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

Percentage Accurate: 100.0% → 100.0%

Time: 5.0s

Alternatives: 9

Speedup: 2.3×

• 0.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.9× speedup

Math

\[\left(0.5641895835477563 \cdot {\left(e^{-x}\right)}^{\left(-x\right)}\right) \cdot \frac{\left(\frac{0.5}{x \cdot x} - -1\right) - \left(\frac{-1.875}{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x} - \frac{0.75}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\right)}{\left|x\right|} \]

FPCore

(FPCore (x)
  :precision binary64
  (*
 (* 0.5641895835477563 (pow (exp (- x)) (- x)))
 (/
  (-
   (- (/ 0.5 (* x x)) -1.0)
   (-
    (/ -1.875 (* (* (* x x) (* (* x x) x)) x))
    (/ 0.75 (* (* x x) (* x x)))))
  (fabs x))))

C

double code(double x) {
	return (0.5641895835477563 * pow(exp(-x), -x)) * ((((0.5 / (x * x)) - -1.0) - ((-1.875 / (((x * x) * ((x * x) * x)) * x)) - (0.75 / ((x * x) * (x * x))))) / fabs(x));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = (0.5641895835477563d0 * (exp(-x) ** -x)) * ((((0.5d0 / (x * x)) - (-1.0d0)) - (((-1.875d0) / (((x * x) * ((x * x) * x)) * x)) - (0.75d0 / ((x * x) * (x * x))))) / abs(x))
end function

Java

public static double code(double x) {
	return (0.5641895835477563 * Math.pow(Math.exp(-x), -x)) * ((((0.5 / (x * x)) - -1.0) - ((-1.875 / (((x * x) * ((x * x) * x)) * x)) - (0.75 / ((x * x) * (x * x))))) / Math.abs(x));
}

Python

def code(x):
	return (0.5641895835477563 * math.pow(math.exp(-x), -x)) * ((((0.5 / (x * x)) - -1.0) - ((-1.875 / (((x * x) * ((x * x) * x)) * x)) - (0.75 / ((x * x) * (x * x))))) / math.fabs(x))

Julia

function code(x)
	return Float64(Float64(0.5641895835477563 * (exp(Float64(-x)) ^ Float64(-x))) * Float64(Float64(Float64(Float64(0.5 / Float64(x * x)) - -1.0) - Float64(Float64(-1.875 / Float64(Float64(Float64(x * x) * Float64(Float64(x * x) * x)) * x)) - Float64(0.75 / Float64(Float64(x * x) * Float64(x * x))))) / abs(x)))
end

MATLAB

function tmp = code(x)
	tmp = (0.5641895835477563 * (exp(-x) ^ -x)) * ((((0.5 / (x * x)) - -1.0) - ((-1.875 / (((x * x) * ((x * x) * x)) * x)) - (0.75 / ((x * x) * (x * x))))) / abs(x));
end

Wolfram

code[x_] := N[(N[(0.5641895835477563 * N[Power[N[Exp[(-x)], $MachinePrecision], (-x)], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] - N[(N[(-1.875 / N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - N[(0.75 / N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. sqr-neg-rev N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-neg.f64 100.0%

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

6. Evaluated real constant 100.0%

   \[\leadsto \left(\color{blue}{0.5641895835477563} \cdot {\left(e^{-x}\right)}^{\left(-x\right)}\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \left(\left(\frac{0.5}{x \cdot x} - -1\right) - \left(\frac{-1.875}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} - \frac{0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}\right)\right)\right) \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\frac{5081767996463981}{9007199254740992} \cdot {\left(e^{-x}\right)}^{\left(-x\right)}\right) \cdot \color{blue}{\left(\frac{1}{\left|x\right|} \cdot \left(\left(\frac{\frac{1}{2}}{x \cdot x} - -1\right) - \left(\frac{\frac{-15}{8}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} - \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}\right)\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\frac{5081767996463981}{9007199254740992} \cdot {\left(e^{-x}\right)}^{\left(-x\right)}\right) \cdot \color{blue}{\left(\left(\left(\frac{\frac{1}{2}}{x \cdot x} - -1\right) - \left(\frac{\frac{-15}{8}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} - \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}\right)\right) \cdot \frac{1}{\left|x\right|}\right)} \]

   3. lift-/.f64 N/A

      \[\leadsto \left(\frac{5081767996463981}{9007199254740992} \cdot {\left(e^{-x}\right)}^{\left(-x\right)}\right) \cdot \left(\left(\left(\frac{\frac{1}{2}}{x \cdot x} - -1\right) - \left(\frac{\frac{-15}{8}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} - \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}\right)\right) \cdot \color{blue}{\frac{1}{\left|x\right|}}\right) \]

   4. mult-flip-rev N/A

      \[\leadsto \left(\frac{5081767996463981}{9007199254740992} \cdot {\left(e^{-x}\right)}^{\left(-x\right)}\right) \cdot \color{blue}{\frac{\left(\frac{\frac{1}{2}}{x \cdot x} - -1\right) - \left(\frac{\frac{-15}{8}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} - \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}\right)}{\left|x\right|}} \]

   5. lower-/.f64 100.0%

      \[\leadsto \left(0.5641895835477563 \cdot {\left(e^{-x}\right)}^{\left(-x\right)}\right) \cdot \color{blue}{\frac{\left(\frac{0.5}{x \cdot x} - -1\right) - \left(\frac{-1.875}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} - \frac{0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}\right)}{\left|x\right|}} \]

8. Applied rewrites 100.0%

   \[\leadsto \left(0.5641895835477563 \cdot {\left(e^{-x}\right)}^{\left(-x\right)}\right) \cdot \color{blue}{\frac{\left(\frac{0.5}{x \cdot x} - -1\right) - \left(\frac{-1.875}{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x} - \frac{0.75}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\right)}{\left|x\right|}} \]
9. Add Preprocessing

Alternative 2: 100.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	double t_0 = (x * x) * x;
	return ((Math.exp((x * x)) / Math.sqrt(Math.PI)) * ((((0.75 / t_0) + (0.5 / x)) / x) + ((1.875 / (((t_0 * x) * x) * x)) - -1.0))) / Math.abs(x);
}

Python

def code(x):
	t_0 = (x * x) * x
	return ((math.exp((x * x)) / math.sqrt(math.pi)) * ((((0.75 / t_0) + (0.5 / x)) / x) + ((1.875 / (((t_0 * x) * x) * x)) - -1.0))) / math.fabs(x)

Julia

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

MATLAB

function tmp = code(x)
	t_0 = (x * x) * x;
	tmp = ((exp((x * x)) / sqrt(pi)) * ((((0.75 / t_0) + (0.5 / x)) / x) + ((1.875 / (((t_0 * x) * x) * x)) - -1.0))) / abs(x);
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. add-flip N/A

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

   3. metadata-eval N/A

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

   4. lower--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-/l/ N/A

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

   8. associate-*l/ N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. unswap-sqr N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 100.0%

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

5. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. div-add-rev N/A

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

7. Applied rewrites 100.0%

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

Alternative 3: 100.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	double t_0 = (x * x) * x;
	return (((((0.75 / t_0) + (0.5 / x)) / x) + ((1.875 / (((t_0 * x) * x) * x)) - -1.0)) / Math.abs(x)) * (Math.exp((x * x)) / Math.sqrt(Math.PI));
}

Python

def code(x):
	t_0 = (x * x) * x
	return (((((0.75 / t_0) + (0.5 / x)) / x) + ((1.875 / (((t_0 * x) * x) * x)) - -1.0)) / math.fabs(x)) * (math.exp((x * x)) / math.sqrt(math.pi))

Julia

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

MATLAB

function tmp = code(x)
	t_0 = (x * x) * x;
	tmp = (((((0.75 / t_0) + (0.5 / x)) / x) + ((1.875 / (((t_0 * x) * x) * x)) - -1.0)) / abs(x)) * (exp((x * x)) / sqrt(pi));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. add-flip N/A

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

   3. metadata-eval N/A

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

   4. lower--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-/l/ N/A

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

   8. associate-*l/ N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. unswap-sqr N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 100.0%

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

5. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 100.0%

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

7. Applied rewrites 100.0%

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

Alternative 4: 100.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	double t_0 = (x * x) * x;
	return ((((((0.75 / t_0) + (0.5 / x)) / x) + ((1.875 / (((t_0 * x) * x) * x)) - -1.0)) * Math.exp((x * x))) / Math.abs(x)) / Math.sqrt(Math.PI);
}

Python

def code(x):
	t_0 = (x * x) * x
	return ((((((0.75 / t_0) + (0.5 / x)) / x) + ((1.875 / (((t_0 * x) * x) * x)) - -1.0)) * math.exp((x * x))) / math.fabs(x)) / math.sqrt(math.pi)

Julia

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

MATLAB

function tmp = code(x)
	t_0 = (x * x) * x;
	tmp = ((((((0.75 / t_0) + (0.5 / x)) / x) + ((1.875 / (((t_0 * x) * x) * x)) - -1.0)) * exp((x * x))) / abs(x)) / sqrt(pi);
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. add-flip N/A

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

   3. metadata-eval N/A

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

   4. lower--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-/l/ N/A

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

   8. associate-*l/ N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. unswap-sqr N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 100.0%

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 100.0%

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

Alternative 5: 99.9% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	double t_0 = (x * x) * x;
	return (((((0.75 / t_0) + (0.5 / x)) / x) + ((1.875 / (((t_0 * x) * x) * x)) - -1.0)) * Math.exp((x * x))) / (Math.abs(x) * Math.sqrt(Math.PI));
}

Python

def code(x):
	t_0 = (x * x) * x
	return (((((0.75 / t_0) + (0.5 / x)) / x) + ((1.875 / (((t_0 * x) * x) * x)) - -1.0)) * math.exp((x * x))) / (math.fabs(x) * math.sqrt(math.pi))

Julia

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

MATLAB

function tmp = code(x)
	t_0 = (x * x) * x;
	tmp = (((((0.75 / t_0) + (0.5 / x)) / x) + ((1.875 / (((t_0 * x) * x) * x)) - -1.0)) * exp((x * x))) / (abs(x) * sqrt(pi));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

   1. lift-fma.f64 N/A

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

   2. add-flip N/A

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

   3. metadata-eval N/A

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

   4. lower--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-/l/ N/A

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

   8. associate-*l/ N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. unswap-sqr N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 100.0%

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 99.9%

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

Alternative 6: 99.7% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around inf

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\color{blue}{\left(1 + \frac{3}{4} \cdot \frac{1}{{x}^{4}}\right)} \cdot \frac{\frac{0.5}{x \cdot x} - -1}{\left|x\right|}\right) \]
5. Step-by-step derivation

   1. metadata-eval N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 N/A

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

   6. lower-pow.f64 99.7%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(1 + 0.75 \cdot \frac{1}{{x}^{\color{blue}{4}}}\right) \cdot \frac{\frac{0.5}{x \cdot x} - -1}{\left|x\right|}\right) \]

6. Applied rewrites 99.7%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\color{blue}{\left(1 + 0.75 \cdot \frac{1}{{x}^{4}}\right)} \cdot \frac{\frac{0.5}{x \cdot x} - -1}{\left|x\right|}\right) \]

8. Applied rewrites 99.6%

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

Alternative 7: 99.6% accurate, 4.4× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (/ (exp (pow x 2.0)) (* (fabs x) (sqrt PI))))

C

double code(double x) {
	return exp(pow(x, 2.0)) / (fabs(x) * sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return Math.exp(Math.pow(x, 2.0)) / (Math.abs(x) * Math.sqrt(Math.PI));
}

Python

def code(x):
	return math.exp(math.pow(x, 2.0)) / (math.fabs(x) * math.sqrt(math.pi))

Julia

function code(x)
	return Float64(exp((x ^ 2.0)) / Float64(abs(x) * sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = exp((x ^ 2.0)) / (abs(x) * sqrt(pi));
end

Wolfram

code[x_] := N[(N[Exp[N[Power[x, 2.0], $MachinePrecision]], $MachinePrecision] / N[(N[Abs[x], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-fabs.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 99.6%

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

6. Applied rewrites 99.6%

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

Alternative 8: 99.6% accurate, 4.6× speedup

Math

\[0.5641895835477563 \cdot \frac{e^{{x}^{2}}}{\left|x\right|} \]

FPCore

(FPCore (x)
  :precision binary64
  (* 0.5641895835477563 (/ (exp (pow x 2.0)) (fabs x))))

C

double code(double x) {
	return 0.5641895835477563 * (exp(pow(x, 2.0)) / fabs(x));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = 0.5641895835477563d0 * (exp((x ** 2.0d0)) / abs(x))
end function

Java

public static double code(double x) {
	return 0.5641895835477563 * (Math.exp(Math.pow(x, 2.0)) / Math.abs(x));
}

Python

def code(x):
	return 0.5641895835477563 * (math.exp(math.pow(x, 2.0)) / math.fabs(x))

Julia

function code(x)
	return Float64(0.5641895835477563 * Float64(exp((x ^ 2.0)) / abs(x)))
end

MATLAB

function tmp = code(x)
	tmp = 0.5641895835477563 * (exp((x ^ 2.0)) / abs(x));
end

Wolfram

code[x_] := N[(0.5641895835477563 * N[(N[Exp[N[Power[x, 2.0], $MachinePrecision]], $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. sqr-neg-rev N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-neg.f64 100.0%

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

6. Evaluated real constant 100.0%

   \[\leadsto \left(\color{blue}{0.5641895835477563} \cdot {\left(e^{-x}\right)}^{\left(-x\right)}\right) \cdot \left(\frac{1}{\left|x\right|} \cdot \left(\left(\frac{0.5}{x \cdot x} - -1\right) - \left(\frac{-1.875}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} - \frac{0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}\right)\right)\right) \]

7. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{5081767996463981}{9007199254740992} \cdot \frac{e^{{x}^{2}}}{\left|x\right|}} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{5081767996463981}{9007199254740992} \cdot \color{blue}{\frac{e^{{x}^{2}}}{\left|x\right|}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{5081767996463981}{9007199254740992} \cdot \frac{e^{{x}^{2}}}{\color{blue}{\left|x\right|}} \]

   3. lower-exp.f64 N/A

      \[\leadsto \frac{5081767996463981}{9007199254740992} \cdot \frac{e^{{x}^{2}}}{\left|\color{blue}{x}\right|} \]

   4. lower-pow.f64 N/A

      \[\leadsto \frac{5081767996463981}{9007199254740992} \cdot \frac{e^{{x}^{2}}}{\left|x\right|} \]

   5. lower-fabs.f64 99.7%

      \[\leadsto 0.5641895835477563 \cdot \frac{e^{{x}^{2}}}{\left|x\right|} \]

9. Applied rewrites 99.7%

   \[\leadsto \color{blue}{0.5641895835477563 \cdot \frac{e^{{x}^{2}}}{\left|x\right|}} \]
10. Add Preprocessing

Alternative 9: 1.7% accurate, 7.2× speedup

Math

\[\frac{\frac{2.625}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}}{\sqrt{\pi}} \]

FPCore

(FPCore (x)
  :precision binary64
  (/ (/ 2.625 (* (* (* x x) (* x x)) (fabs x))) (sqrt PI)))

C

double code(double x) {
	return (2.625 / (((x * x) * (x * x)) * fabs(x))) / sqrt(((double) M_PI));
}

Java

public static double code(double x) {
	return (2.625 / (((x * x) * (x * x)) * Math.abs(x))) / Math.sqrt(Math.PI);
}

Python

def code(x):
	return (2.625 / (((x * x) * (x * x)) * math.fabs(x))) / math.sqrt(math.pi)

Julia

function code(x)
	return Float64(Float64(2.625 / Float64(Float64(Float64(x * x) * Float64(x * x)) * abs(x))) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	tmp = (2.625 / (((x * x) * (x * x)) * abs(x))) / sqrt(pi);
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. lower-/.f64 N/A

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

6. Applied rewrites 1.0%

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

7. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \frac{\frac{21}{8}}{\color{blue}{{x}^{4} \cdot \left|x\right|}} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \frac{\frac{21}{8}}{{x}^{4} \cdot \color{blue}{\left|x\right|}} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \frac{\frac{21}{8}}{{x}^{4} \cdot \left|x\right|} \]

   3. lower-pow.f64 N/A

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \frac{\frac{21}{8}}{{x}^{4} \cdot \left|x\right|} \]

   4. lower-fabs.f64 1.7%

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \frac{2.625}{{x}^{4} \cdot \left|x\right|} \]

9. Applied rewrites 1.7%

   \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \frac{2.625}{\color{blue}{{x}^{4} \cdot \left|x\right|}} \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \color{blue}{\frac{1}{\sqrt{\pi}} \cdot \frac{\frac{21}{8}}{{x}^{4} \cdot \left|x\right|}} \]

    2. *-commutative N/A

       \[\leadsto \color{blue}{\frac{\frac{21}{8}}{{x}^{4} \cdot \left|x\right|} \cdot \frac{1}{\sqrt{\pi}}} \]

    3. lift-/.f64 N/A

       \[\leadsto \frac{\frac{21}{8}}{{x}^{4} \cdot \left|x\right|} \cdot \color{blue}{\frac{1}{\sqrt{\pi}}} \]

11. Applied rewrites 1.7%

    \[\leadsto \color{blue}{\frac{\frac{2.625}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left|x\right|}}{\sqrt{\pi}}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025210 
(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)))))))