Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.2% → 71.9%

Time: 10.3s

Alternatives: 20

Speedup: 0.6×

Specification

Math

\[\begin{array}{l} \\ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (*
  (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
  (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))

C

double code(double d, double h, double l, double M, double D) {
	return (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

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(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

Python

def code(d, h, l, M, D):
	return (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))

Julia

function code(d, h, l, M, D)
	return Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 66.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (*
  (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
  (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))

C

double code(double d, double h, double l, double M, double D) {
	return (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

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(d, h, l, m, d_1)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

Python

def code(d, h, l, M, D):
	return (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))

Julia

function code(d, h, l, M, D)
	return Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 71.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{D\_m \cdot M\_m}{d + d}\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+262}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot t\_0\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(D\_m \cdot M\_m\right) \cdot h}{\left(-2 \cdot d\right) \cdot \ell} \cdot \left(0.25 \cdot \frac{D\_m}{d}\right), M\_m, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ (* D_m M_m) (+ d d))))
   (if (<=
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l))))
        5e+262)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (* (* (* (/ h l) 0.5) t_0) t_0)))
     (*
      (fma
       (* (/ (* (* D_m M_m) h) (* (* -2.0 d) l)) (* 0.25 (/ D_m d)))
       M_m
       1.0)
      (sqrt (* (/ d (* l h)) d))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (D_m * M_m) / (d + d);
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= 5e+262) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((h / l) * 0.5) * t_0) * t_0));
	} else {
		tmp = fma(((((D_m * M_m) * h) / ((-2.0 * d) * l)) * (0.25 * (D_m / d))), M_m, 1.0) * sqrt(((d / (l * h)) * d));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(D_m * M_m) / Float64(d + d))
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 5e+262)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(Float64(h / l) * 0.5) * t_0) * t_0)));
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(D_m * M_m) * h) / Float64(Float64(-2.0 * d) * l)) * Float64(0.25 * Float64(D_m / d))), M_m, 1.0) * sqrt(Float64(Float64(d / Float64(l * h)) * d)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d + d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+262], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(h / l), $MachinePrecision] * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(N[(-2.0 * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(0.25 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * M$95$m + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / N[(l * h), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.00000000000000008e262

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

   3. Applied rewrites 67.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      3. lower-*.f64 67.8

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      5. metadata-eval 67.8

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      8. lift-sqrt.f64 67.8

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      10. metadata-eval 67.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      13. lower-sqrt.f64 67.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   5. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M}{d + d} \cdot D\right)}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{\frac{M}{d + d}} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{M \cdot D}{d + d}}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{\color{blue}{D \cdot M}}{d + d}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{\color{blue}{D \cdot M}}{d + d}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      6. lower-/.f64 67.4

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \color{blue}{\frac{D \cdot M}{d + d}}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   7. Applied rewrites 67.4%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \color{blue}{\frac{D \cdot M}{d + d}}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{D \cdot M}{d + d}\right) \cdot \color{blue}{\left(\frac{M}{d + d} \cdot D\right)}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{D \cdot M}{d + d}\right) \cdot \left(\color{blue}{\frac{M}{d + d}} \cdot D\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{D \cdot M}{d + d}\right) \cdot \color{blue}{\frac{M \cdot D}{d + d}}\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{D \cdot M}{d + d}\right) \cdot \frac{\color{blue}{D \cdot M}}{d + d}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{D \cdot M}{d + d}\right) \cdot \frac{\color{blue}{D \cdot M}}{d + d}\right) \]

      6. lower-/.f64 67.9

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \frac{D \cdot M}{d + d}\right) \cdot \color{blue}{\frac{D \cdot M}{d + d}}\right) \]

   9. Applied rewrites 67.9%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \frac{D \cdot M}{d + d}\right) \cdot \color{blue}{\frac{D \cdot M}{d + d}}\right) \]

   if 5.00000000000000008e262 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

   3. Applied rewrites 67.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)}\right) \]

   5. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot h}{\left(-2 \cdot d\right) \cdot \ell} \cdot \left(0.25 \cdot \frac{D}{d}\right), M, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 71.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{M\_m}{d + d} \cdot D\_m\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+262}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot t\_0\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(D\_m \cdot M\_m\right) \cdot h}{\left(-2 \cdot d\right) \cdot \ell} \cdot \left(0.25 \cdot \frac{D\_m}{d}\right), M\_m, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (/ M_m (+ d d)) D_m)))
   (if (<=
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l))))
        5e+262)
     (*
      (* (sqrt (/ d l)) (sqrt (/ d h)))
      (- 1.0 (* (* (* (/ h l) 0.5) t_0) t_0)))
     (*
      (fma
       (* (/ (* (* D_m M_m) h) (* (* -2.0 d) l)) (* 0.25 (/ D_m d)))
       M_m
       1.0)
      (sqrt (* (/ d (* l h)) d))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (M_m / (d + d)) * D_m;
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= 5e+262) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - ((((h / l) * 0.5) * t_0) * t_0));
	} else {
		tmp = fma(((((D_m * M_m) * h) / ((-2.0 * d) * l)) * (0.25 * (D_m / d))), M_m, 1.0) * sqrt(((d / (l * h)) * d));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(M_m / Float64(d + d)) * D_m)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 5e+262)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(Float64(h / l) * 0.5) * t_0) * t_0)));
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(D_m * M_m) * h) / Float64(Float64(-2.0 * d) * l)) * Float64(0.25 * Float64(D_m / d))), M_m, 1.0) * sqrt(Float64(Float64(d / Float64(l * h)) * d)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(M$95$m / N[(d + d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+262], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(h / l), $MachinePrecision] * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(N[(-2.0 * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(0.25 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * M$95$m + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / N[(l * h), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.00000000000000008e262

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

   3. Applied rewrites 67.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      3. lower-*.f64 67.8

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      5. metadata-eval 67.8

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      8. lift-sqrt.f64 67.8

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      10. metadata-eval 67.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      13. lower-sqrt.f64 67.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   5. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   if 5.00000000000000008e262 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

   3. Applied rewrites 67.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)}\right) \]

   5. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot h}{\left(-2 \cdot d\right) \cdot \ell} \cdot \left(0.25 \cdot \frac{D}{d}\right), M, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 71.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := D\_m \cdot \frac{M\_m}{d + d}\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+262}:\\ \;\;\;\;\left(\mathsf{fma}\left(t\_0 \cdot \left(t\_0 \cdot \frac{h}{\ell}\right), -0.5, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(D\_m \cdot M\_m\right) \cdot h}{\left(-2 \cdot d\right) \cdot \ell} \cdot \left(0.25 \cdot \frac{D\_m}{d}\right), M\_m, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* D_m (/ M_m (+ d d)))))
   (if (<=
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l))))
        5e+262)
     (*
      (* (fma (* t_0 (* t_0 (/ h l))) -0.5 1.0) (sqrt (/ d l)))
      (sqrt (/ d h)))
     (*
      (fma
       (* (/ (* (* D_m M_m) h) (* (* -2.0 d) l)) (* 0.25 (/ D_m d)))
       M_m
       1.0)
      (sqrt (* (/ d (* l h)) d))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = D_m * (M_m / (d + d));
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= 5e+262) {
		tmp = (fma((t_0 * (t_0 * (h / l))), -0.5, 1.0) * sqrt((d / l))) * sqrt((d / h));
	} else {
		tmp = fma(((((D_m * M_m) * h) / ((-2.0 * d) * l)) * (0.25 * (D_m / d))), M_m, 1.0) * sqrt(((d / (l * h)) * d));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(D_m * Float64(M_m / Float64(d + d)))
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 5e+262)
		tmp = Float64(Float64(fma(Float64(t_0 * Float64(t_0 * Float64(h / l))), -0.5, 1.0) * sqrt(Float64(d / l))) * sqrt(Float64(d / h)));
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(D_m * M_m) * h) / Float64(Float64(-2.0 * d) * l)) * Float64(0.25 * Float64(D_m / d))), M_m, 1.0) * sqrt(Float64(Float64(d / Float64(l * h)) * d)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(D$95$m * N[(M$95$m / N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+262], N[(N[(N[(N[(t$95$0 * N[(t$95$0 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(N[(-2.0 * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(0.25 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * M$95$m + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / N[(l * h), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.00000000000000008e262

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 43.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)}, -0.5, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)}}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\color{blue}{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      4. times-frac N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{h}{\ell} \cdot \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{d \cdot d}}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{h}{\ell}} \cdot \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{d \cdot d}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      6. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{d \cdot d} \cdot \frac{h}{\ell}}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

   5. Applied rewrites 67.8%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(D \cdot \frac{M}{d + d}\right) \cdot \left(\left(D \cdot \frac{M}{d + d}\right) \cdot \frac{h}{\ell}\right)}, -0.5, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

   if 5.00000000000000008e262 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

   3. Applied rewrites 67.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)}\right) \]

   5. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot h}{\left(-2 \cdot d\right) \cdot \ell} \cdot \left(0.25 \cdot \frac{D}{d}\right), M, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 71.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 5 \cdot 10^{+262}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \frac{h \cdot \frac{0.25 \cdot \left(D\_m \cdot M\_m\right)}{d}}{\ell} \cdot \left(\frac{M\_m}{d + d} \cdot D\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(D\_m \cdot M\_m\right) \cdot h}{\left(-2 \cdot d\right) \cdot \ell} \cdot \left(0.25 \cdot \frac{D\_m}{d}\right), M\_m, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
       (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l))))
      5e+262)
   (*
    (* (sqrt (/ d l)) (sqrt (/ d h)))
    (- 1.0 (* (/ (* h (/ (* 0.25 (* D_m M_m)) d)) l) (* (/ M_m (+ d d)) D_m))))
   (*
    (fma (* (/ (* (* D_m M_m) h) (* (* -2.0 d) l)) (* 0.25 (/ D_m d))) M_m 1.0)
    (sqrt (* (/ d (* l h)) d)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= 5e+262) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (((h * ((0.25 * (D_m * M_m)) / d)) / l) * ((M_m / (d + d)) * D_m)));
	} else {
		tmp = fma(((((D_m * M_m) * h) / ((-2.0 * d) * l)) * (0.25 * (D_m / d))), M_m, 1.0) * sqrt(((d / (l * h)) * d));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= 5e+262)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(Float64(Float64(h * Float64(Float64(0.25 * Float64(D_m * M_m)) / d)) / l) * Float64(Float64(M_m / Float64(d + d)) * D_m))));
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(D_m * M_m) * h) / Float64(Float64(-2.0 * d) * l)) * Float64(0.25 * Float64(D_m / d))), M_m, 1.0) * sqrt(Float64(Float64(d / Float64(l * h)) * d)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+262], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(h * N[(N[(0.25 * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(N[(M$95$m / N[(d + d), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(N[(-2.0 * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(0.25 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * M$95$m + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / N[(l * h), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.00000000000000008e262

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

   3. Applied rewrites 67.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      3. lower-*.f64 67.8

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      5. metadata-eval 67.8

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      8. lift-sqrt.f64 67.8

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      10. metadata-eval 67.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      13. lower-sqrt.f64 67.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   5. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right)} \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right)} \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot \left(\frac{M}{d + d} \cdot D\right)\right)\right)} \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \left(\color{blue}{\frac{h}{\ell}} \cdot \left(\frac{1}{2} \cdot \left(\frac{M}{d + d} \cdot D\right)\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\frac{h \cdot \left(\frac{1}{2} \cdot \left(\frac{M}{d + d} \cdot D\right)\right)}{\ell}} \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\frac{h \cdot \left(\frac{1}{2} \cdot \left(\frac{M}{d + d} \cdot D\right)\right)}{\ell}} \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   7. Applied rewrites 70.2%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - \color{blue}{\frac{h \cdot \frac{0.25 \cdot \left(D \cdot M\right)}{d}}{\ell}} \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   if 5.00000000000000008e262 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

   3. Applied rewrites 67.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)}\right) \]

   5. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot h}{\left(-2 \cdot d\right) \cdot \ell} \cdot \left(0.25 \cdot \frac{D}{d}\right), M, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 68.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{\left(D\_m \cdot M\_m\right) \cdot h}{\left(-2 \cdot d\right) \cdot \ell} \cdot \left(0.25 \cdot \frac{D\_m}{d}\right), M\_m, 1\right)\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \sqrt{\frac{d}{h}}\\ t_3 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+114}:\\ \;\;\;\;\left(t\_0 \cdot t\_2\right) \cdot t\_3\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-155}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\left(\left(\left(D\_m \cdot M\_m\right) \cdot D\_m\right) \cdot M\_m\right) \cdot \frac{0.25}{d \cdot d}\right) \cdot \frac{h}{\ell}, -0.5, 1\right) \cdot t\_3\right) \cdot t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+262}:\\ \;\;\;\;\left(t\_3 \cdot t\_2\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (fma
          (* (/ (* (* D_m M_m) h) (* (* -2.0 d) l)) (* 0.25 (/ D_m d)))
          M_m
          1.0))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_2 (sqrt (/ d h)))
        (t_3 (sqrt (/ d l))))
   (if (<= t_1 -2e+114)
     (* (* t_0 t_2) t_3)
     (if (<= t_1 -2e-155)
       (*
        (*
         (fma
          (* (* (* (* (* D_m M_m) D_m) M_m) (/ 0.25 (* d d))) (/ h l))
          -0.5
          1.0)
         t_3)
        t_2)
       (if (<= t_1 2e-227)
         (/ (* (sqrt (* d h)) (sqrt (* d l))) (* h l))
         (if (<= t_1 5e+262)
           (* (* t_3 t_2) 1.0)
           (* t_0 (sqrt (* (/ d (* l h)) d)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = fma(((((D_m * M_m) * h) / ((-2.0 * d) * l)) * (0.25 * (D_m / d))), M_m, 1.0);
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = sqrt((d / h));
	double t_3 = sqrt((d / l));
	double tmp;
	if (t_1 <= -2e+114) {
		tmp = (t_0 * t_2) * t_3;
	} else if (t_1 <= -2e-155) {
		tmp = (fma((((((D_m * M_m) * D_m) * M_m) * (0.25 / (d * d))) * (h / l)), -0.5, 1.0) * t_3) * t_2;
	} else if (t_1 <= 2e-227) {
		tmp = (sqrt((d * h)) * sqrt((d * l))) / (h * l);
	} else if (t_1 <= 5e+262) {
		tmp = (t_3 * t_2) * 1.0;
	} else {
		tmp = t_0 * sqrt(((d / (l * h)) * d));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = fma(Float64(Float64(Float64(Float64(D_m * M_m) * h) / Float64(Float64(-2.0 * d) * l)) * Float64(0.25 * Float64(D_m / d))), M_m, 1.0)
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = sqrt(Float64(d / h))
	t_3 = sqrt(Float64(d / l))
	tmp = 0.0
	if (t_1 <= -2e+114)
		tmp = Float64(Float64(t_0 * t_2) * t_3);
	elseif (t_1 <= -2e-155)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(Float64(D_m * M_m) * D_m) * M_m) * Float64(0.25 / Float64(d * d))) * Float64(h / l)), -0.5, 1.0) * t_3) * t_2);
	elseif (t_1 <= 2e-227)
		tmp = Float64(Float64(sqrt(Float64(d * h)) * sqrt(Float64(d * l))) / Float64(h * l));
	elseif (t_1 <= 5e+262)
		tmp = Float64(Float64(t_3 * t_2) * 1.0);
	else
		tmp = Float64(t_0 * sqrt(Float64(Float64(d / Float64(l * h)) * d)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(N[(-2.0 * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(0.25 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * M$95$m + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -2e+114], N[(N[(t$95$0 * t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[t$95$1, -2e-155], N[(N[(N[(N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(0.25 / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 2e-227], N[(N[(N[Sqrt[N[(d * h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+262], N[(N[(t$95$3 * t$95$2), $MachinePrecision] * 1.0), $MachinePrecision], N[(t$95$0 * N[Sqrt[N[(N[(d / N[(l * h), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2e114

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

   3. Applied rewrites 67.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)}\right) \]

   5. Applied rewrites 62.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot h}{\left(-2 \cdot d\right) \cdot \ell} \cdot \left(0.25 \cdot \frac{D}{d}\right), M, 1\right) \cdot \sqrt{\frac{d}{h}}\right) \cdot \sqrt{\frac{d}{\ell}}} \]

   if -2e114 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2.00000000000000003e-155

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 43.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)}, -0.5, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      7. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{1}{4}\right)}}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      8. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\color{blue}{\left(\left(M \cdot M\right) \cdot \left(D \cdot D\right)\right)} \cdot \frac{1}{4}\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(\color{blue}{\left(M \cdot M\right)} \cdot \left(D \cdot D\right)\right) \cdot \frac{1}{4}\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(\left(M \cdot M\right) \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{1}{4}\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      11. unswap-sqr N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\color{blue}{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)} \cdot \frac{1}{4}\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      12. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{2}\right)}\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      13. swap-sqr N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \color{blue}{\left(\left(\left(M \cdot D\right) \cdot \frac{1}{2}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{1}{2}\right)\right)}}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      14. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(\left(M \cdot D\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{1}{2}\right)\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      15. mult-flip N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\color{blue}{\frac{M \cdot D}{2}} \cdot \left(\left(M \cdot D\right) \cdot \frac{1}{2}\right)\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      16. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\frac{M \cdot D}{2} \cdot \left(\left(M \cdot D\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      17. mult-flip N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\frac{M \cdot D}{2} \cdot \color{blue}{\frac{M \cdot D}{2}}\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      18. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \color{blue}{\left(\frac{M \cdot D}{2} \cdot \frac{M \cdot D}{2}\right)}}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      19. mult-flip N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2}\right)} \cdot \frac{M \cdot D}{2}\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      20. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(\left(M \cdot D\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \frac{M \cdot D}{2}\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      21. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2}\right)} \cdot \frac{M \cdot D}{2}\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      22. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2}\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      23. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2}\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      24. mult-flip N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2}\right)}\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      25. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{1}{2}\right) \cdot \left(\left(M \cdot D\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      26. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2}\right)}\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      27. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}\right)\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      28. lower-*.f64 52.7

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(\left(D \cdot M\right) \cdot 0.5\right) \cdot \left(\color{blue}{\left(D \cdot M\right)} \cdot 0.5\right)\right)}{\ell \cdot \left(d \cdot d\right)}, -0.5, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

   5. Applied rewrites 52.7%

      \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \color{blue}{\left(\left(\left(D \cdot M\right) \cdot 0.5\right) \cdot \left(\left(D \cdot M\right) \cdot 0.5\right)\right)}}{\ell \cdot \left(d \cdot d\right)}, -0.5, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{h \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{1}{2}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{1}{2}\right)\right)}{\ell \cdot \left(d \cdot d\right)}}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\color{blue}{h \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{1}{2}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{1}{2}\right)\right)}}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{1}{2}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{1}{2}\right)\right)}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      4. times-frac N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{h}{\ell} \cdot \frac{\left(\left(D \cdot M\right) \cdot \frac{1}{2}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{1}{2}\right)}{d \cdot d}}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{h}{\ell}} \cdot \frac{\left(\left(D \cdot M\right) \cdot \frac{1}{2}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{1}{2}\right)}{d \cdot d}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot \frac{\color{blue}{\left(\left(D \cdot M\right) \cdot \frac{1}{2}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{1}{2}\right)}}{d \cdot d}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot \frac{\left(\left(D \cdot M\right) \cdot \frac{1}{2}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{1}{2}\right)}{\color{blue}{d \cdot d}}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      8. times-frac N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot \color{blue}{\left(\frac{\left(D \cdot M\right) \cdot \frac{1}{2}}{d} \cdot \frac{\left(D \cdot M\right) \cdot \frac{1}{2}}{d}\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\frac{\color{blue}{\left(D \cdot M\right) \cdot \frac{1}{2}}}{d} \cdot \frac{\left(D \cdot M\right) \cdot \frac{1}{2}}{d}\right), \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      10. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\frac{\left(D \cdot M\right) \cdot \color{blue}{\frac{1}{2}}}{d} \cdot \frac{\left(D \cdot M\right) \cdot \frac{1}{2}}{d}\right), \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      11. mult-flip N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\frac{\color{blue}{\frac{D \cdot M}{2}}}{d} \cdot \frac{\left(D \cdot M\right) \cdot \frac{1}{2}}{d}\right), \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      12. associate-/l/ N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\color{blue}{\frac{D \cdot M}{2 \cdot d}} \cdot \frac{\left(D \cdot M\right) \cdot \frac{1}{2}}{d}\right), \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\frac{\color{blue}{D \cdot M}}{2 \cdot d} \cdot \frac{\left(D \cdot M\right) \cdot \frac{1}{2}}{d}\right), \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      14. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\frac{\color{blue}{M \cdot D}}{2 \cdot d} \cdot \frac{\left(D \cdot M\right) \cdot \frac{1}{2}}{d}\right), \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\color{blue}{\left(D \cdot M\right) \cdot \frac{1}{2}}}{d}\right), \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      16. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\left(D \cdot M\right) \cdot \color{blue}{\frac{1}{2}}}{d}\right), \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      17. mult-flip N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\color{blue}{\frac{D \cdot M}{2}}}{d}\right), \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      18. associate-/l/ N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\frac{D \cdot M}{2 \cdot d}}\right), \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      19. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\color{blue}{D \cdot M}}{2 \cdot d}\right), \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      20. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h}{\ell} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\color{blue}{M \cdot D}}{2 \cdot d}\right), \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

   7. Applied rewrites 51.7%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\left(\left(\left(\left(D \cdot M\right) \cdot D\right) \cdot M\right) \cdot \frac{0.25}{d \cdot d}\right) \cdot \frac{h}{\ell}}, -0.5, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

   if -2.00000000000000003e-155 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.99999999999999989e-227

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 22.7

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \color{blue}{\ell}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      7. lower-*.f64 29.6

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

   7. Applied rewrites 29.6%

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]

   if 1.99999999999999989e-227 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.00000000000000008e262

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

   3. Applied rewrites 67.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      3. lower-*.f64 67.8

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      5. metadata-eval 67.8

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      8. lift-sqrt.f64 67.8

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      10. metadata-eval 67.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      13. lower-sqrt.f64 67.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   5. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.5%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 5.00000000000000008e262 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

   3. Applied rewrites 67.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)}\right) \]

   5. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot h}{\left(-2 \cdot d\right) \cdot \ell} \cdot \left(0.25 \cdot \frac{D}{d}\right), M, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 6: 67.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{d}{\ell}}\\ t_2 := \sqrt{\frac{d}{h}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-155}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\frac{\left(h \cdot \left(D\_m \cdot M\_m\right)\right) \cdot \left(0.25 \cdot \left(D\_m \cdot M\_m\right)\right)}{\ell \cdot d}}{d}, -0.5, 1\right) \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+262}:\\ \;\;\;\;\left(t\_1 \cdot t\_2\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(D\_m \cdot M\_m\right) \cdot h}{\left(-2 \cdot d\right) \cdot \ell} \cdot \left(0.25 \cdot \frac{D\_m}{d}\right), M\_m, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (sqrt (/ d l)))
        (t_2 (sqrt (/ d h))))
   (if (<= t_0 -2e-155)
     (*
      (*
       (fma
        (/ (/ (* (* h (* D_m M_m)) (* 0.25 (* D_m M_m))) (* l d)) d)
        -0.5
        1.0)
       t_1)
      t_2)
     (if (<= t_0 2e-227)
       (/ (* (sqrt (* d h)) (sqrt (* d l))) (* h l))
       (if (<= t_0 5e+262)
         (* (* t_1 t_2) 1.0)
         (*
          (fma
           (* (/ (* (* D_m M_m) h) (* (* -2.0 d) l)) (* 0.25 (/ D_m d)))
           M_m
           1.0)
          (sqrt (* (/ d (* l h)) d))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt((d / l));
	double t_2 = sqrt((d / h));
	double tmp;
	if (t_0 <= -2e-155) {
		tmp = (fma(((((h * (D_m * M_m)) * (0.25 * (D_m * M_m))) / (l * d)) / d), -0.5, 1.0) * t_1) * t_2;
	} else if (t_0 <= 2e-227) {
		tmp = (sqrt((d * h)) * sqrt((d * l))) / (h * l);
	} else if (t_0 <= 5e+262) {
		tmp = (t_1 * t_2) * 1.0;
	} else {
		tmp = fma(((((D_m * M_m) * h) / ((-2.0 * d) * l)) * (0.25 * (D_m / d))), M_m, 1.0) * sqrt(((d / (l * h)) * d));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = sqrt(Float64(d / l))
	t_2 = sqrt(Float64(d / h))
	tmp = 0.0
	if (t_0 <= -2e-155)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(Float64(h * Float64(D_m * M_m)) * Float64(0.25 * Float64(D_m * M_m))) / Float64(l * d)) / d), -0.5, 1.0) * t_1) * t_2);
	elseif (t_0 <= 2e-227)
		tmp = Float64(Float64(sqrt(Float64(d * h)) * sqrt(Float64(d * l))) / Float64(h * l));
	elseif (t_0 <= 5e+262)
		tmp = Float64(Float64(t_1 * t_2) * 1.0);
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(D_m * M_m) * h) / Float64(Float64(-2.0 * d) * l)) * Float64(0.25 * Float64(D_m / d))), M_m, 1.0) * sqrt(Float64(Float64(d / Float64(l * h)) * d)));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -2e-155], N[(N[(N[(N[(N[(N[(N[(h * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(0.25 * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$0, 2e-227], N[(N[(N[Sqrt[N[(d * h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+262], N[(N[(t$95$1 * t$95$2), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(N[(-2.0 * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(0.25 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * M$95$m + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / N[(l * h), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2.00000000000000003e-155

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 43.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)\right)}{\ell \cdot \left(d \cdot d\right)}, -0.5, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \color{blue}{\left(\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(D \cdot D\right) \cdot \color{blue}{\left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\color{blue}{\left(D \cdot D\right)} \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(D \cdot D\right) \cdot \left(\color{blue}{\left(M \cdot M\right)} \cdot \frac{1}{4}\right)\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      7. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(M \cdot M\right)\right) \cdot \frac{1}{4}\right)}}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      8. *-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\color{blue}{\left(\left(M \cdot M\right) \cdot \left(D \cdot D\right)\right)} \cdot \frac{1}{4}\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(\color{blue}{\left(M \cdot M\right)} \cdot \left(D \cdot D\right)\right) \cdot \frac{1}{4}\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(\left(M \cdot M\right) \cdot \color{blue}{\left(D \cdot D\right)}\right) \cdot \frac{1}{4}\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      11. unswap-sqr N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\color{blue}{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right)} \cdot \frac{1}{4}\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      12. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{2}\right)}\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      13. swap-sqr N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \color{blue}{\left(\left(\left(M \cdot D\right) \cdot \frac{1}{2}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{1}{2}\right)\right)}}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      14. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(\left(M \cdot D\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{1}{2}\right)\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      15. mult-flip N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\color{blue}{\frac{M \cdot D}{2}} \cdot \left(\left(M \cdot D\right) \cdot \frac{1}{2}\right)\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      16. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\frac{M \cdot D}{2} \cdot \left(\left(M \cdot D\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      17. mult-flip N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\frac{M \cdot D}{2} \cdot \color{blue}{\frac{M \cdot D}{2}}\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      18. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \color{blue}{\left(\frac{M \cdot D}{2} \cdot \frac{M \cdot D}{2}\right)}}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      19. mult-flip N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2}\right)} \cdot \frac{M \cdot D}{2}\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      20. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(\left(M \cdot D\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \frac{M \cdot D}{2}\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      21. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2}\right)} \cdot \frac{M \cdot D}{2}\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      22. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2}\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      23. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2}\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      24. mult-flip N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2}\right)}\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      25. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{1}{2}\right) \cdot \left(\left(M \cdot D\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      26. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\left(M \cdot D\right) \cdot \frac{1}{2}\right)}\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      27. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{1}{2}\right) \cdot \left(\color{blue}{\left(D \cdot M\right)} \cdot \frac{1}{2}\right)\right)}{\ell \cdot \left(d \cdot d\right)}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      28. lower-*.f64 52.7

          \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(\left(D \cdot M\right) \cdot 0.5\right) \cdot \left(\color{blue}{\left(D \cdot M\right)} \cdot 0.5\right)\right)}{\ell \cdot \left(d \cdot d\right)}, -0.5, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

   5. Applied rewrites 52.7%

      \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \color{blue}{\left(\left(\left(D \cdot M\right) \cdot 0.5\right) \cdot \left(\left(D \cdot M\right) \cdot 0.5\right)\right)}}{\ell \cdot \left(d \cdot d\right)}, -0.5, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{h \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{1}{2}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{1}{2}\right)\right)}{\ell \cdot \left(d \cdot d\right)}}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{1}{2}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{1}{2}\right)\right)}{\color{blue}{\ell \cdot \left(d \cdot d\right)}}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{1}{2}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{1}{2}\right)\right)}{\ell \cdot \color{blue}{\left(d \cdot d\right)}}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      4. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{h \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{1}{2}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{1}{2}\right)\right)}{\color{blue}{\left(\ell \cdot d\right) \cdot d}}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      5. associate-/r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{\frac{h \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{1}{2}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{1}{2}\right)\right)}{\ell \cdot d}}{d}}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{\frac{h \cdot \left(\left(\left(D \cdot M\right) \cdot \frac{1}{2}\right) \cdot \left(\left(D \cdot M\right) \cdot \frac{1}{2}\right)\right)}{\ell \cdot d}}{d}}, \frac{-1}{2}, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

   7. Applied rewrites 63.1%

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{\frac{\left(h \cdot \left(D \cdot M\right)\right) \cdot \left(0.25 \cdot \left(D \cdot M\right)\right)}{\ell \cdot d}}{d}}, -0.5, 1\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}} \]

   if -2.00000000000000003e-155 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.99999999999999989e-227

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 22.7

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \color{blue}{\ell}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      7. lower-*.f64 29.6

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

   7. Applied rewrites 29.6%

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]

   if 1.99999999999999989e-227 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.00000000000000008e262

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

   3. Applied rewrites 67.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      3. lower-*.f64 67.8

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      5. metadata-eval 67.8

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      8. lift-sqrt.f64 67.8

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      10. metadata-eval 67.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      13. lower-sqrt.f64 67.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   5. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.5%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 5.00000000000000008e262 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

   3. Applied rewrites 67.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)}\right) \]

   5. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot h}{\left(-2 \cdot d\right) \cdot \ell} \cdot \left(0.25 \cdot \frac{D}{d}\right), M, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 66.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell \cdot h} \cdot d}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := 0.25 \cdot \frac{D\_m}{d}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-88}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{h}{\left(d + d\right) \cdot \ell} \cdot \left(D\_m \cdot M\_m\right), t\_2 \cdot M\_m, -1\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+262}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(D\_m \cdot M\_m\right) \cdot h}{\left(-2 \cdot d\right) \cdot \ell} \cdot t\_2, M\_m, 1\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* (/ d (* l h)) d)))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_2 (* 0.25 (/ D_m d))))
   (if (<= t_1 -5e-88)
     (- (* (fma (* (/ h (* (+ d d) l)) (* D_m M_m)) (* t_2 M_m) -1.0) t_0))
     (if (<= t_1 2e-227)
       (/ (* (sqrt (* d h)) (sqrt (* d l))) (* h l))
       (if (<= t_1 5e+262)
         (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
         (*
          (fma (* (/ (* (* D_m M_m) h) (* (* -2.0 d) l)) t_2) M_m 1.0)
          t_0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt(((d / (l * h)) * d));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = 0.25 * (D_m / d);
	double tmp;
	if (t_1 <= -5e-88) {
		tmp = -(fma(((h / ((d + d) * l)) * (D_m * M_m)), (t_2 * M_m), -1.0) * t_0);
	} else if (t_1 <= 2e-227) {
		tmp = (sqrt((d * h)) * sqrt((d * l))) / (h * l);
	} else if (t_1 <= 5e+262) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = fma(((((D_m * M_m) * h) / ((-2.0 * d) * l)) * t_2), M_m, 1.0) * t_0;
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(Float64(d / Float64(l * h)) * d))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = Float64(0.25 * Float64(D_m / d))
	tmp = 0.0
	if (t_1 <= -5e-88)
		tmp = Float64(-Float64(fma(Float64(Float64(h / Float64(Float64(d + d) * l)) * Float64(D_m * M_m)), Float64(t_2 * M_m), -1.0) * t_0));
	elseif (t_1 <= 2e-227)
		tmp = Float64(Float64(sqrt(Float64(d * h)) * sqrt(Float64(d * l))) / Float64(h * l));
	elseif (t_1 <= 5e+262)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = Float64(fma(Float64(Float64(Float64(Float64(D_m * M_m) * h) / Float64(Float64(-2.0 * d) * l)) * t_2), M_m, 1.0) * t_0);
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(N[(d / N[(l * h), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.25 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-88], (-N[(N[(N[(N[(h / N[(N[(d + d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * M$95$m), $MachinePrecision] + -1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), If[LessEqual[t$95$1, 2e-227], N[(N[(N[Sqrt[N[(d * h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+262], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(N[(-2.0 * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * M$95$m + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.00000000000000009e-88

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

   3. Applied rewrites 67.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)}\right) \]

   5. Applied rewrites 50.0%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(\frac{h}{\left(d + d\right) \cdot \ell} \cdot \left(D \cdot M\right), \left(0.25 \cdot \frac{D}{d}\right) \cdot M, -1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}} \]

   if -5.00000000000000009e-88 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.99999999999999989e-227

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 22.7

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \color{blue}{\ell}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      7. lower-*.f64 29.6

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

   7. Applied rewrites 29.6%

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]

   if 1.99999999999999989e-227 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.00000000000000008e262

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

   3. Applied rewrites 67.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      3. lower-*.f64 67.8

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      5. metadata-eval 67.8

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      8. lift-sqrt.f64 67.8

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      10. metadata-eval 67.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      13. lower-sqrt.f64 67.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   5. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.5%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 5.00000000000000008e262 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

   3. Applied rewrites 67.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)}\right) \]

   5. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot h}{\left(-2 \cdot d\right) \cdot \ell} \cdot \left(0.25 \cdot \frac{D}{d}\right), M, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 64.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\left(\left(D\_m \cdot D\_m\right) \cdot M\_m\right) \cdot M\_m}{d} \cdot 0.125}{d}, \frac{-h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+262}:\\ \;\;\;\;\left(t\_0 \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{\left(-d\right) \cdot \sqrt{\frac{h}{d}}}{h}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_1 0.0)
     (*
      (fma (/ (* (/ (* (* (* D_m D_m) M_m) M_m) d) 0.125) d) (/ (- h) l) 1.0)
      (sqrt (* (/ d (* l h)) d)))
     (if (<= t_1 5e+262)
       (* (* t_0 (sqrt (/ d h))) 1.0)
       (if (<= t_1 INFINITY)
         (/ (/ (* (sqrt (* d h)) (sqrt (* d l))) l) h)
         (* t_0 (/ (* (- d) (sqrt (/ h d))) h)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = fma(((((((D_m * D_m) * M_m) * M_m) / d) * 0.125) / d), (-h / l), 1.0) * sqrt(((d / (l * h)) * d));
	} else if (t_1 <= 5e+262) {
		tmp = (t_0 * sqrt((d / h))) * 1.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((sqrt((d * h)) * sqrt((d * l))) / l) / h;
	} else {
		tmp = t_0 * ((-d * sqrt((h / d))) / h);
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(Float64(Float64(D_m * D_m) * M_m) * M_m) / d) * 0.125) / d), Float64(Float64(-h) / l), 1.0) * sqrt(Float64(Float64(d / Float64(l * h)) * d)));
	elseif (t_1 <= 5e+262)
		tmp = Float64(Float64(t_0 * sqrt(Float64(d / h))) * 1.0);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(sqrt(Float64(d * h)) * sqrt(Float64(d * l))) / l) / h);
	else
		tmp = Float64(t_0 * Float64(Float64(Float64(-d) * sqrt(Float64(h / d))) / h));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(N[(N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / d), $MachinePrecision] * 0.125), $MachinePrecision] / d), $MachinePrecision] * N[((-h) / l), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / N[(l * h), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+262], N[(N[(t$95$0 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(N[Sqrt[N[(d * h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] / h), $MachinePrecision], N[(t$95$0 * N[(N[((-d) * N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 24.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-h}{\ell}, \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)}{\left(d + d\right) \cdot d} \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}, \sqrt{\frac{d}{\ell \cdot h} \cdot d}\right)} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\frac{-h}{\ell} \cdot \left(\frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{\left(d + d\right) \cdot d} \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}\right) + \sqrt{\frac{d}{\ell \cdot h} \cdot d}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-h}{\ell} \cdot \color{blue}{\left(\frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{\left(d + d\right) \cdot d} \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}\right)} + \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-h}{\ell} \cdot \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{\left(d + d\right) \cdot d}\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}} + \sqrt{\frac{d}{\ell \cdot h} \cdot d} \]

      4. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left(\frac{-h}{\ell} \cdot \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{\left(d + d\right) \cdot d} + 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{-h}{\ell} \cdot \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot \frac{1}{4}\right)}{\left(d + d\right) \cdot d} + 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}} \]

   5. Applied rewrites 43.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{\left(\left(D \cdot D\right) \cdot M\right) \cdot M}{d} \cdot 0.125}{d}, \frac{-h}{\ell}, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.00000000000000008e262

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

   3. Applied rewrites 67.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      3. lower-*.f64 67.8

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      5. metadata-eval 67.8

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      8. lift-sqrt.f64 67.8

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      10. metadata-eval 67.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      13. lower-sqrt.f64 67.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   5. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.5%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 5.00000000000000008e262 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 22.7

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 31.9

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 31.9%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 22.7

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in d around -inf

      \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-/.f64 11.8

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   7. Applied rewrites 11.8%

      \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. mult-flip N/A

         \[\leadsto \left(\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\frac{1}{h}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{\color{blue}{1}}{h} \]

      4. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right)\right) \cdot \frac{\color{blue}{1}}{h} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right)\right) \cdot \frac{1}{\color{blue}{h}} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \frac{1}{h}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \frac{1}{h}\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \frac{1}{\color{blue}{h}}\right) \]

      9. mult-flip-rev N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)}{\color{blue}{h}} \]

      10. lower-/.f64 11.4

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)}{\color{blue}{h}} \]

   9. Applied rewrites 11.4%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\frac{\left(-d\right) \cdot \sqrt{\frac{h}{d}}}{h}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 63.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \mathsf{fma}\left(\frac{\left(D\_m \cdot M\_m\right) \cdot h}{\left(-2 \cdot d\right) \cdot \ell} \cdot \left(0.25 \cdot \frac{D\_m}{d}\right), M\_m, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+262}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1
         (*
          (fma
           (* (/ (* (* D_m M_m) h) (* (* -2.0 d) l)) (* 0.25 (/ D_m d)))
           M_m
           1.0)
          (sqrt (* (/ d (* l h)) d)))))
   (if (<= t_0 0.0)
     t_1
     (if (<= t_0 5e+262) (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0) t_1))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = fma(((((D_m * M_m) * h) / ((-2.0 * d) * l)) * (0.25 * (D_m / d))), M_m, 1.0) * sqrt(((d / (l * h)) * d));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+262) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(fma(Float64(Float64(Float64(Float64(D_m * M_m) * h) / Float64(Float64(-2.0 * d) * l)) * Float64(0.25 * Float64(D_m / d))), M_m, 1.0) * sqrt(Float64(Float64(d / Float64(l * h)) * d)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+262)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(N[(-2.0 * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(0.25 * N[(D$95$m / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * M$95$m + 1.0), $MachinePrecision] * N[Sqrt[N[(N[(d / N[(l * h), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 5e+262], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 0.0 or 5.00000000000000008e262 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

   3. Applied rewrites 67.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)}\right) \]

   5. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(D \cdot M\right) \cdot h}{\left(-2 \cdot d\right) \cdot \ell} \cdot \left(0.25 \cdot \frac{D}{d}\right), M, 1\right) \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}} \]

   if 0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.00000000000000008e262

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

   3. Applied rewrites 67.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      3. lower-*.f64 67.8

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      5. metadata-eval 67.8

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      8. lift-sqrt.f64 67.8

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      10. metadata-eval 67.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      13. lower-sqrt.f64 67.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   5. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.5%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 55.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-88}:\\ \;\;\;\;\frac{\left(-1 \cdot \left(d \cdot \sqrt{\sqrt{\frac{h}{d} \cdot \frac{h}{d}}}\right)\right) \cdot t\_2}{h}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-227}:\\ \;\;\;\;\frac{t\_0}{h \cdot \ell}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+262}:\\ \;\;\;\;\left(t\_2 \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\frac{t\_0}{\ell}}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \frac{\left(-d\right) \cdot \sqrt{\frac{h}{d}}}{h}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (* (sqrt (* d h)) (sqrt (* d l))))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_2 (sqrt (/ d l))))
   (if (<= t_1 -5e-88)
     (/ (* (* -1.0 (* d (sqrt (sqrt (* (/ h d) (/ h d)))))) t_2) h)
     (if (<= t_1 2e-227)
       (/ t_0 (* h l))
       (if (<= t_1 5e+262)
         (* (* t_2 (sqrt (/ d h))) 1.0)
         (if (<= t_1 INFINITY)
           (/ (/ t_0 l) h)
           (* t_2 (/ (* (- d) (sqrt (/ h d))) h))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d * h)) * sqrt((d * l));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = sqrt((d / l));
	double tmp;
	if (t_1 <= -5e-88) {
		tmp = ((-1.0 * (d * sqrt(sqrt(((h / d) * (h / d)))))) * t_2) / h;
	} else if (t_1 <= 2e-227) {
		tmp = t_0 / (h * l);
	} else if (t_1 <= 5e+262) {
		tmp = (t_2 * sqrt((d / h))) * 1.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_0 / l) / h;
	} else {
		tmp = t_2 * ((-d * sqrt((h / d))) / h);
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((d * h)) * Math.sqrt((d * l));
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = Math.sqrt((d / l));
	double tmp;
	if (t_1 <= -5e-88) {
		tmp = ((-1.0 * (d * Math.sqrt(Math.sqrt(((h / d) * (h / d)))))) * t_2) / h;
	} else if (t_1 <= 2e-227) {
		tmp = t_0 / (h * l);
	} else if (t_1 <= 5e+262) {
		tmp = (t_2 * Math.sqrt((d / h))) * 1.0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 / l) / h;
	} else {
		tmp = t_2 * ((-d * Math.sqrt((h / d))) / h);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((d * h)) * math.sqrt((d * l))
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	t_2 = math.sqrt((d / l))
	tmp = 0
	if t_1 <= -5e-88:
		tmp = ((-1.0 * (d * math.sqrt(math.sqrt(((h / d) * (h / d)))))) * t_2) / h
	elif t_1 <= 2e-227:
		tmp = t_0 / (h * l)
	elif t_1 <= 5e+262:
		tmp = (t_2 * math.sqrt((d / h))) * 1.0
	elif t_1 <= math.inf:
		tmp = (t_0 / l) / h
	else:
		tmp = t_2 * ((-d * math.sqrt((h / d))) / h)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(sqrt(Float64(d * h)) * sqrt(Float64(d * l)))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (t_1 <= -5e-88)
		tmp = Float64(Float64(Float64(-1.0 * Float64(d * sqrt(sqrt(Float64(Float64(h / d) * Float64(h / d)))))) * t_2) / h);
	elseif (t_1 <= 2e-227)
		tmp = Float64(t_0 / Float64(h * l));
	elseif (t_1 <= 5e+262)
		tmp = Float64(Float64(t_2 * sqrt(Float64(d / h))) * 1.0);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_0 / l) / h);
	else
		tmp = Float64(t_2 * Float64(Float64(Float64(-d) * sqrt(Float64(h / d))) / h));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((d * h)) * sqrt((d * l));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_2 = sqrt((d / l));
	tmp = 0.0;
	if (t_1 <= -5e-88)
		tmp = ((-1.0 * (d * sqrt(sqrt(((h / d) * (h / d)))))) * t_2) / h;
	elseif (t_1 <= 2e-227)
		tmp = t_0 / (h * l);
	elseif (t_1 <= 5e+262)
		tmp = (t_2 * sqrt((d / h))) * 1.0;
	elseif (t_1 <= Inf)
		tmp = (t_0 / l) / h;
	else
		tmp = t_2 * ((-d * sqrt((h / d))) / h);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[Sqrt[N[(d * h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -5e-88], N[(N[(N[(-1.0 * N[(d * N[Sqrt[N[Sqrt[N[(N[(h / d), $MachinePrecision] * N[(h / d), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[t$95$1, 2e-227], N[(t$95$0 / N[(h * l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+262], N[(N[(t$95$2 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$0 / l), $MachinePrecision] / h), $MachinePrecision], N[(t$95$2 * N[(N[((-d) * N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -5.00000000000000009e-88

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 22.7

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in d around -inf

      \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-/.f64 11.8

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   7. Applied rewrites 11.8%

      \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]
   8. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\sqrt{\frac{h}{d}} \cdot \sqrt{\frac{h}{d}}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\sqrt{\frac{h}{d} \cdot \frac{h}{d}}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\sqrt{\frac{h}{d} \cdot \frac{h}{d}}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 15.6

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\sqrt{\frac{h}{d} \cdot \frac{h}{d}}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   9. Applied rewrites 15.6%

      \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\sqrt{\frac{h}{d} \cdot \frac{h}{d}}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   if -5.00000000000000009e-88 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.99999999999999989e-227

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 22.7

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \color{blue}{\ell}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      7. lower-*.f64 29.6

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

   7. Applied rewrites 29.6%

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]

   if 1.99999999999999989e-227 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.00000000000000008e262

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

   3. Applied rewrites 67.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      3. lower-*.f64 67.8

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      5. metadata-eval 67.8

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      8. lift-sqrt.f64 67.8

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      10. metadata-eval 67.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      13. lower-sqrt.f64 67.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   5. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.5%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 5.00000000000000008e262 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 22.7

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 31.9

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 31.9%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 22.7

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in d around -inf

      \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-/.f64 11.8

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   7. Applied rewrites 11.8%

      \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. mult-flip N/A

         \[\leadsto \left(\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\frac{1}{h}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{\color{blue}{1}}{h} \]

      4. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right)\right) \cdot \frac{\color{blue}{1}}{h} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right)\right) \cdot \frac{1}{\color{blue}{h}} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \frac{1}{h}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \frac{1}{h}\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \frac{1}{\color{blue}{h}}\right) \]

      9. mult-flip-rev N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)}{\color{blue}{h}} \]

      10. lower-/.f64 11.4

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)}{\color{blue}{h}} \]

   9. Applied rewrites 11.4%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\frac{\left(-d\right) \cdot \sqrt{\frac{h}{d}}}{h}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 11: 53.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-155}:\\ \;\;\;\;-1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-227}:\\ \;\;\;\;\frac{t\_2}{h \cdot \ell}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+262}:\\ \;\;\;\;\left(t\_0 \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\frac{t\_2}{\ell}}{h}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{\left(-d\right) \cdot \sqrt{\frac{h}{d}}}{h}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_2 (* (sqrt (* d h)) (sqrt (* d l)))))
   (if (<= t_1 -2e-155)
     (* -1.0 (/ (* d (sqrt (/ h l))) h))
     (if (<= t_1 2e-227)
       (/ t_2 (* h l))
       (if (<= t_1 5e+262)
         (* (* t_0 (sqrt (/ d h))) 1.0)
         (if (<= t_1 INFINITY)
           (/ (/ t_2 l) h)
           (* t_0 (/ (* (- d) (sqrt (/ h d))) h))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = sqrt((d * h)) * sqrt((d * l));
	double tmp;
	if (t_1 <= -2e-155) {
		tmp = -1.0 * ((d * sqrt((h / l))) / h);
	} else if (t_1 <= 2e-227) {
		tmp = t_2 / (h * l);
	} else if (t_1 <= 5e+262) {
		tmp = (t_0 * sqrt((d / h))) * 1.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_2 / l) / h;
	} else {
		tmp = t_0 * ((-d * sqrt((h / d))) / h);
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = Math.sqrt((d * h)) * Math.sqrt((d * l));
	double tmp;
	if (t_1 <= -2e-155) {
		tmp = -1.0 * ((d * Math.sqrt((h / l))) / h);
	} else if (t_1 <= 2e-227) {
		tmp = t_2 / (h * l);
	} else if (t_1 <= 5e+262) {
		tmp = (t_0 * Math.sqrt((d / h))) * 1.0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (t_2 / l) / h;
	} else {
		tmp = t_0 * ((-d * Math.sqrt((h / d))) / h);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((d / l))
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	t_2 = math.sqrt((d * h)) * math.sqrt((d * l))
	tmp = 0
	if t_1 <= -2e-155:
		tmp = -1.0 * ((d * math.sqrt((h / l))) / h)
	elif t_1 <= 2e-227:
		tmp = t_2 / (h * l)
	elif t_1 <= 5e+262:
		tmp = (t_0 * math.sqrt((d / h))) * 1.0
	elif t_1 <= math.inf:
		tmp = (t_2 / l) / h
	else:
		tmp = t_0 * ((-d * math.sqrt((h / d))) / h)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = Float64(sqrt(Float64(d * h)) * sqrt(Float64(d * l)))
	tmp = 0.0
	if (t_1 <= -2e-155)
		tmp = Float64(-1.0 * Float64(Float64(d * sqrt(Float64(h / l))) / h));
	elseif (t_1 <= 2e-227)
		tmp = Float64(t_2 / Float64(h * l));
	elseif (t_1 <= 5e+262)
		tmp = Float64(Float64(t_0 * sqrt(Float64(d / h))) * 1.0);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_2 / l) / h);
	else
		tmp = Float64(t_0 * Float64(Float64(Float64(-d) * sqrt(Float64(h / d))) / h));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((d / l));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_2 = sqrt((d * h)) * sqrt((d * l));
	tmp = 0.0;
	if (t_1 <= -2e-155)
		tmp = -1.0 * ((d * sqrt((h / l))) / h);
	elseif (t_1 <= 2e-227)
		tmp = t_2 / (h * l);
	elseif (t_1 <= 5e+262)
		tmp = (t_0 * sqrt((d / h))) * 1.0;
	elseif (t_1 <= Inf)
		tmp = (t_2 / l) / h;
	else
		tmp = t_0 * ((-d * sqrt((h / d))) / h);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(d * h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-155], N[(-1.0 * N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-227], N[(t$95$2 / N[(h * l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+262], N[(N[(t$95$0 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$2 / l), $MachinePrecision] / h), $MachinePrecision], N[(t$95$0 * N[(N[((-d) * N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2.00000000000000003e-155

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 24.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-h}{\ell}, \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)}{\left(d + d\right) \cdot d} \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}, \sqrt{\frac{d}{\ell \cdot h} \cdot d}\right)} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 27.1

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 27.1%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   7. Taylor expanded in h around 0

      \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      4. lower-/.f64 13.5

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 13.5%

      \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   if -2.00000000000000003e-155 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.99999999999999989e-227

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 22.7

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \color{blue}{\ell}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      7. lower-*.f64 29.6

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

   7. Applied rewrites 29.6%

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]

   if 1.99999999999999989e-227 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.00000000000000008e262

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

   3. Applied rewrites 67.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      3. lower-*.f64 67.8

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      5. metadata-eval 67.8

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      8. lift-sqrt.f64 67.8

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      10. metadata-eval 67.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      13. lower-sqrt.f64 67.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   5. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.5%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 5.00000000000000008e262 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 22.7

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 31.9

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 31.9%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 22.7

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in d around -inf

      \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-/.f64 11.8

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   7. Applied rewrites 11.8%

      \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. mult-flip N/A

         \[\leadsto \left(\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\frac{1}{h}} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{\color{blue}{1}}{h} \]

      4. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right)\right) \cdot \frac{\color{blue}{1}}{h} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right)\right) \cdot \frac{1}{\color{blue}{h}} \]

      6. associate-*l* N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \frac{1}{h}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \frac{1}{h}\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \frac{1}{\color{blue}{h}}\right) \]

      9. mult-flip-rev N/A

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)}{\color{blue}{h}} \]

      10. lower-/.f64 11.4

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \frac{-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)}{\color{blue}{h}} \]

   9. Applied rewrites 11.4%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\frac{\left(-d\right) \cdot \sqrt{\frac{h}{d}}}{h}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 12: 53.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-155}:\\ \;\;\;\;-1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-227}:\\ \;\;\;\;\frac{t\_2}{h \cdot \ell}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+262}:\\ \;\;\;\;\left(t\_0 \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\frac{t\_2}{\ell}}{h}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \left(\left(-d\right) \cdot \sqrt{\frac{h}{d}}\right)}{h}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ d l)))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_2 (* (sqrt (* d h)) (sqrt (* d l)))))
   (if (<= t_1 -2e-155)
     (* -1.0 (/ (* d (sqrt (/ h l))) h))
     (if (<= t_1 2e-227)
       (/ t_2 (* h l))
       (if (<= t_1 5e+262)
         (* (* t_0 (sqrt (/ d h))) 1.0)
         (if (<= t_1 INFINITY)
           (/ (/ t_2 l) h)
           (/ (* t_0 (* (- d) (sqrt (/ h d)))) h)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((d / l));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = sqrt((d * h)) * sqrt((d * l));
	double tmp;
	if (t_1 <= -2e-155) {
		tmp = -1.0 * ((d * sqrt((h / l))) / h);
	} else if (t_1 <= 2e-227) {
		tmp = t_2 / (h * l);
	} else if (t_1 <= 5e+262) {
		tmp = (t_0 * sqrt((d / h))) * 1.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (t_2 / l) / h;
	} else {
		tmp = (t_0 * (-d * sqrt((h / d)))) / h;
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((d / l));
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = Math.sqrt((d * h)) * Math.sqrt((d * l));
	double tmp;
	if (t_1 <= -2e-155) {
		tmp = -1.0 * ((d * Math.sqrt((h / l))) / h);
	} else if (t_1 <= 2e-227) {
		tmp = t_2 / (h * l);
	} else if (t_1 <= 5e+262) {
		tmp = (t_0 * Math.sqrt((d / h))) * 1.0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (t_2 / l) / h;
	} else {
		tmp = (t_0 * (-d * Math.sqrt((h / d)))) / h;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((d / l))
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	t_2 = math.sqrt((d * h)) * math.sqrt((d * l))
	tmp = 0
	if t_1 <= -2e-155:
		tmp = -1.0 * ((d * math.sqrt((h / l))) / h)
	elif t_1 <= 2e-227:
		tmp = t_2 / (h * l)
	elif t_1 <= 5e+262:
		tmp = (t_0 * math.sqrt((d / h))) * 1.0
	elif t_1 <= math.inf:
		tmp = (t_2 / l) / h
	else:
		tmp = (t_0 * (-d * math.sqrt((h / d)))) / h
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(d / l))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = Float64(sqrt(Float64(d * h)) * sqrt(Float64(d * l)))
	tmp = 0.0
	if (t_1 <= -2e-155)
		tmp = Float64(-1.0 * Float64(Float64(d * sqrt(Float64(h / l))) / h));
	elseif (t_1 <= 2e-227)
		tmp = Float64(t_2 / Float64(h * l));
	elseif (t_1 <= 5e+262)
		tmp = Float64(Float64(t_0 * sqrt(Float64(d / h))) * 1.0);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(t_2 / l) / h);
	else
		tmp = Float64(Float64(t_0 * Float64(Float64(-d) * sqrt(Float64(h / d)))) / h);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((d / l));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_2 = sqrt((d * h)) * sqrt((d * l));
	tmp = 0.0;
	if (t_1 <= -2e-155)
		tmp = -1.0 * ((d * sqrt((h / l))) / h);
	elseif (t_1 <= 2e-227)
		tmp = t_2 / (h * l);
	elseif (t_1 <= 5e+262)
		tmp = (t_0 * sqrt((d / h))) * 1.0;
	elseif (t_1 <= Inf)
		tmp = (t_2 / l) / h;
	else
		tmp = (t_0 * (-d * sqrt((h / d)))) / h;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(d * h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-155], N[(-1.0 * N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-227], N[(t$95$2 / N[(h * l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+262], N[(N[(t$95$0 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(t$95$2 / l), $MachinePrecision] / h), $MachinePrecision], N[(N[(t$95$0 * N[((-d) * N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2.00000000000000003e-155

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 24.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-h}{\ell}, \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)}{\left(d + d\right) \cdot d} \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}, \sqrt{\frac{d}{\ell \cdot h} \cdot d}\right)} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 27.1

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 27.1%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   7. Taylor expanded in h around 0

      \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      4. lower-/.f64 13.5

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 13.5%

      \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   if -2.00000000000000003e-155 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.99999999999999989e-227

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 22.7

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \color{blue}{\ell}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      7. lower-*.f64 29.6

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

   7. Applied rewrites 29.6%

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]

   if 1.99999999999999989e-227 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.00000000000000008e262

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

   3. Applied rewrites 67.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      3. lower-*.f64 67.8

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      5. metadata-eval 67.8

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      8. lift-sqrt.f64 67.8

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      10. metadata-eval 67.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      13. lower-sqrt.f64 67.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   5. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.5%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 5.00000000000000008e262 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 22.7

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 31.9

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 31.9%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 22.7

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in d around -inf

      \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-/.f64 11.8

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   7. Applied rewrites 11.8%

      \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right) \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right)}{h} \]

      3. lower-*.f64 11.8

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right)}{h} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \left(-1 \cdot \left(d \cdot \sqrt{\frac{h}{d}}\right)\right)}{h} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \left(\mathsf{neg}\left(d \cdot \sqrt{\frac{h}{d}}\right)\right)}{h} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \left(\mathsf{neg}\left(d \cdot \sqrt{\frac{h}{d}}\right)\right)}{h} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \left(\left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{h}{d}}\right)}{h} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \left(\left(\mathsf{neg}\left(d\right)\right) \cdot \sqrt{\frac{h}{d}}\right)}{h} \]

      9. lower-neg.f64 11.8

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \left(\left(-d\right) \cdot \sqrt{\frac{h}{d}}\right)}{h} \]

   9. Applied rewrites 11.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{d}{\ell}} \cdot \left(\left(-d\right) \cdot \sqrt{\frac{h}{d}}\right)}{h}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 13: 50.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-155}:\\ \;\;\;\;-1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-227}:\\ \;\;\;\;\frac{t\_1}{h \cdot \ell}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+262}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\frac{t\_1}{\ell}}{h}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_1 (* (sqrt (* d h)) (sqrt (* d l)))))
   (if (<= t_0 -2e-155)
     (* -1.0 (/ (* d (sqrt (/ h l))) h))
     (if (<= t_0 2e-227)
       (/ t_1 (* h l))
       (if (<= t_0 5e+262)
         (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
         (if (<= t_0 INFINITY)
           (/ (/ t_1 l) h)
           (* d (sqrt (/ 1.0 (* h l))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = sqrt((d * h)) * sqrt((d * l));
	double tmp;
	if (t_0 <= -2e-155) {
		tmp = -1.0 * ((d * sqrt((h / l))) / h);
	} else if (t_0 <= 2e-227) {
		tmp = t_1 / (h * l);
	} else if (t_0 <= 5e+262) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (t_1 / l) / h;
	} else {
		tmp = d * sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_1 = Math.sqrt((d * h)) * Math.sqrt((d * l));
	double tmp;
	if (t_0 <= -2e-155) {
		tmp = -1.0 * ((d * Math.sqrt((h / l))) / h);
	} else if (t_0 <= 2e-227) {
		tmp = t_1 / (h * l);
	} else if (t_0 <= 5e+262) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (t_1 / l) / h;
	} else {
		tmp = d * Math.sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	t_1 = math.sqrt((d * h)) * math.sqrt((d * l))
	tmp = 0
	if t_0 <= -2e-155:
		tmp = -1.0 * ((d * math.sqrt((h / l))) / h)
	elif t_0 <= 2e-227:
		tmp = t_1 / (h * l)
	elif t_0 <= 5e+262:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	elif t_0 <= math.inf:
		tmp = (t_1 / l) / h
	else:
		tmp = d * math.sqrt((1.0 / (h * l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_1 = Float64(sqrt(Float64(d * h)) * sqrt(Float64(d * l)))
	tmp = 0.0
	if (t_0 <= -2e-155)
		tmp = Float64(-1.0 * Float64(Float64(d * sqrt(Float64(h / l))) / h));
	elseif (t_0 <= 2e-227)
		tmp = Float64(t_1 / Float64(h * l));
	elseif (t_0 <= 5e+262)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(t_1 / l) / h);
	else
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_1 = sqrt((d * h)) * sqrt((d * l));
	tmp = 0.0;
	if (t_0 <= -2e-155)
		tmp = -1.0 * ((d * sqrt((h / l))) / h);
	elseif (t_0 <= 2e-227)
		tmp = t_1 / (h * l);
	elseif (t_0 <= 5e+262)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	elseif (t_0 <= Inf)
		tmp = (t_1 / l) / h;
	else
		tmp = d * sqrt((1.0 / (h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(d * h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-155], N[(-1.0 * N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-227], N[(t$95$1 / N[(h * l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+262], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(t$95$1 / l), $MachinePrecision] / h), $MachinePrecision], N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2.00000000000000003e-155

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 24.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-h}{\ell}, \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)}{\left(d + d\right) \cdot d} \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}, \sqrt{\frac{d}{\ell \cdot h} \cdot d}\right)} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 27.1

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 27.1%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   7. Taylor expanded in h around 0

      \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      4. lower-/.f64 13.5

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 13.5%

      \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   if -2.00000000000000003e-155 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.99999999999999989e-227

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 22.7

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \color{blue}{\ell}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      7. lower-*.f64 29.6

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

   7. Applied rewrites 29.6%

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]

   if 1.99999999999999989e-227 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.00000000000000008e262

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

   3. Applied rewrites 67.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      3. lower-*.f64 67.8

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      5. metadata-eval 67.8

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      8. lift-sqrt.f64 67.8

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      10. metadata-eval 67.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      13. lower-sqrt.f64 67.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   5. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.5%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if 5.00000000000000008e262 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 22.7

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

      6. lower-*.f64 31.9

         \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   7. Applied rewrites 31.9%

      \[\leadsto \frac{\frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\ell}}{h} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 24.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-h}{\ell}, \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)}{\left(d + d\right) \cdot d} \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}, \sqrt{\frac{d}{\ell \cdot h} \cdot d}\right)} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 27.1

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 27.1%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      10. lower-neg.f64 27.1

          \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right) \]

   8. Applied rewrites 27.1%

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \color{blue}{\left(-d\right)} \]

   9. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       3. lower-/.f64 N/A

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       4. lower-*.f64 25.8

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   11. Applied rewrites 25.8%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 14: 49.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-155}:\\ \;\;\;\;-1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-227}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+262}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ (* (sqrt (* d h)) (sqrt (* d l))) (* h l)))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_1 -2e-155)
     (* -1.0 (/ (* d (sqrt (/ h l))) h))
     (if (<= t_1 2e-227)
       t_0
       (if (<= t_1 5e+262)
         (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)
         (if (<= t_1 INFINITY) t_0 (* d (sqrt (/ 1.0 (* h l))))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (sqrt((d * h)) * sqrt((d * l))) / (h * l);
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -2e-155) {
		tmp = -1.0 * ((d * sqrt((h / l))) / h);
	} else if (t_1 <= 2e-227) {
		tmp = t_0;
	} else if (t_1 <= 5e+262) {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = d * sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (Math.sqrt((d * h)) * Math.sqrt((d * l))) / (h * l);
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -2e-155) {
		tmp = -1.0 * ((d * Math.sqrt((h / l))) / h);
	} else if (t_1 <= 2e-227) {
		tmp = t_0;
	} else if (t_1 <= 5e+262) {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else {
		tmp = d * Math.sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (math.sqrt((d * h)) * math.sqrt((d * l))) / (h * l)
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_1 <= -2e-155:
		tmp = -1.0 * ((d * math.sqrt((h / l))) / h)
	elif t_1 <= 2e-227:
		tmp = t_0
	elif t_1 <= 5e+262:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	elif t_1 <= math.inf:
		tmp = t_0
	else:
		tmp = d * math.sqrt((1.0 / (h * l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(sqrt(Float64(d * h)) * sqrt(Float64(d * l))) / Float64(h * l))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= -2e-155)
		tmp = Float64(-1.0 * Float64(Float64(d * sqrt(Float64(h / l))) / h));
	elseif (t_1 <= 2e-227)
		tmp = t_0;
	elseif (t_1 <= 5e+262)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	elseif (t_1 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (sqrt((d * h)) * sqrt((d * l))) / (h * l);
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_1 <= -2e-155)
		tmp = -1.0 * ((d * sqrt((h / l))) / h);
	elseif (t_1 <= 2e-227)
		tmp = t_0;
	elseif (t_1 <= 5e+262)
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	elseif (t_1 <= Inf)
		tmp = t_0;
	else
		tmp = d * sqrt((1.0 / (h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(d * h), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(h * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-155], N[(-1.0 * N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-227], t$95$0, If[LessEqual[t$95$1, 5e+262], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$0, N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2.00000000000000003e-155

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 24.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-h}{\ell}, \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)}{\left(d + d\right) \cdot d} \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}, \sqrt{\frac{d}{\ell \cdot h} \cdot d}\right)} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 27.1

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 27.1%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   7. Taylor expanded in h around 0

      \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      4. lower-/.f64 13.5

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 13.5%

      \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   if -2.00000000000000003e-155 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.99999999999999989e-227 or 5.00000000000000008e262 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 22.7

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \color{blue}{\ell}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

      7. lower-*.f64 29.6

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{h \cdot \ell} \]

   7. Applied rewrites 29.6%

      \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{d \cdot \ell}}{\color{blue}{h \cdot \ell}} \]

   if 1.99999999999999989e-227 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 5.00000000000000008e262

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

   3. Applied rewrites 67.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      3. lower-*.f64 67.8

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      5. metadata-eval 67.8

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      8. lift-sqrt.f64 67.8

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      10. metadata-eval 67.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      13. lower-sqrt.f64 67.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   5. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.5%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 24.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-h}{\ell}, \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)}{\left(d + d\right) \cdot d} \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}, \sqrt{\frac{d}{\ell \cdot h} \cdot d}\right)} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 27.1

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 27.1%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      10. lower-neg.f64 27.1

          \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right) \]

   8. Applied rewrites 27.1%

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \color{blue}{\left(-d\right)} \]

   9. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       3. lower-/.f64 N/A

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       4. lower-*.f64 25.8

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   11. Applied rewrites 25.8%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 46.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-155}:\\ \;\;\;\;-1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-227}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 -2e-155)
     (* -1.0 (/ (* d (sqrt (/ h l))) h))
     (if (<= t_0 2e-227)
       (* d (sqrt (/ 1.0 (* h l))))
       (* (* (sqrt (/ d l)) (sqrt (/ d h))) 1.0)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -2e-155) {
		tmp = -1.0 * ((d * sqrt((h / l))) / h);
	} else if (t_0 <= 2e-227) {
		tmp = d * sqrt((1.0 / (h * l)));
	} else {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
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(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m_m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= (-2d-155)) then
        tmp = (-1.0d0) * ((d * sqrt((h / l))) / h)
    else if (t_0 <= 2d-227) then
        tmp = d * sqrt((1.0d0 / (h * l)))
    else
        tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0d0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -2e-155) {
		tmp = -1.0 * ((d * Math.sqrt((h / l))) / h);
	} else if (t_0 <= 2e-227) {
		tmp = d * Math.sqrt((1.0 / (h * l)));
	} else {
		tmp = (Math.sqrt((d / l)) * Math.sqrt((d / h))) * 1.0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= -2e-155:
		tmp = -1.0 * ((d * math.sqrt((h / l))) / h)
	elif t_0 <= 2e-227:
		tmp = d * math.sqrt((1.0 / (h * l)))
	else:
		tmp = (math.sqrt((d / l)) * math.sqrt((d / h))) * 1.0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -2e-155)
		tmp = Float64(-1.0 * Float64(Float64(d * sqrt(Float64(h / l))) / h));
	elseif (t_0 <= 2e-227)
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	else
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * 1.0);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= -2e-155)
		tmp = -1.0 * ((d * sqrt((h / l))) / h);
	elseif (t_0 <= 2e-227)
		tmp = d * sqrt((1.0 / (h * l)));
	else
		tmp = (sqrt((d / l)) * sqrt((d / h))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-155], N[(-1.0 * N[(N[(d * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-227], N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2.00000000000000003e-155

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 24.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-h}{\ell}, \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)}{\left(d + d\right) \cdot d} \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}, \sqrt{\frac{d}{\ell \cdot h} \cdot d}\right)} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 27.1

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 27.1%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   7. Taylor expanded in h around 0

      \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

      4. lower-/.f64 13.5

         \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]

   9. Applied rewrites 13.5%

      \[\leadsto -1 \cdot \frac{d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]

   if -2.00000000000000003e-155 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 1.99999999999999989e-227

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 24.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-h}{\ell}, \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)}{\left(d + d\right) \cdot d} \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}, \sqrt{\frac{d}{\ell \cdot h} \cdot d}\right)} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 27.1

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 27.1%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      10. lower-neg.f64 27.1

          \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right) \]

   8. Applied rewrites 27.1%

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \color{blue}{\left(-d\right)} \]

   9. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       3. lower-/.f64 N/A

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       4. lower-*.f64 25.8

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   11. Applied rewrites 25.8%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]

   if 1.99999999999999989e-227 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \color{blue}{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      5. lift-pow.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \]

      6. unpow2 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{M \cdot D}{2 \cdot d}}\right) \]

   3. Applied rewrites 67.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      3. lower-*.f64 67.8

         \[\leadsto \color{blue}{\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      5. metadata-eval 67.8

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      7. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      8. lift-sqrt.f64 67.8

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{\ell}}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      10. metadata-eval 67.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\frac{1}{2}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      12. unpow1/2 N/A

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot \frac{1}{2}\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

      13. lower-sqrt.f64 67.8

          \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}}\right) \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   5. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)} \cdot \left(1 - \left(\left(\frac{h}{\ell} \cdot 0.5\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \cdot \left(\frac{M}{d + d} \cdot D\right)\right) \]

   6. Taylor expanded in d around inf

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.5%

      \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 42.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 7.4 \cdot 10^{-175}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)\\ \mathbf{elif}\;d \leq 5.7 \cdot 10^{+119}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{h \cdot d}}{h}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= d 7.4e-175)
   (* (sqrt (/ (/ 1.0 l) h)) (- d))
   (if (<= d 5.7e+119)
     (* (sqrt (/ d l)) (/ (sqrt (* h d)) h))
     (* d (sqrt (/ 1.0 (* h l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 7.4e-175) {
		tmp = sqrt(((1.0 / l) / h)) * -d;
	} else if (d <= 5.7e+119) {
		tmp = sqrt((d / l)) * (sqrt((h * d)) / h);
	} else {
		tmp = d * sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
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(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (d <= 7.4d-175) then
        tmp = sqrt(((1.0d0 / l) / h)) * -d
    else if (d <= 5.7d+119) then
        tmp = sqrt((d / l)) * (sqrt((h * d)) / h)
    else
        tmp = d * sqrt((1.0d0 / (h * l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (d <= 7.4e-175) {
		tmp = Math.sqrt(((1.0 / l) / h)) * -d;
	} else if (d <= 5.7e+119) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((h * d)) / h);
	} else {
		tmp = d * Math.sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if d <= 7.4e-175:
		tmp = math.sqrt(((1.0 / l) / h)) * -d
	elif d <= 5.7e+119:
		tmp = math.sqrt((d / l)) * (math.sqrt((h * d)) / h)
	else:
		tmp = d * math.sqrt((1.0 / (h * l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (d <= 7.4e-175)
		tmp = Float64(sqrt(Float64(Float64(1.0 / l) / h)) * Float64(-d));
	elseif (d <= 5.7e+119)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(h * d)) / h));
	else
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (d <= 7.4e-175)
		tmp = sqrt(((1.0 / l) / h)) * -d;
	elseif (d <= 5.7e+119)
		tmp = sqrt((d / l)) * (sqrt((h * d)) / h);
	else
		tmp = d * sqrt((1.0 / (h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[d, 7.4e-175], N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], If[LessEqual[d, 5.7e+119], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(h * d), $MachinePrecision]], $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < 7.39999999999999997e-175

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 24.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-h}{\ell}, \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)}{\left(d + d\right) \cdot d} \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}, \sqrt{\frac{d}{\ell \cdot h} \cdot d}\right)} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 27.1

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 27.1%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      10. lower-neg.f64 27.1

          \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right) \]

   8. Applied rewrites 27.1%

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \color{blue}{\left(-d\right)} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right) \]

      3. associate-/r* N/A

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right) \]

      5. lower-/.f64 27.2

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right) \]

   10. Applied rewrites 27.2%

       \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right) \]

   if 7.39999999999999997e-175 < d < 5.7000000000000002e119

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lower-/.f64 22.7

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

   4. Applied rewrites 22.7%

      \[\leadsto \color{blue}{\frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{\color{blue}{h}} \]

      2. mult-flip N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\frac{1}{h}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \frac{1}{h} \]

      4. pow1/2 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right) \cdot \frac{1}{h} \]

      5. lift-pow.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}}\right) \cdot \frac{1}{h} \]

      6. metadata-eval N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{h} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{1}{h} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\sqrt{d \cdot h} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \frac{\color{blue}{1}}{h} \]

      9. *-commutative N/A

         \[\leadsto \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{d \cdot h}\right) \cdot \frac{\color{blue}{1}}{h} \]

      10. associate-*l* N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right)} \]

      12. lift-/.f64 N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right) \]

      13. metadata-eval N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(\sqrt{d \cdot h} \cdot \frac{1}{h}\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto {\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(\color{blue}{\sqrt{d \cdot h}} \cdot \frac{1}{h}\right) \]

      15. pow1/2 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\sqrt{d \cdot h}} \cdot \frac{1}{h}\right) \]

      16. lift-sqrt.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\color{blue}{\sqrt{d \cdot h}} \cdot \frac{1}{h}\right) \]

      17. mult-flip-rev N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d \cdot h}}{\color{blue}{h}} \]

      18. lower-/.f64 22.7

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d \cdot h}}{\color{blue}{h}} \]

      19. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{d \cdot h}}{h} \]

      20. *-commutative N/A

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{h \cdot d}}{h} \]

      21. lower-*.f64 22.7

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \frac{\sqrt{h \cdot d}}{h} \]

   6. Applied rewrites 22.7%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\frac{\sqrt{h \cdot d}}{h}} \]

   if 5.7000000000000002e119 < d

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 24.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-h}{\ell}, \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)}{\left(d + d\right) \cdot d} \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}, \sqrt{\frac{d}{\ell \cdot h} \cdot d}\right)} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 27.1

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 27.1%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      10. lower-neg.f64 27.1

          \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right) \]

   8. Applied rewrites 27.1%

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \color{blue}{\left(-d\right)} \]

   9. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       3. lower-/.f64 N/A

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       4. lower-*.f64 25.8

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   11. Applied rewrites 25.8%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 42.7% accurate, 5.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.7 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -6.7e-276)
   (* (sqrt (/ (/ 1.0 l) h)) (- d))
   (* d (sqrt (/ 1.0 (* h l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -6.7e-276) {
		tmp = sqrt(((1.0 / l) / h)) * -d;
	} else {
		tmp = d * sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
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(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-6.7d-276)) then
        tmp = sqrt(((1.0d0 / l) / h)) * -d
    else
        tmp = d * sqrt((1.0d0 / (h * l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -6.7e-276) {
		tmp = Math.sqrt(((1.0 / l) / h)) * -d;
	} else {
		tmp = d * Math.sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -6.7e-276:
		tmp = math.sqrt(((1.0 / l) / h)) * -d
	else:
		tmp = d * math.sqrt((1.0 / (h * l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -6.7e-276)
		tmp = Float64(sqrt(Float64(Float64(1.0 / l) / h)) * Float64(-d));
	else
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -6.7e-276)
		tmp = sqrt(((1.0 / l) / h)) * -d;
	else
		tmp = d * sqrt((1.0 / (h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -6.7e-276], N[(N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -6.69999999999999983e-276

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 24.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-h}{\ell}, \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)}{\left(d + d\right) \cdot d} \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}, \sqrt{\frac{d}{\ell \cdot h} \cdot d}\right)} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 27.1

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 27.1%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      10. lower-neg.f64 27.1

          \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right) \]

   8. Applied rewrites 27.1%

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \color{blue}{\left(-d\right)} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right) \]

      3. associate-/r* N/A

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right) \]

      5. lower-/.f64 27.2

         \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right) \]

   10. Applied rewrites 27.2%

       \[\leadsto \sqrt{\frac{\frac{1}{\ell}}{h}} \cdot \left(-d\right) \]

   if -6.69999999999999983e-276 < l

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 24.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-h}{\ell}, \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)}{\left(d + d\right) \cdot d} \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}, \sqrt{\frac{d}{\ell \cdot h} \cdot d}\right)} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 27.1

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 27.1%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      10. lower-neg.f64 27.1

          \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right) \]

   8. Applied rewrites 27.1%

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \color{blue}{\left(-d\right)} \]

   9. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       3. lower-/.f64 N/A

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       4. lower-*.f64 25.8

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   11. Applied rewrites 25.8%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 42.5% accurate, 5.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.7 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -6.7e-276)
   (* (sqrt (/ (/ 1.0 h) l)) (- d))
   (* d (sqrt (/ 1.0 (* h l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -6.7e-276) {
		tmp = sqrt(((1.0 / h) / l)) * -d;
	} else {
		tmp = d * sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
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(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-6.7d-276)) then
        tmp = sqrt(((1.0d0 / h) / l)) * -d
    else
        tmp = d * sqrt((1.0d0 / (h * l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -6.7e-276) {
		tmp = Math.sqrt(((1.0 / h) / l)) * -d;
	} else {
		tmp = d * Math.sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -6.7e-276:
		tmp = math.sqrt(((1.0 / h) / l)) * -d
	else:
		tmp = d * math.sqrt((1.0 / (h * l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -6.7e-276)
		tmp = Float64(sqrt(Float64(Float64(1.0 / h) / l)) * Float64(-d));
	else
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -6.7e-276)
		tmp = sqrt(((1.0 / h) / l)) * -d;
	else
		tmp = d * sqrt((1.0 / (h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -6.7e-276], N[(N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -6.69999999999999983e-276

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 24.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-h}{\ell}, \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)}{\left(d + d\right) \cdot d} \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}, \sqrt{\frac{d}{\ell \cdot h} \cdot d}\right)} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 27.1

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 27.1%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      10. lower-neg.f64 27.1

          \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right) \]

   8. Applied rewrites 27.1%

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \color{blue}{\left(-d\right)} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(-d\right) \]

      4. associate-/r* N/A

         \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right) \]

      6. lower-/.f64 27.2

         \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right) \]

   10. Applied rewrites 27.2%

       \[\leadsto \sqrt{\frac{\frac{1}{h}}{\ell}} \cdot \left(-d\right) \]

   if -6.69999999999999983e-276 < l

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 24.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-h}{\ell}, \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)}{\left(d + d\right) \cdot d} \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}, \sqrt{\frac{d}{\ell \cdot h} \cdot d}\right)} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 27.1

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 27.1%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      10. lower-neg.f64 27.1

          \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right) \]

   8. Applied rewrites 27.1%

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \color{blue}{\left(-d\right)} \]

   9. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       3. lower-/.f64 N/A

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       4. lower-*.f64 25.8

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   11. Applied rewrites 25.8%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 42.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.7 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= l -6.7e-276)
   (* (sqrt (/ 1.0 (* l h))) (- d))
   (* d (sqrt (/ 1.0 (* h l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -6.7e-276) {
		tmp = sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d * sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
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(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (l <= (-6.7d-276)) then
        tmp = sqrt((1.0d0 / (l * h))) * -d
    else
        tmp = d * sqrt((1.0d0 / (h * l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -6.7e-276) {
		tmp = Math.sqrt((1.0 / (l * h))) * -d;
	} else {
		tmp = d * Math.sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if l <= -6.7e-276:
		tmp = math.sqrt((1.0 / (l * h))) * -d
	else:
		tmp = d * math.sqrt((1.0 / (h * l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -6.7e-276)
		tmp = Float64(sqrt(Float64(1.0 / Float64(l * h))) * Float64(-d));
	else
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (l <= -6.7e-276)
		tmp = sqrt((1.0 / (l * h))) * -d;
	else
		tmp = d * sqrt((1.0 / (h * l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -6.7e-276], N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-d)), $MachinePrecision], N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -6.69999999999999983e-276

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 24.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-h}{\ell}, \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)}{\left(d + d\right) \cdot d} \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}, \sqrt{\frac{d}{\ell \cdot h} \cdot d}\right)} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 27.1

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 27.1%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      10. lower-neg.f64 27.1

          \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right) \]

   8. Applied rewrites 27.1%

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \color{blue}{\left(-d\right)} \]

   if -6.69999999999999983e-276 < l

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 24.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-h}{\ell}, \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)}{\left(d + d\right) \cdot d} \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}, \sqrt{\frac{d}{\ell \cdot h} \cdot d}\right)} \]

   4. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      5. lower-*.f64 27.1

         \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   6. Applied rewrites 27.1%

      \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

      10. lower-neg.f64 27.1

          \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right) \]

   8. Applied rewrites 27.1%

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \color{blue}{\left(-d\right)} \]

   9. Taylor expanded in d around inf

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

       2. lower-sqrt.f64 N/A

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       3. lower-/.f64 N/A

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

       4. lower-*.f64 25.8

          \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

   11. Applied rewrites 25.8%

       \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 25.8% accurate, 7.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ d \cdot \sqrt{\frac{1}{h \cdot \ell}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (* d (sqrt (/ 1.0 (* h l)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return d * sqrt((1.0 / (h * l)));
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
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(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = d * sqrt((1.0d0 / (h * l)))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return d * Math.sqrt((1.0 / (h * l)));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return d * math.sqrt((1.0 / (h * l)))

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(d * sqrt(Float64(1.0 / Float64(h * l))))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = d * sqrt((1.0 / (h * l)));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.2%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Applied rewrites 24.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-h}{\ell}, \frac{\left(D \cdot D\right) \cdot \left(\left(M \cdot M\right) \cdot 0.25\right)}{\left(d + d\right) \cdot d} \cdot \sqrt{\frac{d}{\ell \cdot h} \cdot d}, \sqrt{\frac{d}{\ell \cdot h} \cdot d}\right)} \]

4. Taylor expanded in d around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto -1 \cdot \left(d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}}\right) \]

   3. lower-sqrt.f64 N/A

      \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   4. lower-/.f64 N/A

      \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   5. lower-*.f64 27.1

      \[\leadsto -1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

6. Applied rewrites 27.1%

   \[\leadsto \color{blue}{-1 \cdot \left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)} \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(d \cdot \sqrt{\frac{1}{h \cdot \ell}}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\frac{1}{h \cdot \ell}} \cdot d\right) \]

   5. distribute-rgt-neg-in N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \color{blue}{\left(\mathsf{neg}\left(d\right)\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{h \cdot \ell}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(\mathsf{neg}\left(d\right)\right) \]

   10. lower-neg.f64 27.1

       \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \left(-d\right) \]

8. Applied rewrites 27.1%

   \[\leadsto \sqrt{\frac{1}{\ell \cdot h}} \cdot \color{blue}{\left(-d\right)} \]

9. Taylor expanded in d around inf

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto d \cdot \color{blue}{\sqrt{\frac{1}{h \cdot \ell}}} \]

    2. lower-sqrt.f64 N/A

       \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

    3. lower-/.f64 N/A

       \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

    4. lower-*.f64 25.8

       \[\leadsto d \cdot \sqrt{\frac{1}{h \cdot \ell}} \]

11. Applied rewrites 25.8%

    \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  :precision binary64
  (* (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))) (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))