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

Percentage Accurate: 100.0% → 100.0%

Time: 5.1s

Alternatives: 9

Speedup: 2.9×

• 99.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Python

def code(x):
	return ((1.0 / math.sqrt(math.pi)) * math.pow(math.exp(x), x)) * (((((((1.875 / (x * x)) - -0.75) / (x * x)) - -0.5) / (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(Float64(1.875 / Float64(x * x)) - -0.75) / Float64(x * x)) - -0.5) / Float64(x * x)) - -1.0) / abs(x)))
end

MATLAB

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

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

7. Applied rewrites 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. lower--.f64 N/A

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

   5. metadata-eval 100.0

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

9. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 3: 100.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. sqr-abs N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 100.0

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-exp.f64 N/A

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

   5. pow-exp N/A

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

   6. lift-*.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

7. Applied rewrites 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. lower--.f64 N/A

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

   5. metadata-eval 100.0

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

9. Applied rewrites 100.0%

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

Alternative 4: 99.9% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

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

7. Applied rewrites 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. lower--.f64 N/A

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

   5. metadata-eval 99.9

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

9. Applied rewrites 99.9%

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

Alternative 5: 99.5% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 99.5%

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

Alternative 6: 99.4% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 99.4%

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

Alternative 7: 99.4% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-fabs.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-PI.f64 99.4

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

6. Applied rewrites 99.4%

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

Alternative 8: 1.8% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{2.625}{{x}^{5}}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ 2.625 (pow x 5.0)) (sqrt PI)))

C

double code(double x) {
	return (2.625 / pow(x, 5.0)) / sqrt(((double) M_PI));
}

Java

public static double code(double x) {
	return (2.625 / Math.pow(x, 5.0)) / Math.sqrt(Math.PI);
}

Python

def code(x):
	return (2.625 / math.pow(x, 5.0)) / math.sqrt(math.pi)

Julia

function code(x)
	return Float64(Float64(2.625 / (x ^ 5.0)) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	tmp = (2.625 / (x ^ 5.0)) / sqrt(pi);
end

Wolfram

code[x_] := N[(N[(2.625 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{\frac{\frac{21}{8} \cdot \frac{{x}^{2}}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{\left|x\right|}}{{x}^{6}}}}{\sqrt{\pi}} \]
5. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{\frac{\frac{21}{8} \cdot \frac{{x}^{2}}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{\left|x\right|}}{{x}^{6}}}{\sqrt{\pi}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\frac{21}{8} \cdot \frac{{x}^{2}}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{\left|x\right|}}{\color{blue}{{x}^{6}}}}{\sqrt{\pi}} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{21}{8}, \frac{{x}^{2}}{\left|x\right|}, \frac{15}{8} \cdot \frac{1}{\left|x\right|}\right)}{{\color{blue}{x}}^{6}}}{\sqrt{\pi}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{21}{8}, \frac{{x}^{2}}{\left|x\right|}, \frac{15}{8} \cdot \frac{1}{\left|x\right|}\right)}{{x}^{6}}}{\sqrt{\pi}} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{21}{8}, \frac{{x}^{2}}{\left|x\right|}, \frac{15}{8} \cdot \frac{1}{\left|x\right|}\right)}{{x}^{6}}}{\sqrt{\pi}} \]

   6. lower-fabs.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{21}{8}, \frac{{x}^{2}}{\left|x\right|}, \frac{15}{8} \cdot \frac{1}{\left|x\right|}\right)}{{x}^{6}}}{\sqrt{\pi}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{21}{8}, \frac{{x}^{2}}{\left|x\right|}, \frac{15}{8} \cdot \frac{1}{\left|x\right|}\right)}{{x}^{6}}}{\sqrt{\pi}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{21}{8}, \frac{{x}^{2}}{\left|x\right|}, \frac{15}{8} \cdot \frac{1}{\left|x\right|}\right)}{{x}^{6}}}{\sqrt{\pi}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{21}{8}, \frac{{x}^{2}}{\left|x\right|}, \frac{15}{8} \cdot \frac{1}{\left|x\right|}\right)}{{x}^{6}}}{\sqrt{\pi}} \]

   10. lower-fabs.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{21}{8}, \frac{{x}^{2}}{\left|x\right|}, \frac{15}{8} \cdot \frac{1}{\left|x\right|}\right)}{{x}^{6}}}{\sqrt{\pi}} \]

   11. lower-pow.f64 1.0

       \[\leadsto \frac{\frac{\mathsf{fma}\left(2.625, \frac{{x}^{2}}{\left|x\right|}, 1.875 \cdot \frac{1}{\left|x\right|}\right)}{{x}^{\color{blue}{6}}}}{\sqrt{\pi}} \]

6. Applied rewrites 1.0%

   \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(2.625, \frac{{x}^{2}}{\left|x\right|}, 1.875 \cdot \frac{1}{\left|x\right|}\right)}{{x}^{6}}}}{\sqrt{\pi}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{21}{8}, \frac{{x}^{2}}{\left|x\right|}, \frac{15}{8} \cdot \frac{1}{\left|x\right|}\right)}{\color{blue}{{x}^{6}}}}{\sqrt{\pi}} \]

   2. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{\frac{21}{8} \cdot \frac{{x}^{2}}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{\left|x\right|}}{{\color{blue}{x}}^{6}}}{\sqrt{\pi}} \]

   3. div-add N/A

      \[\leadsto \frac{\frac{\frac{21}{8} \cdot \frac{{x}^{2}}{\left|x\right|}}{{x}^{6}} + \color{blue}{\frac{\frac{15}{8} \cdot \frac{1}{\left|x\right|}}{{x}^{6}}}}{\sqrt{\pi}} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{\frac{\frac{21}{8} \cdot \frac{{x}^{2}}{\left|x\right|}}{{x}^{6}} + \frac{\frac{15}{8} \cdot \frac{1}{\left|x\right|}}{{x}^{6}}}{\sqrt{\pi}} \]

   5. lift-pow.f64 N/A

      \[\leadsto \frac{\frac{\frac{21}{8} \cdot \frac{{x}^{2}}{\left|x\right|}}{{x}^{6}} + \frac{\frac{15}{8} \cdot \frac{1}{\left|x\right|}}{{x}^{6}}}{\sqrt{\pi}} \]

   6. pow2 N/A

      \[\leadsto \frac{\frac{\frac{21}{8} \cdot \frac{x \cdot x}{\left|x\right|}}{{x}^{6}} + \frac{\frac{15}{8} \cdot \frac{1}{\left|x\right|}}{{x}^{6}}}{\sqrt{\pi}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{21}{8} \cdot \frac{x \cdot x}{\left|x\right|}}{{x}^{6}} + \frac{\frac{15}{8} \cdot \frac{1}{\left|x\right|}}{{x}^{6}}}{\sqrt{\pi}} \]

   8. associate-*r/ N/A

      \[\leadsto \frac{\frac{\frac{\frac{21}{8} \cdot \left(x \cdot x\right)}{\left|x\right|}}{{x}^{6}} + \frac{\color{blue}{\frac{15}{8}} \cdot \frac{1}{\left|x\right|}}{{x}^{6}}}{\sqrt{\pi}} \]

   9. associate-/l/ N/A

      \[\leadsto \frac{\frac{\frac{21}{8} \cdot \left(x \cdot x\right)}{\left|x\right| \cdot {x}^{6}} + \frac{\color{blue}{\frac{15}{8} \cdot \frac{1}{\left|x\right|}}}{{x}^{6}}}{\sqrt{\pi}} \]

   10. *-commutative N/A

       \[\leadsto \frac{\frac{\frac{21}{8} \cdot \left(x \cdot x\right)}{{x}^{6} \cdot \left|x\right|} + \frac{\frac{15}{8} \cdot \color{blue}{\frac{1}{\left|x\right|}}}{{x}^{6}}}{\sqrt{\pi}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{21}{8} \cdot \left(x \cdot x\right)}{{x}^{6} \cdot \left|x\right|} + \frac{\frac{15}{8} \cdot \color{blue}{\frac{1}{\left|x\right|}}}{{x}^{6}}}{\sqrt{\pi}} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{21}{8} \cdot \left(x \cdot x\right)}{{x}^{6} \cdot \left|x\right|} + \frac{\frac{15}{8} \cdot \frac{1}{\left|x\right|}}{{\color{blue}{x}}^{6}}}{\sqrt{\pi}} \]

   13. lift-/.f64 N/A

       \[\leadsto \frac{\frac{\frac{21}{8} \cdot \left(x \cdot x\right)}{{x}^{6} \cdot \left|x\right|} + \frac{\frac{15}{8} \cdot \frac{1}{\left|x\right|}}{{x}^{6}}}{\sqrt{\pi}} \]

   14. mult-flip-rev N/A

       \[\leadsto \frac{\frac{\frac{21}{8} \cdot \left(x \cdot x\right)}{{x}^{6} \cdot \left|x\right|} + \frac{\frac{\frac{15}{8}}{\left|x\right|}}{{\color{blue}{x}}^{6}}}{\sqrt{\pi}} \]

   15. associate-/l/ N/A

       \[\leadsto \frac{\frac{\frac{21}{8} \cdot \left(x \cdot x\right)}{{x}^{6} \cdot \left|x\right|} + \frac{\frac{15}{8}}{\color{blue}{\left|x\right| \cdot {x}^{6}}}}{\sqrt{\pi}} \]

   16. *-commutative N/A

       \[\leadsto \frac{\frac{\frac{21}{8} \cdot \left(x \cdot x\right)}{{x}^{6} \cdot \left|x\right|} + \frac{\frac{15}{8}}{{x}^{6} \cdot \color{blue}{\left|x\right|}}}{\sqrt{\pi}} \]

   17. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{21}{8} \cdot \left(x \cdot x\right)}{{x}^{6} \cdot \left|x\right|} + \frac{\frac{15}{8}}{{x}^{6} \cdot \color{blue}{\left|x\right|}}}{\sqrt{\pi}} \]

8. Applied rewrites 1.0%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(2.625, x \cdot x, 1.875\right)}{\color{blue}{{\left(x \cdot x\right)}^{3.5}}}}{\sqrt{\pi}} \]

9. Taylor expanded in x around inf

   \[\leadsto \frac{\frac{\frac{21}{8}}{\color{blue}{{x}^{5}}}}{\sqrt{\pi}} \]
10. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \frac{\frac{\frac{21}{8}}{{x}^{\color{blue}{5}}}}{\sqrt{\pi}} \]

    2. lower-pow.f64 1.8

       \[\leadsto \frac{\frac{2.625}{{x}^{5}}}{\sqrt{\pi}} \]

11. Applied rewrites 1.8%

    \[\leadsto \frac{\frac{2.625}{\color{blue}{{x}^{5}}}}{\sqrt{\pi}} \]
12. Add Preprocessing

Alternative 9: 1.7% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1.875}{{x}^{7}}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ 1.875 (pow x 7.0)) (sqrt PI)))

C

double code(double x) {
	return (1.875 / pow(x, 7.0)) / sqrt(((double) M_PI));
}

Java

public static double code(double x) {
	return (1.875 / Math.pow(x, 7.0)) / Math.sqrt(Math.PI);
}

Python

def code(x):
	return (1.875 / math.pow(x, 7.0)) / math.sqrt(math.pi)

Julia

function code(x)
	return Float64(Float64(1.875 / (x ^ 7.0)) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	tmp = (1.875 / (x ^ 7.0)) / sqrt(pi);
end

Wolfram

code[x_] := N[(N[(1.875 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{\frac{\frac{21}{8} \cdot \frac{{x}^{2}}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{\left|x\right|}}{{x}^{6}}}}{\sqrt{\pi}} \]
5. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{\frac{\frac{21}{8} \cdot \frac{{x}^{2}}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{\left|x\right|}}{{x}^{6}}}{\sqrt{\pi}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\frac{21}{8} \cdot \frac{{x}^{2}}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{\left|x\right|}}{\color{blue}{{x}^{6}}}}{\sqrt{\pi}} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{21}{8}, \frac{{x}^{2}}{\left|x\right|}, \frac{15}{8} \cdot \frac{1}{\left|x\right|}\right)}{{\color{blue}{x}}^{6}}}{\sqrt{\pi}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{21}{8}, \frac{{x}^{2}}{\left|x\right|}, \frac{15}{8} \cdot \frac{1}{\left|x\right|}\right)}{{x}^{6}}}{\sqrt{\pi}} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{21}{8}, \frac{{x}^{2}}{\left|x\right|}, \frac{15}{8} \cdot \frac{1}{\left|x\right|}\right)}{{x}^{6}}}{\sqrt{\pi}} \]

   6. lower-fabs.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{21}{8}, \frac{{x}^{2}}{\left|x\right|}, \frac{15}{8} \cdot \frac{1}{\left|x\right|}\right)}{{x}^{6}}}{\sqrt{\pi}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{21}{8}, \frac{{x}^{2}}{\left|x\right|}, \frac{15}{8} \cdot \frac{1}{\left|x\right|}\right)}{{x}^{6}}}{\sqrt{\pi}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{21}{8}, \frac{{x}^{2}}{\left|x\right|}, \frac{15}{8} \cdot \frac{1}{\left|x\right|}\right)}{{x}^{6}}}{\sqrt{\pi}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{21}{8}, \frac{{x}^{2}}{\left|x\right|}, \frac{15}{8} \cdot \frac{1}{\left|x\right|}\right)}{{x}^{6}}}{\sqrt{\pi}} \]

   10. lower-fabs.f64 N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{21}{8}, \frac{{x}^{2}}{\left|x\right|}, \frac{15}{8} \cdot \frac{1}{\left|x\right|}\right)}{{x}^{6}}}{\sqrt{\pi}} \]

   11. lower-pow.f64 1.0

       \[\leadsto \frac{\frac{\mathsf{fma}\left(2.625, \frac{{x}^{2}}{\left|x\right|}, 1.875 \cdot \frac{1}{\left|x\right|}\right)}{{x}^{\color{blue}{6}}}}{\sqrt{\pi}} \]

6. Applied rewrites 1.0%

   \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(2.625, \frac{{x}^{2}}{\left|x\right|}, 1.875 \cdot \frac{1}{\left|x\right|}\right)}{{x}^{6}}}}{\sqrt{\pi}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{21}{8}, \frac{{x}^{2}}{\left|x\right|}, \frac{15}{8} \cdot \frac{1}{\left|x\right|}\right)}{\color{blue}{{x}^{6}}}}{\sqrt{\pi}} \]

   2. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{\frac{21}{8} \cdot \frac{{x}^{2}}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{\left|x\right|}}{{\color{blue}{x}}^{6}}}{\sqrt{\pi}} \]

   3. div-add N/A

      \[\leadsto \frac{\frac{\frac{21}{8} \cdot \frac{{x}^{2}}{\left|x\right|}}{{x}^{6}} + \color{blue}{\frac{\frac{15}{8} \cdot \frac{1}{\left|x\right|}}{{x}^{6}}}}{\sqrt{\pi}} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{\frac{\frac{21}{8} \cdot \frac{{x}^{2}}{\left|x\right|}}{{x}^{6}} + \frac{\frac{15}{8} \cdot \frac{1}{\left|x\right|}}{{x}^{6}}}{\sqrt{\pi}} \]

   5. lift-pow.f64 N/A

      \[\leadsto \frac{\frac{\frac{21}{8} \cdot \frac{{x}^{2}}{\left|x\right|}}{{x}^{6}} + \frac{\frac{15}{8} \cdot \frac{1}{\left|x\right|}}{{x}^{6}}}{\sqrt{\pi}} \]

   6. pow2 N/A

      \[\leadsto \frac{\frac{\frac{21}{8} \cdot \frac{x \cdot x}{\left|x\right|}}{{x}^{6}} + \frac{\frac{15}{8} \cdot \frac{1}{\left|x\right|}}{{x}^{6}}}{\sqrt{\pi}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\frac{21}{8} \cdot \frac{x \cdot x}{\left|x\right|}}{{x}^{6}} + \frac{\frac{15}{8} \cdot \frac{1}{\left|x\right|}}{{x}^{6}}}{\sqrt{\pi}} \]

   8. associate-*r/ N/A

      \[\leadsto \frac{\frac{\frac{\frac{21}{8} \cdot \left(x \cdot x\right)}{\left|x\right|}}{{x}^{6}} + \frac{\color{blue}{\frac{15}{8}} \cdot \frac{1}{\left|x\right|}}{{x}^{6}}}{\sqrt{\pi}} \]

   9. associate-/l/ N/A

      \[\leadsto \frac{\frac{\frac{21}{8} \cdot \left(x \cdot x\right)}{\left|x\right| \cdot {x}^{6}} + \frac{\color{blue}{\frac{15}{8} \cdot \frac{1}{\left|x\right|}}}{{x}^{6}}}{\sqrt{\pi}} \]

   10. *-commutative N/A

       \[\leadsto \frac{\frac{\frac{21}{8} \cdot \left(x \cdot x\right)}{{x}^{6} \cdot \left|x\right|} + \frac{\frac{15}{8} \cdot \color{blue}{\frac{1}{\left|x\right|}}}{{x}^{6}}}{\sqrt{\pi}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{21}{8} \cdot \left(x \cdot x\right)}{{x}^{6} \cdot \left|x\right|} + \frac{\frac{15}{8} \cdot \color{blue}{\frac{1}{\left|x\right|}}}{{x}^{6}}}{\sqrt{\pi}} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{21}{8} \cdot \left(x \cdot x\right)}{{x}^{6} \cdot \left|x\right|} + \frac{\frac{15}{8} \cdot \frac{1}{\left|x\right|}}{{\color{blue}{x}}^{6}}}{\sqrt{\pi}} \]

   13. lift-/.f64 N/A

       \[\leadsto \frac{\frac{\frac{21}{8} \cdot \left(x \cdot x\right)}{{x}^{6} \cdot \left|x\right|} + \frac{\frac{15}{8} \cdot \frac{1}{\left|x\right|}}{{x}^{6}}}{\sqrt{\pi}} \]

   14. mult-flip-rev N/A

       \[\leadsto \frac{\frac{\frac{21}{8} \cdot \left(x \cdot x\right)}{{x}^{6} \cdot \left|x\right|} + \frac{\frac{\frac{15}{8}}{\left|x\right|}}{{\color{blue}{x}}^{6}}}{\sqrt{\pi}} \]

   15. associate-/l/ N/A

       \[\leadsto \frac{\frac{\frac{21}{8} \cdot \left(x \cdot x\right)}{{x}^{6} \cdot \left|x\right|} + \frac{\frac{15}{8}}{\color{blue}{\left|x\right| \cdot {x}^{6}}}}{\sqrt{\pi}} \]

   16. *-commutative N/A

       \[\leadsto \frac{\frac{\frac{21}{8} \cdot \left(x \cdot x\right)}{{x}^{6} \cdot \left|x\right|} + \frac{\frac{15}{8}}{{x}^{6} \cdot \color{blue}{\left|x\right|}}}{\sqrt{\pi}} \]

   17. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{21}{8} \cdot \left(x \cdot x\right)}{{x}^{6} \cdot \left|x\right|} + \frac{\frac{15}{8}}{{x}^{6} \cdot \color{blue}{\left|x\right|}}}{\sqrt{\pi}} \]

8. Applied rewrites 1.0%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(2.625, x \cdot x, 1.875\right)}{\color{blue}{{\left(x \cdot x\right)}^{3.5}}}}{\sqrt{\pi}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{\frac{15}{8}}{\color{blue}{{x}^{7}}}}{\sqrt{\pi}} \]
10. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \frac{\frac{\frac{15}{8}}{{x}^{\color{blue}{7}}}}{\sqrt{\pi}} \]

    2. lower-pow.f64 1.7

       \[\leadsto \frac{\frac{1.875}{{x}^{7}}}{\sqrt{\pi}} \]

11. Applied rewrites 1.7%

    \[\leadsto \frac{\frac{1.875}{\color{blue}{{x}^{7}}}}{\sqrt{\pi}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(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)))))))