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

Percentage Accurate: 100.0% → 100.0%

Time: 6.4s

Alternatives: 11

Speedup: 2.4×

• 0.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

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

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

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

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, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \left(\frac{1}{\sqrt[3]{\sqrt{\left(\pi \cdot \pi\right) \cdot \pi}}} \cdot {\left(e^{x}\right)}^{x}\right) \cdot \left(\mathsf{fma}\left(\frac{\frac{1.875}{\left(x \cdot x\right) \cdot t\_0} + \frac{0.75}{t\_0}}{\left|x\right| \cdot \left(1 - \frac{-0.5}{x \cdot x}\right)}, \left|x\right|, 1\right) \cdot \left(\left(\frac{0.5}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Applied rewrites 100.0%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. add-cbrt-cube N/A

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

   4. sqrt-cbrt N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lower-*.f64 100.0%

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

7. Applied rewrites 100.0%

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

9. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Applied rewrites 100.0%

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

Alternative 3: 100.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = ((1.0 / sqrt(pi)) * (exp(x) ^ x)) * ((((((0.75 / (x * x)) - -0.5) + (1.875 / ((x * x) * (x * x)))) / (x * x)) / abs(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[(N[(N[(N[(N[(N[(0.75 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision] + N[(1.875 / N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 100.0%

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

Alternative 4: 100.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = ((1.0 / sqrt(pi)) * (exp(x) ^ x)) * ((((((0.75 / (x * x)) - -0.5) + (1.875 / ((x * x) * (x * x)))) / (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[(N[(N[(N[(N[(N[(0.75 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision] + N[(1.875 / N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 100.0%

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

Alternative 5: 100.0% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

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

5. Applied rewrites 100.0%

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

Alternative 6: 99.9% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

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

5. Applied rewrites 99.9%

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

7. Applied rewrites 99.9%

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

Alternative 7: 99.7% accurate, 4.5× speedup

Math

\[\frac{e^{x \cdot x} \cdot \frac{\frac{0.5}{x \cdot x} - -1}{\left|x\right|}}{\sqrt{\pi}} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / N[Abs[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 100.0%

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

4. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. lower-+.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-fabs.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-fabs.f64 99.7%

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

6. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.7%

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

   4. lift-+.f64 N/A

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

8. Applied rewrites 99.7%

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

Alternative 8: 99.6% accurate, 6.6× speedup

Math

\[\frac{\frac{e^{x \cdot x}}{\sqrt{\pi}}}{\left|x\right|} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-fabs.f64 N/A

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

   6. lower-sqrt.f64 N/A

      \[\leadsto \frac{e^{{x}^{2}}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

   7. lower-PI.f64 99.6%

      \[\leadsto \frac{e^{{x}^{2}}}{\left|x\right| \cdot \sqrt{\pi}} \]

6. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. *-lft-identity N/A

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

   9. associate-*l/ N/A

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

   10. lift-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-exp.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. pow-exp N/A

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

   15. lift-exp.f64 N/A

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

   16. lift-pow.f64 N/A

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

   17. lower-/.f64 99.6%

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

8. Applied rewrites 99.6%

   \[\leadsto \frac{\frac{e^{x \cdot x}}{\sqrt{\pi}}}{\color{blue}{\left|x\right|}} \]
9. Add Preprocessing

Alternative 9: 99.6% accurate, 6.8× speedup

Math

\[\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\pi}} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-fabs.f64 N/A

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

   6. lower-sqrt.f64 N/A

      \[\leadsto \frac{e^{{x}^{2}}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

   7. lower-PI.f64 99.6%

      \[\leadsto \frac{e^{{x}^{2}}}{\left|x\right| \cdot \sqrt{\pi}} \]

6. Applied rewrites 99.6%

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

   1. lift-pow.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 99.6%

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

8. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\pi}}} \]
9. Add Preprocessing

Alternative 10: 2.3% accurate, 14.6× speedup

Math

\[\frac{\frac{1}{\left|x\right|}}{\sqrt{\pi}} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 / N[Abs[x], $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 100.0%

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

4. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-fabs.f64 N/A

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

   6. lower-sqrt.f64 N/A

      \[\leadsto \frac{e^{{x}^{2}}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

   7. lower-PI.f64 99.6%

      \[\leadsto \frac{e^{{x}^{2}}}{\left|x\right| \cdot \sqrt{\pi}} \]

6. Applied rewrites 99.6%

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

7. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-fabs.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-PI.f64 2.3%

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

9. Applied rewrites 2.3%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-/r* N/A

       \[\leadsto \frac{\frac{1}{\left|x\right|}}{\sqrt{\pi}} \]

    4. lift-/.f64 N/A

       \[\leadsto \frac{\frac{1}{\left|x\right|}}{\sqrt{\pi}} \]

    5. lower-/.f64 2.3%

       \[\leadsto \frac{\frac{1}{\left|x\right|}}{\sqrt{\pi}} \]

11. Applied rewrites 2.3%

    \[\leadsto \frac{\frac{1}{\left|x\right|}}{\sqrt{\pi}} \]
12. Add Preprocessing

Alternative 11: 2.3% accurate, 15.7× speedup

Math

\[\frac{1}{\left|x\right| \cdot \sqrt{\pi}} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-fabs.f64 N/A

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

   6. lower-sqrt.f64 N/A

      \[\leadsto \frac{e^{{x}^{2}}}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

   7. lower-PI.f64 99.6%

      \[\leadsto \frac{e^{{x}^{2}}}{\left|x\right| \cdot \sqrt{\pi}} \]

6. Applied rewrites 99.6%

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

7. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-fabs.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-PI.f64 2.3%

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

9. Applied rewrites 2.3%

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

Reproduce

herbie shell --seed 2025196 
(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)))))))