cdot \frac{3}{4} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\frac{e^{0 + x \cdot x}}{\sqrt{\pi}}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)} \cdot \frac{3}{4} \]

   3. associate-/l/ N/A

      \[\leadsto \frac{e^{0 + x \cdot x}}{\sqrt{\pi} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot \frac{3}{4} \]

   4. lift-exp.f64 N/A

      \[\leadsto \frac{e^{0 + x \cdot x}}{\sqrt{\pi} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot \frac{3}{4} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{e^{0 + x \cdot x}}{\sqrt{\pi} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot \frac{3}{4} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{e^{0 + x \cdot x}}{\sqrt{\pi} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot \frac{3}{4} \]

   7. pow2 N/A

      \[\leadsto \frac{e^{0 + {x}^{2}}}{\sqrt{\pi} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot \frac{3}{4} \]

   8. +-lft-identity N/A

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

   9. lower-/.f64 N/A

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

   10. pow2 N/A

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

   11. lift-exp.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

8. Applied rewrites 18.5%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right)} \cdot 0.75 \]
9. Add Preprocessing

Alternative 13: 1.7% accurate, 7.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ 0.75 (* (* (* x x) x) (* x x))) (sqrt PI)))

C

double code(double x) {
	return (0.75 / (((x * x) * x) * (x * x))) / sqrt(((double) M_PI));
}

Java

public static double code(double x) {
	return (0.75 / (((x * x) * x) * (x * x))) / Math.sqrt(Math.PI);
}

Python

def code(x):
	return (0.75 / (((x * x) * x) * (x * x))) / math.sqrt(math.pi)

Julia

function code(x)
	return Float64(Float64(0.75 / Float64(Float64(Float64(x * x) * x) * Float64(x * x))) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	tmp = (0.75 / (((x * x) * x) * (x * x))) / sqrt(pi);
end

Wolfram

code[x_] := N[(N[(0.75 / N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $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. Applied rewrites 99.6%

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

4. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{\frac{\frac{3}{4}}{{x}^{5}}}}{\sqrt{\pi}} \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{\color{blue}{{x}^{5}}}}{\sqrt{\pi}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{{x}^{\left(3 + \color{blue}{2}\right)}}}{\sqrt{\pi}} \]

   3. pow-prod-up N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{{x}^{3} \cdot \color{blue}{{x}^{2}}}}{\sqrt{\pi}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{{x}^{3} \cdot \color{blue}{{x}^{2}}}}{\sqrt{\pi}} \]

   5. pow3 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot {\color{blue}{x}}^{2}}}{\sqrt{\pi}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot {\color{blue}{x}}^{2}}}{\sqrt{\pi}} \]

   7. lift-*.f64 N/A

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

   8. pow2 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)}}{\sqrt{\pi}} \]

   9. lift-*.f64 1.7

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

6. Applied rewrites 1.7%

   \[\leadsto \frac{\color{blue}{\frac{0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)}}}{\sqrt{\pi}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025132 
(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)))))))