Henrywood and Agarwal, Equation (12)

Percentage Accurate: 35.4% → 75.8%

Time: 12.1s

Alternatives: 18

Speedup: 10.2×

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: 35.4% 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: 75.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{h} \cdot \ell}\\ \mathbf{if}\;h \leq -1 \cdot 10^{+225}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(M\_m \cdot M\_m\right) \cdot \left(D\_m \cdot D\_m\right)}{\left(\ell \cdot d\_m\right) \cdot t\_0}, 0.125, \frac{-d\_m}{t\_0 \cdot h}\right)\\ \mathbf{elif}\;h \leq 2.7 \cdot 10^{-298}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\frac{M\_m}{\ell \cdot d\_m} \cdot \frac{M\_m}{t\_0}\right) \cdot -0.125\right) \cdot D\_m, D\_m, \frac{d\_m}{\sqrt{\ell} \cdot \sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* (/ 1.0 h) l))))
   (if (<= h -1e+225)
     (fma
      (/ (* (* M_m M_m) (* D_m D_m)) (* (* l d_m) t_0))
      0.125
      (/ (- d_m) (* t_0 h)))
     (if (<= h 2.7e-298)
       (*
        (* (sqrt (/ 1.0 (* l h))) d_m)
        (-
         1.0
         (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0)) (/ h l))))
       (fma
        (* (* (* (/ M_m (* l d_m)) (/ M_m t_0)) -0.125) D_m)
        D_m
        (/ d_m (* (sqrt l) (sqrt h))))))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt(((1.0 / h) * l));
	double tmp;
	if (h <= -1e+225) {
		tmp = fma((((M_m * M_m) * (D_m * D_m)) / ((l * d_m) * t_0)), 0.125, (-d_m / (t_0 * h)));
	} else if (h <= 2.7e-298) {
		tmp = (sqrt((1.0 / (l * h))) * d_m) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) * (h / l)));
	} else {
		tmp = fma(((((M_m / (l * d_m)) * (M_m / t_0)) * -0.125) * D_m), D_m, (d_m / (sqrt(l) * sqrt(h))));
	}
	return tmp;
}

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	t_0 = sqrt(Float64(Float64(1.0 / h) * l))
	tmp = 0.0
	if (h <= -1e+225)
		tmp = fma(Float64(Float64(Float64(M_m * M_m) * Float64(D_m * D_m)) / Float64(Float64(l * d_m) * t_0)), 0.125, Float64(Float64(-d_m) / Float64(t_0 * h)));
	elseif (h <= 2.7e-298)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0)) * Float64(h / l))));
	else
		tmp = fma(Float64(Float64(Float64(Float64(M_m / Float64(l * d_m)) * Float64(M_m / t_0)) * -0.125) * D_m), D_m, Float64(d_m / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -1e+225], N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(l * d$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.125 + N[((-d$95$m) / N[(t$95$0 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 2.7e-298], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $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$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(M$95$m / N[(l * d$95$m), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / t$95$0), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m + N[(d$95$m / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -9.99999999999999928e224

   1. Initial program 35.4%

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

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

   4. Applied rewrites 28.1%

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

   5. Taylor expanded in h around -inf

      \[\leadsto -1 \cdot \frac{d}{h \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} + \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} + -1 \cdot \color{blue}{\frac{d}{h \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{1}{8} + -1 \cdot \frac{\color{blue}{d}}{h \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)}, \frac{1}{8}, -1 \cdot \frac{d}{h \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)}\right) \]

   7. Applied rewrites 27.1%

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

   if -9.99999999999999928e224 < h < 2.7000000000000001e-298

   1. Initial program 35.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. frac-times N/A

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

      12. unpow2 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 42.3

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

   3. Applied rewrites 42.3%

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

   4. Taylor expanded in d around 0

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

      1. times-frac N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. frac-2neg N/A

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

      5. sqrt-undiv N/A

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

      6. pow1/2 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 69.3%

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

   if 2.7000000000000001e-298 < h

   1. Initial program 35.4%

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

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

   4. Applied rewrites 28.1%

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

   5. Taylor expanded in h around inf

      \[\leadsto \mathsf{fma}\left(\left(\frac{{M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{{M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(d \cdot \ell\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(d \cdot \ell\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\frac{1}{h}} \cdot \sqrt{\ell}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      9. sqrt-unprod N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      12. lower-/.f64 37.6

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

   7. Applied rewrites 37.6%

      \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot -0.125\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{d \cdot \ell} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{d \cdot \ell} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      17. lift-sqrt.f64 41.1

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

   9. Applied rewrites 41.1%

      \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot -0.125\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 75.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{h} \cdot \ell}\\ \mathbf{if}\;h \leq -1 \cdot 10^{+225}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(M\_m \cdot M\_m\right) \cdot \left(D\_m \cdot D\_m\right)}{\left(\ell \cdot d\_m\right) \cdot t\_0}, 0.125, \frac{-d\_m}{t\_0 \cdot h}\right)\\ \mathbf{elif}\;h \leq 2.7 \cdot 10^{-298}:\\ \;\;\;\;\frac{\left|d\_m\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\frac{M\_m}{\ell \cdot d\_m} \cdot \frac{M\_m}{t\_0}\right) \cdot -0.125\right) \cdot D\_m, D\_m, \frac{d\_m}{\sqrt{\ell} \cdot \sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* (/ 1.0 h) l))))
   (if (<= h -1e+225)
     (fma
      (/ (* (* M_m M_m) (* D_m D_m)) (* (* l d_m) t_0))
      0.125
      (/ (- d_m) (* t_0 h)))
     (if (<= h 2.7e-298)
       (*
        (/ (fabs d_m) (sqrt (* l h)))
        (-
         1.0
         (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0)) (/ h l))))
       (fma
        (* (* (* (/ M_m (* l d_m)) (/ M_m t_0)) -0.125) D_m)
        D_m
        (/ d_m (* (sqrt l) (sqrt h))))))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt(((1.0 / h) * l));
	double tmp;
	if (h <= -1e+225) {
		tmp = fma((((M_m * M_m) * (D_m * D_m)) / ((l * d_m) * t_0)), 0.125, (-d_m / (t_0 * h)));
	} else if (h <= 2.7e-298) {
		tmp = (fabs(d_m) / sqrt((l * h))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) * (h / l)));
	} else {
		tmp = fma(((((M_m / (l * d_m)) * (M_m / t_0)) * -0.125) * D_m), D_m, (d_m / (sqrt(l) * sqrt(h))));
	}
	return tmp;
}

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	t_0 = sqrt(Float64(Float64(1.0 / h) * l))
	tmp = 0.0
	if (h <= -1e+225)
		tmp = fma(Float64(Float64(Float64(M_m * M_m) * Float64(D_m * D_m)) / Float64(Float64(l * d_m) * t_0)), 0.125, Float64(Float64(-d_m) / Float64(t_0 * h)));
	elseif (h <= 2.7e-298)
		tmp = Float64(Float64(abs(d_m) / sqrt(Float64(l * h))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d_m)) ^ 2.0)) * Float64(h / l))));
	else
		tmp = fma(Float64(Float64(Float64(Float64(M_m / Float64(l * d_m)) * Float64(M_m / t_0)) * -0.125) * D_m), D_m, Float64(d_m / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -1e+225], N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(l * d$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.125 + N[((-d$95$m) / N[(t$95$0 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 2.7e-298], N[(N[(N[Abs[d$95$m], $MachinePrecision] / N[Sqrt[N[(l * h), $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$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(M$95$m / N[(l * d$95$m), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / t$95$0), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m + N[(d$95$m / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -9.99999999999999928e224

   1. Initial program 35.4%

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

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

   4. Applied rewrites 28.1%

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

   5. Taylor expanded in h around -inf

      \[\leadsto -1 \cdot \frac{d}{h \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} + \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} + -1 \cdot \color{blue}{\frac{d}{h \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{1}{8} + -1 \cdot \frac{\color{blue}{d}}{h \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)}, \frac{1}{8}, -1 \cdot \frac{d}{h \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)}\right) \]

   7. Applied rewrites 27.1%

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

   if -9.99999999999999928e224 < h < 2.7000000000000001e-298

   1. Initial program 35.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. frac-times N/A

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

      12. unpow2 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 42.3

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

   3. Applied rewrites 42.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. sqrt-unprod N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. pow2 N/A

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

      14. rem-sqrt-square-rev N/A

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

      15. lower-fabs.f64 N/A

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

      16. sqrt-unprod N/A

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

      17. lower-sqrt.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 69.6

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

   5. Applied rewrites 69.6%

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

   if 2.7000000000000001e-298 < h

   1. Initial program 35.4%

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

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

   4. Applied rewrites 28.1%

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

   5. Taylor expanded in h around inf

      \[\leadsto \mathsf{fma}\left(\left(\frac{{M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{{M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(d \cdot \ell\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(d \cdot \ell\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\frac{1}{h}} \cdot \sqrt{\ell}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      9. sqrt-unprod N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      12. lower-/.f64 37.6

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

   7. Applied rewrites 37.6%

      \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot -0.125\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{d \cdot \ell} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{d \cdot \ell} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      17. lift-sqrt.f64 41.1

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

   9. Applied rewrites 41.1%

      \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot -0.125\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 75.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{h} \cdot \ell}\\ t_1 := M\_m \cdot \frac{D\_m}{d\_m + d\_m}\\ \mathbf{if}\;h \leq -1 \cdot 10^{+225}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(M\_m \cdot M\_m\right) \cdot \left(D\_m \cdot D\_m\right)}{\left(\ell \cdot d\_m\right) \cdot t\_0}, 0.125, \frac{-d\_m}{t\_0 \cdot h}\right)\\ \mathbf{elif}\;h \leq 2.7 \cdot 10^{-298}:\\ \;\;\;\;\frac{\left|d\_m\right|}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(t\_1 \cdot t\_1\right)\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\frac{M\_m}{\ell \cdot d\_m} \cdot \frac{M\_m}{t\_0}\right) \cdot -0.125\right) \cdot D\_m, D\_m, \frac{d\_m}{\sqrt{\ell} \cdot \sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* (/ 1.0 h) l))) (t_1 (* M_m (/ D_m (+ d_m d_m)))))
   (if (<= h -1e+225)
     (fma
      (/ (* (* M_m M_m) (* D_m D_m)) (* (* l d_m) t_0))
      0.125
      (/ (- d_m) (* t_0 h)))
     (if (<= h 2.7e-298)
       (*
        (/ (fabs d_m) (sqrt (* l h)))
        (- 1.0 (* (* (/ 1.0 2.0) (* t_1 t_1)) (/ h l))))
       (fma
        (* (* (* (/ M_m (* l d_m)) (/ M_m t_0)) -0.125) D_m)
        D_m
        (/ d_m (* (sqrt l) (sqrt h))))))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt(((1.0 / h) * l));
	double t_1 = M_m * (D_m / (d_m + d_m));
	double tmp;
	if (h <= -1e+225) {
		tmp = fma((((M_m * M_m) * (D_m * D_m)) / ((l * d_m) * t_0)), 0.125, (-d_m / (t_0 * h)));
	} else if (h <= 2.7e-298) {
		tmp = (fabs(d_m) / sqrt((l * h))) * (1.0 - (((1.0 / 2.0) * (t_1 * t_1)) * (h / l)));
	} else {
		tmp = fma(((((M_m / (l * d_m)) * (M_m / t_0)) * -0.125) * D_m), D_m, (d_m / (sqrt(l) * sqrt(h))));
	}
	return tmp;
}

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	t_0 = sqrt(Float64(Float64(1.0 / h) * l))
	t_1 = Float64(M_m * Float64(D_m / Float64(d_m + d_m)))
	tmp = 0.0
	if (h <= -1e+225)
		tmp = fma(Float64(Float64(Float64(M_m * M_m) * Float64(D_m * D_m)) / Float64(Float64(l * d_m) * t_0)), 0.125, Float64(Float64(-d_m) / Float64(t_0 * h)));
	elseif (h <= 2.7e-298)
		tmp = Float64(Float64(abs(d_m) / sqrt(Float64(l * h))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * Float64(t_1 * t_1)) * Float64(h / l))));
	else
		tmp = fma(Float64(Float64(Float64(Float64(M_m / Float64(l * d_m)) * Float64(M_m / t_0)) * -0.125) * D_m), D_m, Float64(d_m / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(M$95$m * N[(D$95$m / N[(d$95$m + d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[h, -1e+225], N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(l * d$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.125 + N[((-d$95$m) / N[(t$95$0 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, 2.7e-298], N[(N[(N[Abs[d$95$m], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(M$95$m / N[(l * d$95$m), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / t$95$0), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m + N[(d$95$m / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if h < -9.99999999999999928e224

   1. Initial program 35.4%

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

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

   4. Applied rewrites 28.1%

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

   5. Taylor expanded in h around -inf

      \[\leadsto -1 \cdot \frac{d}{h \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} + \color{blue}{\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} + -1 \cdot \color{blue}{\frac{d}{h \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{1}{8} + -1 \cdot \frac{\color{blue}{d}}{h \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)}, \frac{1}{8}, -1 \cdot \frac{d}{h \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)}\right) \]

   7. Applied rewrites 27.1%

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

   if -9.99999999999999928e224 < h < 2.7000000000000001e-298

   1. Initial program 35.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. frac-times N/A

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

      12. unpow2 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 42.3

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

   3. Applied rewrites 42.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. sqrt-unprod N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. pow2 N/A

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

      14. rem-sqrt-square-rev N/A

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

      15. lower-fabs.f64 N/A

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

      16. sqrt-unprod N/A

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

      17. lower-sqrt.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 69.6

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

   5. Applied rewrites 69.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 68.9

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

   7. Applied rewrites 68.9%

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

   if 2.7000000000000001e-298 < h

   1. Initial program 35.4%

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

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

   4. Applied rewrites 28.1%

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

   5. Taylor expanded in h around inf

      \[\leadsto \mathsf{fma}\left(\left(\frac{{M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{{M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(d \cdot \ell\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(d \cdot \ell\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\frac{1}{h}} \cdot \sqrt{\ell}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      9. sqrt-unprod N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      12. lower-/.f64 37.6

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

   7. Applied rewrites 37.6%

      \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot -0.125\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{d \cdot \ell} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{d \cdot \ell} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      17. lift-sqrt.f64 41.1

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

   9. Applied rewrites 41.1%

      \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot -0.125\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 69.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;h \leq -5 \cdot 10^{-168}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot D\_m\right) \cdot D\_m}{t\_0 \cdot d\_m}, -0.125, \frac{\ell \cdot d\_m}{t\_0}\right)}{\ell}\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-226}:\\ \;\;\;\;\frac{\left|d\_m\right|}{t\_0} \cdot \mathsf{fma}\left(\frac{\left(M\_m \cdot M\_m\right) \cdot \left(D\_m \cdot D\_m\right)}{\left(d\_m \cdot d\_m\right) \cdot \ell} \cdot -0.125, h, 1\right)\\ \mathbf{elif}\;h \leq -2 \cdot 10^{-302}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\frac{M\_m}{\ell \cdot d\_m} \cdot \frac{M\_m}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot -0.125\right) \cdot D\_m, D\_m, \frac{d\_m}{\sqrt{\ell} \cdot \sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<= h -5e-168)
     (/
      (fma
       (/ (* (* (* (* M_m M_m) h) D_m) D_m) (* t_0 d_m))
       -0.125
       (/ (* l d_m) t_0))
      l)
     (if (<= h -2e-226)
       (*
        (/ (fabs d_m) t_0)
        (fma
         (* (/ (* (* M_m M_m) (* D_m D_m)) (* (* d_m d_m) l)) -0.125)
         h
         1.0))
       (if (<= h -2e-302)
         (* (* (sqrt (/ 1.0 (* l h))) d_m) 1.0)
         (fma
          (*
           (* (* (/ M_m (* l d_m)) (/ M_m (sqrt (* (/ 1.0 h) l)))) -0.125)
           D_m)
          D_m
          (/ d_m (* (sqrt l) (sqrt h)))))))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (h <= -5e-168) {
		tmp = fma((((((M_m * M_m) * h) * D_m) * D_m) / (t_0 * d_m)), -0.125, ((l * d_m) / t_0)) / l;
	} else if (h <= -2e-226) {
		tmp = (fabs(d_m) / t_0) * fma(((((M_m * M_m) * (D_m * D_m)) / ((d_m * d_m) * l)) * -0.125), h, 1.0);
	} else if (h <= -2e-302) {
		tmp = (sqrt((1.0 / (l * h))) * d_m) * 1.0;
	} else {
		tmp = fma(((((M_m / (l * d_m)) * (M_m / sqrt(((1.0 / h) * l)))) * -0.125) * D_m), D_m, (d_m / (sqrt(l) * sqrt(h))));
	}
	return tmp;
}

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (h <= -5e-168)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(Float64(M_m * M_m) * h) * D_m) * D_m) / Float64(t_0 * d_m)), -0.125, Float64(Float64(l * d_m) / t_0)) / l);
	elseif (h <= -2e-226)
		tmp = Float64(Float64(abs(d_m) / t_0) * fma(Float64(Float64(Float64(Float64(M_m * M_m) * Float64(D_m * D_m)) / Float64(Float64(d_m * d_m) * l)) * -0.125), h, 1.0));
	elseif (h <= -2e-302)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m) * 1.0);
	else
		tmp = fma(Float64(Float64(Float64(Float64(M_m / Float64(l * d_m)) * Float64(M_m / sqrt(Float64(Float64(1.0 / h) * l)))) * -0.125) * D_m), D_m, Float64(d_m / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -5e-168], N[(N[(N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] / N[(t$95$0 * d$95$m), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(N[(l * d$95$m), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], If[LessEqual[h, -2e-226], N[(N[(N[Abs[d$95$m], $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[h, -2e-302], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[(N[(N[(M$95$m / N[(l * d$95$m), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m + N[(d$95$m / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if h < -5.00000000000000001e-168

   1. Initial program 35.4%

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

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

   4. Applied rewrites 28.1%

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

   5. Taylor expanded in D around inf

      \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right) \cdot {D}^{\color{blue}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right) \cdot {D}^{\color{blue}{2}} \]

   7. Applied rewrites 43.5%

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

   8. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)} + \frac{d \cdot \ell}{\sqrt{h} \cdot \sqrt{\ell}}}{\color{blue}{\ell}} \]

   10. Applied rewrites 55.2%

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

   if -5.00000000000000001e-168 < h < -1.99999999999999984e-226

   1. Initial program 35.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. frac-times N/A

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

      12. unpow2 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 42.3

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

   3. Applied rewrites 42.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. sqrt-unprod N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. pow2 N/A

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

      14. rem-sqrt-square-rev N/A

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

      15. lower-fabs.f64 N/A

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

      16. sqrt-unprod N/A

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

      17. lower-sqrt.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 69.6

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

   5. Applied rewrites 69.6%

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

   6. Taylor expanded in h around inf

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{\left(h \cdot \left(\frac{1}{h} - \frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)\right)} \]
   7. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(h \cdot \left(\frac{1}{h} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(h \cdot \left(\frac{1}{h} + \frac{-1}{8} \cdot \frac{\color{blue}{{D}^{2} \cdot {M}^{2}}}{{d}^{2} \cdot \ell}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} + \color{blue}{\frac{1}{h}}\right)\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + \color{blue}{\frac{1}{h} \cdot h}\right) \]

      5. inv-pow N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + {h}^{-1} \cdot h\right) \]

      6. pow-plus N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + {h}^{\color{blue}{\left(-1 + 1\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + {h}^{0}\right) \]

      8. metadata-eval N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + 1\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}, \color{blue}{h}, 1\right) \]

   8. Applied rewrites 49.1%

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

   if -1.99999999999999984e-226 < h < -1.9999999999999999e-302

   1. Initial program 35.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. frac-times N/A

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

      12. unpow2 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 42.3

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

   3. Applied rewrites 42.3%

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

   4. Taylor expanded in d around 0

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

      1. times-frac N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. frac-2neg N/A

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

      5. sqrt-undiv N/A

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

      6. pow1/2 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 69.3%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 42.7%

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

   if -1.9999999999999999e-302 < h

   1. Initial program 35.4%

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

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

   4. Applied rewrites 28.1%

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

   5. Taylor expanded in h around inf

      \[\leadsto \mathsf{fma}\left(\left(\frac{{M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{{M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(d \cdot \ell\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(d \cdot \ell\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\frac{1}{h}} \cdot \sqrt{\ell}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      9. sqrt-unprod N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      12. lower-/.f64 37.6

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

   7. Applied rewrites 37.6%

      \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot -0.125\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{d \cdot \ell} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{d \cdot \ell} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      17. lift-sqrt.f64 41.1

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

   9. Applied rewrites 41.1%

      \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot -0.125\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 69.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+85}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\right) \cdot 1\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-122}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot \left(D\_m \cdot D\_m\right)}{t\_0 \cdot d\_m}, -0.125, \frac{\ell \cdot d\_m}{t\_0}\right)}{\ell}\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{-308}:\\ \;\;\;\;\frac{\left|d\_m\right|}{t\_0} \cdot \mathsf{fma}\left(\frac{\left(M\_m \cdot M\_m\right) \cdot \left(D\_m \cdot D\_m\right)}{\left(d\_m \cdot d\_m\right) \cdot \ell} \cdot -0.125, h, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\frac{M\_m}{\ell \cdot d\_m} \cdot \frac{M\_m}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot -0.125\right) \cdot D\_m, D\_m, \frac{d\_m}{\sqrt{\ell} \cdot \sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<= l -1e+85)
     (* (* (sqrt (/ 1.0 (* l h))) d_m) 1.0)
     (if (<= l -5e-122)
       (/
        (fma
         (/ (* (* (* M_m M_m) h) (* D_m D_m)) (* t_0 d_m))
         -0.125
         (/ (* l d_m) t_0))
        l)
       (if (<= l 9.5e-308)
         (*
          (/ (fabs d_m) t_0)
          (fma
           (* (/ (* (* M_m M_m) (* D_m D_m)) (* (* d_m d_m) l)) -0.125)
           h
           1.0))
         (fma
          (*
           (* (* (/ M_m (* l d_m)) (/ M_m (sqrt (* (/ 1.0 h) l)))) -0.125)
           D_m)
          D_m
          (/ d_m (* (sqrt l) (sqrt h)))))))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (l <= -1e+85) {
		tmp = (sqrt((1.0 / (l * h))) * d_m) * 1.0;
	} else if (l <= -5e-122) {
		tmp = fma(((((M_m * M_m) * h) * (D_m * D_m)) / (t_0 * d_m)), -0.125, ((l * d_m) / t_0)) / l;
	} else if (l <= 9.5e-308) {
		tmp = (fabs(d_m) / t_0) * fma(((((M_m * M_m) * (D_m * D_m)) / ((d_m * d_m) * l)) * -0.125), h, 1.0);
	} else {
		tmp = fma(((((M_m / (l * d_m)) * (M_m / sqrt(((1.0 / h) * l)))) * -0.125) * D_m), D_m, (d_m / (sqrt(l) * sqrt(h))));
	}
	return tmp;
}

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (l <= -1e+85)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m) * 1.0);
	elseif (l <= -5e-122)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(M_m * M_m) * h) * Float64(D_m * D_m)) / Float64(t_0 * d_m)), -0.125, Float64(Float64(l * d_m) / t_0)) / l);
	elseif (l <= 9.5e-308)
		tmp = Float64(Float64(abs(d_m) / t_0) * fma(Float64(Float64(Float64(Float64(M_m * M_m) * Float64(D_m * D_m)) / Float64(Float64(d_m * d_m) * l)) * -0.125), h, 1.0));
	else
		tmp = fma(Float64(Float64(Float64(Float64(M_m / Float64(l * d_m)) * Float64(M_m / sqrt(Float64(Float64(1.0 / h) * l)))) * -0.125) * D_m), D_m, Float64(d_m / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1e+85], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[l, -5e-122], N[(N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * d$95$m), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(N[(l * d$95$m), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], If[LessEqual[l, 9.5e-308], N[(N[(N[Abs[d$95$m], $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(M$95$m / N[(l * d$95$m), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m + N[(d$95$m / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1e85

   1. Initial program 35.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. frac-times N/A

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

      12. unpow2 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 42.3

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

   3. Applied rewrites 42.3%

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

   4. Taylor expanded in d around 0

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

      1. times-frac N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. frac-2neg N/A

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

      5. sqrt-undiv N/A

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

      6. pow1/2 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 69.3%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 42.7%

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

   if -1e85 < l < -4.9999999999999999e-122

   1. Initial program 35.4%

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

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

   4. Applied rewrites 28.1%

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

   5. Taylor expanded in l around 0

      \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)} + \frac{d \cdot \ell}{\sqrt{h} \cdot \sqrt{\ell}}}{\color{blue}{\ell}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)} + \frac{d \cdot \ell}{\sqrt{h} \cdot \sqrt{\ell}}}{\ell} \]

   7. Applied rewrites 49.0%

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

   if -4.9999999999999999e-122 < l < 9.49999999999999963e-308

   1. Initial program 35.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. frac-times N/A

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

      12. unpow2 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 42.3

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

   3. Applied rewrites 42.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. sqrt-unprod N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. pow2 N/A

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

      14. rem-sqrt-square-rev N/A

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

      15. lower-fabs.f64 N/A

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

      16. sqrt-unprod N/A

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

      17. lower-sqrt.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 69.6

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

   5. Applied rewrites 69.6%

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

   6. Taylor expanded in h around inf

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{\left(h \cdot \left(\frac{1}{h} - \frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)\right)} \]
   7. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(h \cdot \left(\frac{1}{h} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(h \cdot \left(\frac{1}{h} + \frac{-1}{8} \cdot \frac{\color{blue}{{D}^{2} \cdot {M}^{2}}}{{d}^{2} \cdot \ell}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} + \color{blue}{\frac{1}{h}}\right)\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + \color{blue}{\frac{1}{h} \cdot h}\right) \]

      5. inv-pow N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + {h}^{-1} \cdot h\right) \]

      6. pow-plus N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + {h}^{\color{blue}{\left(-1 + 1\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + {h}^{0}\right) \]

      8. metadata-eval N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + 1\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}, \color{blue}{h}, 1\right) \]

   8. Applied rewrites 49.1%

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

   if 9.49999999999999963e-308 < l

   1. Initial program 35.4%

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

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

   4. Applied rewrites 28.1%

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

   5. Taylor expanded in h around inf

      \[\leadsto \mathsf{fma}\left(\left(\frac{{M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{{M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(d \cdot \ell\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(d \cdot \ell\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\frac{1}{h}} \cdot \sqrt{\ell}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      9. sqrt-unprod N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      12. lower-/.f64 37.6

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

   7. Applied rewrites 37.6%

      \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot -0.125\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{d \cdot \ell} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{d \cdot \ell} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      17. lift-sqrt.f64 41.1

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

   9. Applied rewrites 41.1%

      \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot -0.125\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 67.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-21}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\right) \cdot 1\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{-308}:\\ \;\;\;\;\frac{\left|d\_m\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{\left(M\_m \cdot M\_m\right) \cdot \left(D\_m \cdot D\_m\right)}{\left(d\_m \cdot d\_m\right) \cdot \ell} \cdot -0.125, h, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\frac{M\_m}{\ell \cdot d\_m} \cdot \frac{M\_m}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot -0.125\right) \cdot D\_m, D\_m, \frac{d\_m}{\sqrt{\ell} \cdot \sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (if (<= l -1e-21)
   (* (* (sqrt (/ 1.0 (* l h))) d_m) 1.0)
   (if (<= l 9.5e-308)
     (*
      (/ (fabs d_m) (sqrt (* l h)))
      (fma (* (/ (* (* M_m M_m) (* D_m D_m)) (* (* d_m d_m) l)) -0.125) h 1.0))
     (fma
      (* (* (* (/ M_m (* l d_m)) (/ M_m (sqrt (* (/ 1.0 h) l)))) -0.125) D_m)
      D_m
      (/ d_m (* (sqrt l) (sqrt h)))))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1e-21) {
		tmp = (sqrt((1.0 / (l * h))) * d_m) * 1.0;
	} else if (l <= 9.5e-308) {
		tmp = (fabs(d_m) / sqrt((l * h))) * fma(((((M_m * M_m) * (D_m * D_m)) / ((d_m * d_m) * l)) * -0.125), h, 1.0);
	} else {
		tmp = fma(((((M_m / (l * d_m)) * (M_m / sqrt(((1.0 / h) * l)))) * -0.125) * D_m), D_m, (d_m / (sqrt(l) * sqrt(h))));
	}
	return tmp;
}

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -1e-21)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m) * 1.0);
	elseif (l <= 9.5e-308)
		tmp = Float64(Float64(abs(d_m) / sqrt(Float64(l * h))) * fma(Float64(Float64(Float64(Float64(M_m * M_m) * Float64(D_m * D_m)) / Float64(Float64(d_m * d_m) * l)) * -0.125), h, 1.0));
	else
		tmp = fma(Float64(Float64(Float64(Float64(M_m / Float64(l * d_m)) * Float64(M_m / sqrt(Float64(Float64(1.0 / h) * l)))) * -0.125) * D_m), D_m, Float64(d_m / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -1e-21], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[l, 9.5e-308], N[(N[(N[Abs[d$95$m], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(M$95$m / N[(l * d$95$m), $MachinePrecision]), $MachinePrecision] * N[(M$95$m / N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m + N[(d$95$m / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.99999999999999908e-22

   1. Initial program 35.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. frac-times N/A

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

      12. unpow2 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 42.3

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

   3. Applied rewrites 42.3%

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

   4. Taylor expanded in d around 0

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

      1. times-frac N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. frac-2neg N/A

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

      5. sqrt-undiv N/A

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

      6. pow1/2 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 69.3%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 42.7%

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

   if -9.99999999999999908e-22 < l < 9.49999999999999963e-308

   1. Initial program 35.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. frac-times N/A

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

      12. unpow2 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 42.3

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

   3. Applied rewrites 42.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. sqrt-unprod N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. pow2 N/A

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

      14. rem-sqrt-square-rev N/A

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

      15. lower-fabs.f64 N/A

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

      16. sqrt-unprod N/A

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

      17. lower-sqrt.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 69.6

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

   5. Applied rewrites 69.6%

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

   6. Taylor expanded in h around inf

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{\left(h \cdot \left(\frac{1}{h} - \frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)\right)} \]
   7. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(h \cdot \left(\frac{1}{h} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(h \cdot \left(\frac{1}{h} + \frac{-1}{8} \cdot \frac{\color{blue}{{D}^{2} \cdot {M}^{2}}}{{d}^{2} \cdot \ell}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} + \color{blue}{\frac{1}{h}}\right)\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + \color{blue}{\frac{1}{h} \cdot h}\right) \]

      5. inv-pow N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + {h}^{-1} \cdot h\right) \]

      6. pow-plus N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + {h}^{\color{blue}{\left(-1 + 1\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + {h}^{0}\right) \]

      8. metadata-eval N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + 1\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}, \color{blue}{h}, 1\right) \]

   8. Applied rewrites 49.1%

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

   if 9.49999999999999963e-308 < l

   1. Initial program 35.4%

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

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

   4. Applied rewrites 28.1%

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

   5. Taylor expanded in h around inf

      \[\leadsto \mathsf{fma}\left(\left(\frac{{M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{{M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(d \cdot \ell\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(d \cdot \ell\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\frac{1}{h}} \cdot \sqrt{\ell}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      9. sqrt-unprod N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      12. lower-/.f64 37.6

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

   7. Applied rewrites 37.6%

      \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot -0.125\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      8. times-frac N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{d \cdot \ell} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{d \cdot \ell} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      17. lift-sqrt.f64 41.1

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

   9. Applied rewrites 41.1%

      \[\leadsto \mathsf{fma}\left(\left(\left(\frac{M}{\ell \cdot d} \cdot \frac{M}{\sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot -0.125\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 65.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-21}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\right) \cdot 1\\ \mathbf{elif}\;\ell \leq 1.06 \cdot 10^{-259}:\\ \;\;\;\;\frac{\left|d\_m\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{\left(M\_m \cdot M\_m\right) \cdot \left(D\_m \cdot D\_m\right)}{\left(d\_m \cdot d\_m\right) \cdot \ell} \cdot -0.125, h, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(M\_m \cdot \frac{M\_m}{\left(\sqrt{\frac{1}{h} \cdot \ell} \cdot \ell\right) \cdot d\_m}\right) \cdot -0.125\right) \cdot D\_m, D\_m, \frac{d\_m}{\sqrt{\ell} \cdot \sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (if (<= l -1e-21)
   (* (* (sqrt (/ 1.0 (* l h))) d_m) 1.0)
   (if (<= l 1.06e-259)
     (*
      (/ (fabs d_m) (sqrt (* l h)))
      (fma (* (/ (* (* M_m M_m) (* D_m D_m)) (* (* d_m d_m) l)) -0.125) h 1.0))
     (fma
      (* (* (* M_m (/ M_m (* (* (sqrt (* (/ 1.0 h) l)) l) d_m))) -0.125) D_m)
      D_m
      (/ d_m (* (sqrt l) (sqrt h)))))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1e-21) {
		tmp = (sqrt((1.0 / (l * h))) * d_m) * 1.0;
	} else if (l <= 1.06e-259) {
		tmp = (fabs(d_m) / sqrt((l * h))) * fma(((((M_m * M_m) * (D_m * D_m)) / ((d_m * d_m) * l)) * -0.125), h, 1.0);
	} else {
		tmp = fma((((M_m * (M_m / ((sqrt(((1.0 / h) * l)) * l) * d_m))) * -0.125) * D_m), D_m, (d_m / (sqrt(l) * sqrt(h))));
	}
	return tmp;
}

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -1e-21)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m) * 1.0);
	elseif (l <= 1.06e-259)
		tmp = Float64(Float64(abs(d_m) / sqrt(Float64(l * h))) * fma(Float64(Float64(Float64(Float64(M_m * M_m) * Float64(D_m * D_m)) / Float64(Float64(d_m * d_m) * l)) * -0.125), h, 1.0));
	else
		tmp = fma(Float64(Float64(Float64(M_m * Float64(M_m / Float64(Float64(sqrt(Float64(Float64(1.0 / h) * l)) * l) * d_m))) * -0.125) * D_m), D_m, Float64(d_m / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -1e-21], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[l, 1.06e-259], N[(N[(N[Abs[d$95$m], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(M$95$m * N[(M$95$m / N[(N[(N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] * l), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] * d$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m + N[(d$95$m / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.99999999999999908e-22

   1. Initial program 35.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. frac-times N/A

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

      12. unpow2 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 42.3

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

   3. Applied rewrites 42.3%

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

   4. Taylor expanded in d around 0

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

      1. times-frac N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. frac-2neg N/A

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

      5. sqrt-undiv N/A

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

      6. pow1/2 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 69.3%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 42.7%

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

   if -9.99999999999999908e-22 < l < 1.06e-259

   1. Initial program 35.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. frac-times N/A

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

      12. unpow2 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 42.3

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

   3. Applied rewrites 42.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. sqrt-unprod N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. pow2 N/A

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

      14. rem-sqrt-square-rev N/A

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

      15. lower-fabs.f64 N/A

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

      16. sqrt-unprod N/A

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

      17. lower-sqrt.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 69.6

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

   5. Applied rewrites 69.6%

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

   6. Taylor expanded in h around inf

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{\left(h \cdot \left(\frac{1}{h} - \frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)\right)} \]
   7. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(h \cdot \left(\frac{1}{h} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(h \cdot \left(\frac{1}{h} + \frac{-1}{8} \cdot \frac{\color{blue}{{D}^{2} \cdot {M}^{2}}}{{d}^{2} \cdot \ell}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} + \color{blue}{\frac{1}{h}}\right)\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + \color{blue}{\frac{1}{h} \cdot h}\right) \]

      5. inv-pow N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + {h}^{-1} \cdot h\right) \]

      6. pow-plus N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + {h}^{\color{blue}{\left(-1 + 1\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + {h}^{0}\right) \]

      8. metadata-eval N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + 1\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}, \color{blue}{h}, 1\right) \]

   8. Applied rewrites 49.1%

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

   if 1.06e-259 < l

   1. Initial program 35.4%

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

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

   4. Applied rewrites 28.1%

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

   5. Taylor expanded in h around inf

      \[\leadsto \mathsf{fma}\left(\left(\frac{{M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{{M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(d \cdot \ell\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(d \cdot \ell\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\frac{1}{h}} \cdot \sqrt{\ell}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      9. sqrt-unprod N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      12. lower-/.f64 37.6

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

   7. Applied rewrites 37.6%

      \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot -0.125\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(M \cdot \frac{M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(M \cdot \frac{M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(M \cdot \frac{M}{\left(d \cdot \ell\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      11. sqrt-prod N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(M \cdot \frac{M}{\left(d \cdot \ell\right) \cdot \left(\sqrt{\frac{1}{h}} \cdot \sqrt{\ell}\right)}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(M \cdot \frac{M}{\left(d \cdot \ell\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(M \cdot \frac{M}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(M \cdot \frac{M}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(M \cdot \frac{M}{\left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right) \cdot d}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(M \cdot \frac{M}{\left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right) \cdot d}\right) \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

   9. Applied rewrites 39.1%

      \[\leadsto \mathsf{fma}\left(\left(\left(M \cdot \frac{M}{\left(\sqrt{\frac{1}{h} \cdot \ell} \cdot \ell\right) \cdot d}\right) \cdot -0.125\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 64.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-21}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\right) \cdot 1\\ \mathbf{elif}\;\ell \leq 1.62 \cdot 10^{-257}:\\ \;\;\;\;\frac{\left|d\_m\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{\left(M\_m \cdot M\_m\right) \cdot \left(D\_m \cdot D\_m\right)}{\left(d\_m \cdot d\_m\right) \cdot \ell} \cdot -0.125, h, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{M\_m \cdot M\_m}{\left(\ell \cdot d\_m\right) \cdot \sqrt{\frac{\ell}{h}}} \cdot -0.125\right) \cdot D\_m, D\_m, \frac{d\_m}{\sqrt{\ell} \cdot \sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (if (<= l -1e-21)
   (* (* (sqrt (/ 1.0 (* l h))) d_m) 1.0)
   (if (<= l 1.62e-257)
     (*
      (/ (fabs d_m) (sqrt (* l h)))
      (fma (* (/ (* (* M_m M_m) (* D_m D_m)) (* (* d_m d_m) l)) -0.125) h 1.0))
     (fma
      (* (* (/ (* M_m M_m) (* (* l d_m) (sqrt (/ l h)))) -0.125) D_m)
      D_m
      (/ d_m (* (sqrt l) (sqrt h)))))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1e-21) {
		tmp = (sqrt((1.0 / (l * h))) * d_m) * 1.0;
	} else if (l <= 1.62e-257) {
		tmp = (fabs(d_m) / sqrt((l * h))) * fma(((((M_m * M_m) * (D_m * D_m)) / ((d_m * d_m) * l)) * -0.125), h, 1.0);
	} else {
		tmp = fma(((((M_m * M_m) / ((l * d_m) * sqrt((l / h)))) * -0.125) * D_m), D_m, (d_m / (sqrt(l) * sqrt(h))));
	}
	return tmp;
}

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -1e-21)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m) * 1.0);
	elseif (l <= 1.62e-257)
		tmp = Float64(Float64(abs(d_m) / sqrt(Float64(l * h))) * fma(Float64(Float64(Float64(Float64(M_m * M_m) * Float64(D_m * D_m)) / Float64(Float64(d_m * d_m) * l)) * -0.125), h, 1.0));
	else
		tmp = fma(Float64(Float64(Float64(Float64(M_m * M_m) / Float64(Float64(l * d_m) * sqrt(Float64(l / h)))) * -0.125) * D_m), D_m, Float64(d_m / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -1e-21], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[l, 1.62e-257], N[(N[(N[Abs[d$95$m], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * h + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / N[(N[(l * d$95$m), $MachinePrecision] * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m + N[(d$95$m / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.99999999999999908e-22

   1. Initial program 35.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. frac-times N/A

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

      12. unpow2 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 42.3

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

   3. Applied rewrites 42.3%

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

   4. Taylor expanded in d around 0

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

      1. times-frac N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. frac-2neg N/A

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

      5. sqrt-undiv N/A

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

      6. pow1/2 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 69.3%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 42.7%

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

   if -9.99999999999999908e-22 < l < 1.6200000000000001e-257

   1. Initial program 35.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. frac-times N/A

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

      12. unpow2 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 42.3

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

   3. Applied rewrites 42.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. sqrt-unprod N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. pow2 N/A

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

      14. rem-sqrt-square-rev N/A

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

      15. lower-fabs.f64 N/A

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

      16. sqrt-unprod N/A

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

      17. lower-sqrt.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 69.6

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

   5. Applied rewrites 69.6%

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

   6. Taylor expanded in h around inf

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{\left(h \cdot \left(\frac{1}{h} - \frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)\right)} \]
   7. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(h \cdot \left(\frac{1}{h} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(h \cdot \left(\frac{1}{h} + \frac{-1}{8} \cdot \frac{\color{blue}{{D}^{2} \cdot {M}^{2}}}{{d}^{2} \cdot \ell}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} + \color{blue}{\frac{1}{h}}\right)\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + \color{blue}{\frac{1}{h} \cdot h}\right) \]

      5. inv-pow N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + {h}^{-1} \cdot h\right) \]

      6. pow-plus N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + {h}^{\color{blue}{\left(-1 + 1\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + {h}^{0}\right) \]

      8. metadata-eval N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + 1\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}, \color{blue}{h}, 1\right) \]

   8. Applied rewrites 49.1%

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

   if 1.6200000000000001e-257 < l

   1. Initial program 35.4%

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

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

   4. Applied rewrites 28.1%

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

   5. Taylor expanded in h around inf

      \[\leadsto \mathsf{fma}\left(\left(\frac{{M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{{M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(d \cdot \ell\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(d \cdot \ell\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\frac{1}{h}} \cdot \sqrt{\ell}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      9. sqrt-unprod N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      12. lower-/.f64 37.6

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

   7. Applied rewrites 37.6%

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

   8. Taylor expanded in h around 0

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

      1. lower-/.f64 37.6

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

   10. Applied rewrites 37.6%

       \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{\ell}{h}}} \cdot -0.125\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 64.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-21}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell \cdot h}} \cdot d\_m\right) \cdot 1\\ \mathbf{elif}\;\ell \leq 1.62 \cdot 10^{-257}:\\ \;\;\;\;\frac{\left|d\_m\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\left(\left(M\_m \cdot M\_m\right) \cdot \frac{h}{\left(d\_m \cdot d\_m\right) \cdot \ell}\right) \cdot -0.125, D\_m \cdot D\_m, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{M\_m \cdot M\_m}{\left(\ell \cdot d\_m\right) \cdot \sqrt{\frac{\ell}{h}}} \cdot -0.125\right) \cdot D\_m, D\_m, \frac{d\_m}{\sqrt{\ell} \cdot \sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (if (<= l -1e-21)
   (* (* (sqrt (/ 1.0 (* l h))) d_m) 1.0)
   (if (<= l 1.62e-257)
     (*
      (/ (fabs d_m) (sqrt (* l h)))
      (fma (* (* (* M_m M_m) (/ h (* (* d_m d_m) l))) -0.125) (* D_m D_m) 1.0))
     (fma
      (* (* (/ (* M_m M_m) (* (* l d_m) (sqrt (/ l h)))) -0.125) D_m)
      D_m
      (/ d_m (* (sqrt l) (sqrt h)))))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double tmp;
	if (l <= -1e-21) {
		tmp = (sqrt((1.0 / (l * h))) * d_m) * 1.0;
	} else if (l <= 1.62e-257) {
		tmp = (fabs(d_m) / sqrt((l * h))) * fma((((M_m * M_m) * (h / ((d_m * d_m) * l))) * -0.125), (D_m * D_m), 1.0);
	} else {
		tmp = fma(((((M_m * M_m) / ((l * d_m) * sqrt((l / h)))) * -0.125) * D_m), D_m, (d_m / (sqrt(l) * sqrt(h))));
	}
	return tmp;
}

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	tmp = 0.0
	if (l <= -1e-21)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(l * h))) * d_m) * 1.0);
	elseif (l <= 1.62e-257)
		tmp = Float64(Float64(abs(d_m) / sqrt(Float64(l * h))) * fma(Float64(Float64(Float64(M_m * M_m) * Float64(h / Float64(Float64(d_m * d_m) * l))) * -0.125), Float64(D_m * D_m), 1.0));
	else
		tmp = fma(Float64(Float64(Float64(Float64(M_m * M_m) / Float64(Float64(l * d_m) * sqrt(Float64(l / h)))) * -0.125) * D_m), D_m, Float64(d_m / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[l, -1e-21], N[(N[(N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * d$95$m), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[l, 1.62e-257], N[(N[(N[Abs[d$95$m], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * N[(h / N[(N[(d$95$m * d$95$m), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / N[(N[(l * d$95$m), $MachinePrecision] * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m + N[(d$95$m / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.99999999999999908e-22

   1. Initial program 35.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. frac-times N/A

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

      12. unpow2 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 42.3

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

   3. Applied rewrites 42.3%

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

   4. Taylor expanded in d around 0

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

      1. times-frac N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. frac-2neg N/A

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

      5. sqrt-undiv N/A

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

      6. pow1/2 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 69.3%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 42.7%

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

   if -9.99999999999999908e-22 < l < 1.6200000000000001e-257

   1. Initial program 35.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. frac-times N/A

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

      12. unpow2 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 42.3

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

   3. Applied rewrites 42.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. sqrt-div N/A

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

      7. pow1/2 N/A

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

      8. metadata-eval N/A

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

      9. sqrt-unprod N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow1/2 N/A

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

      13. pow2 N/A

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

      14. rem-sqrt-square-rev N/A

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

      15. lower-fabs.f64 N/A

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

      16. sqrt-unprod N/A

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

      17. lower-sqrt.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 69.6

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

   5. Applied rewrites 69.6%

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

   6. Taylor expanded in D around inf

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \color{blue}{\left({D}^{2} \cdot \left(\frac{1}{{D}^{2}} - \frac{1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right)\right)} \]
   7. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left({D}^{2} \cdot \left(\frac{1}{{D}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left({D}^{2} \cdot \left(\frac{1}{{D}^{2}} + \frac{-1}{8} \cdot \frac{\color{blue}{{M}^{2} \cdot h}}{{d}^{2} \cdot \ell}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left({D}^{2} \cdot \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell} + \color{blue}{\frac{1}{{D}^{2}}}\right)\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot {D}^{2} + \color{blue}{\frac{1}{{D}^{2}} \cdot {D}^{2}}\right) \]

      5. inv-pow N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot {D}^{2} + {\left({D}^{2}\right)}^{-1} \cdot {\color{blue}{D}}^{2}\right) \]

      6. pow-plus N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot {D}^{2} + {\left({D}^{2}\right)}^{\color{blue}{\left(-1 + 1\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot {D}^{2} + {\left({D}^{2}\right)}^{0}\right) \]

      8. metadata-eval N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}\right) \cdot {D}^{2} + 1\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{{d}^{2} \cdot \ell}, \color{blue}{{D}^{2}}, 1\right) \]

   8. Applied rewrites 48.1%

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

   if 1.6200000000000001e-257 < l

   1. Initial program 35.4%

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

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

   4. Applied rewrites 28.1%

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

   5. Taylor expanded in h around inf

      \[\leadsto \mathsf{fma}\left(\left(\frac{{M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{{M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(d \cdot \ell\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(d \cdot \ell\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\frac{1}{h}} \cdot \sqrt{\ell}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      9. sqrt-unprod N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      12. lower-/.f64 37.6

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

   7. Applied rewrites 37.6%

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

   8. Taylor expanded in h around 0

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

      1. lower-/.f64 37.6

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

   10. Applied rewrites 37.6%

       \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{\ell}{h}}} \cdot -0.125\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 60.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\ell \cdot h}}\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{-21}:\\ \;\;\;\;\left(t\_0 \cdot d\_m\right) \cdot 1\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot t\_0\right) \cdot \left(D\_m \cdot D\_m\right)}{\ell \cdot d\_m} \cdot -0.125\\ \mathbf{elif}\;\ell \leq 4.3 \cdot 10^{-260}:\\ \;\;\;\;\frac{\sqrt{h \cdot \left(d\_m \cdot \frac{d\_m}{\ell}\right)}}{h}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{M\_m \cdot M\_m}{\left(\ell \cdot d\_m\right) \cdot \sqrt{\frac{\ell}{h}}} \cdot -0.125\right) \cdot D\_m, D\_m, \frac{d\_m}{\sqrt{\ell} \cdot \sqrt{h}}\right)\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (* l h)))))
   (if (<= l -1e-21)
     (* (* t_0 d_m) 1.0)
     (if (<= l -2e-304)
       (* (/ (* (* (* (* M_m M_m) h) t_0) (* D_m D_m)) (* l d_m)) -0.125)
       (if (<= l 4.3e-260)
         (/ (sqrt (* h (* d_m (/ d_m l)))) h)
         (fma
          (* (* (/ (* M_m M_m) (* (* l d_m) (sqrt (/ l h)))) -0.125) D_m)
          D_m
          (/ d_m (* (sqrt l) (sqrt h)))))))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((1.0 / (l * h)));
	double tmp;
	if (l <= -1e-21) {
		tmp = (t_0 * d_m) * 1.0;
	} else if (l <= -2e-304) {
		tmp = (((((M_m * M_m) * h) * t_0) * (D_m * D_m)) / (l * d_m)) * -0.125;
	} else if (l <= 4.3e-260) {
		tmp = sqrt((h * (d_m * (d_m / l)))) / h;
	} else {
		tmp = fma(((((M_m * M_m) / ((l * d_m) * sqrt((l / h)))) * -0.125) * D_m), D_m, (d_m / (sqrt(l) * sqrt(h))));
	}
	return tmp;
}

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	t_0 = sqrt(Float64(1.0 / Float64(l * h)))
	tmp = 0.0
	if (l <= -1e-21)
		tmp = Float64(Float64(t_0 * d_m) * 1.0);
	elseif (l <= -2e-304)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M_m * M_m) * h) * t_0) * Float64(D_m * D_m)) / Float64(l * d_m)) * -0.125);
	elseif (l <= 4.3e-260)
		tmp = Float64(sqrt(Float64(h * Float64(d_m * Float64(d_m / l)))) / h);
	else
		tmp = fma(Float64(Float64(Float64(Float64(M_m * M_m) / Float64(Float64(l * d_m) * sqrt(Float64(l / h)))) * -0.125) * D_m), D_m, Float64(d_m / Float64(sqrt(l) * sqrt(h))));
	end
	return tmp
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1e-21], N[(N[(t$95$0 * d$95$m), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[l, -2e-304], N[(N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * d$95$m), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision], If[LessEqual[l, 4.3e-260], N[(N[Sqrt[N[(h * N[(d$95$m * N[(d$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / h), $MachinePrecision], N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] / N[(N[(l * d$95$m), $MachinePrecision] * N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m + N[(d$95$m / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -9.99999999999999908e-22

   1. Initial program 35.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. frac-times N/A

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

      12. unpow2 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 42.3

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

   3. Applied rewrites 42.3%

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

   4. Taylor expanded in d around 0

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

      1. times-frac N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. frac-2neg N/A

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

      5. sqrt-undiv N/A

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

      6. pow1/2 N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 69.3%

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

   7. Taylor expanded in d around inf

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

   9. Applied rewrites 42.7%

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

   if -9.99999999999999908e-22 < l < -1.99999999999999994e-304

   1. Initial program 35.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. frac-times N/A

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

      12. unpow2 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 42.3

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

   3. Applied rewrites 42.3%

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

   4. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}{d \cdot \ell}} \]

   6. Applied rewrites 31.1%

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

   if -1.99999999999999994e-304 < l < 4.30000000000000021e-260

   1. Initial program 35.4%

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

   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. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 21.4

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

   4. Applied rewrites 21.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 20.8

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

   6. Applied rewrites 20.8%

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

   if 4.30000000000000021e-260 < l

   1. Initial program 35.4%

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

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

   4. Applied rewrites 28.1%

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

   5. Taylor expanded in h around inf

      \[\leadsto \mathsf{fma}\left(\left(\frac{{M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{{M}^{2}}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{d \cdot \left(\ell \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(d \cdot \ell\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(d \cdot \ell\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\ell} \cdot \sqrt{\frac{1}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \left(\sqrt{\frac{1}{h}} \cdot \sqrt{\ell}\right)} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      9. sqrt-unprod N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{1}{h} \cdot \ell}} \cdot \frac{-1}{8}\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]

      12. lower-/.f64 37.6

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

   7. Applied rewrites 37.6%

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

   8. Taylor expanded in h around 0

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

      1. lower-/.f64 37.6

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

   10. Applied rewrites 37.6%

       \[\leadsto \mathsf{fma}\left(\left(\frac{M \cdot M}{\left(\ell \cdot d\right) \cdot \sqrt{\frac{\ell}{h}}} \cdot -0.125\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 11: 59.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 7.6 \cdot 10^{-50}:\\ \;\;\;\;\frac{d\_m}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(D\_m \cdot D\_m\right)}{\ell \cdot d\_m} \cdot -0.125\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (if (<= M_m 7.6e-50)
   (/ d_m (sqrt (* l h)))
   (*
    (/ (* (* (* (* M_m M_m) h) (sqrt (/ 1.0 (* l h)))) (* D_m D_m)) (* l d_m))
    -0.125)))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double tmp;
	if (M_m <= 7.6e-50) {
		tmp = d_m / sqrt((l * h));
	} else {
		tmp = (((((M_m * M_m) * h) * sqrt((1.0 / (l * h)))) * (D_m * D_m)) / (l * d_m)) * -0.125;
	}
	return tmp;
}

Fortran

d_m =     private
M_m =     private
D_m =     private
NOTE: d_m, 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_m, h, l, m_m, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m_1
    real(8) :: tmp
    if (m_m <= 7.6d-50) then
        tmp = d_m / sqrt((l * h))
    else
        tmp = (((((m_m * m_m) * h) * sqrt((1.0d0 / (l * h)))) * (d_m_1 * d_m_1)) / (l * d_m)) * (-0.125d0)
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
M_m = Math.abs(M);
D_m = Math.abs(D);
assert d_m < h && h < l && l < M_m && M_m < D_m;
public static double code(double d_m, double h, double l, double M_m, double D_m) {
	double tmp;
	if (M_m <= 7.6e-50) {
		tmp = d_m / Math.sqrt((l * h));
	} else {
		tmp = (((((M_m * M_m) * h) * Math.sqrt((1.0 / (l * h)))) * (D_m * D_m)) / (l * d_m)) * -0.125;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
M_m = math.fabs(M)
D_m = math.fabs(D)
[d_m, h, l, M_m, D_m] = sort([d_m, h, l, M_m, D_m])
def code(d_m, h, l, M_m, D_m):
	tmp = 0
	if M_m <= 7.6e-50:
		tmp = d_m / math.sqrt((l * h))
	else:
		tmp = (((((M_m * M_m) * h) * math.sqrt((1.0 / (l * h)))) * (D_m * D_m)) / (l * d_m)) * -0.125
	return tmp

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	tmp = 0.0
	if (M_m <= 7.6e-50)
		tmp = Float64(d_m / sqrt(Float64(l * h)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(M_m * M_m) * h) * sqrt(Float64(1.0 / Float64(l * h)))) * Float64(D_m * D_m)) / Float64(l * d_m)) * -0.125);
	end
	return tmp
end

MATLAB

d_m = abs(d);
M_m = abs(M);
D_m = abs(D);
d_m, h, l, M_m, D_m = num2cell(sort([d_m, h, l, M_m, D_m])){:}
function tmp_2 = code(d_m, h, l, M_m, D_m)
	tmp = 0.0;
	if (M_m <= 7.6e-50)
		tmp = d_m / sqrt((l * h));
	else
		tmp = (((((M_m * M_m) * h) * sqrt((1.0 / (l * h)))) * (D_m * D_m)) / (l * d_m)) * -0.125;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[M$95$m, 7.6e-50], N[(d$95$m / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * d$95$m), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 7.5999999999999998e-50

   1. Initial program 35.4%

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

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

   4. Applied rewrites 28.1%

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

   5. Taylor expanded in D around inf

      \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right) \cdot {D}^{\color{blue}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right) \cdot {D}^{\color{blue}{2}} \]

   7. Applied rewrites 43.5%

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

   8. Taylor expanded in d around inf

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

      1. sqrt-prod N/A

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

      2. lower-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 42.9

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

   10. Applied rewrites 42.9%

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

   if 7.5999999999999998e-50 < M

   1. Initial program 35.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-down N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow1/2 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. frac-times N/A

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

      12. unpow2 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 42.3

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

   3. Applied rewrites 42.3%

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

   4. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \sqrt{\frac{1}{h \cdot \ell}}\right)\right)}{d \cdot \ell}} \]

   6. Applied rewrites 31.1%

      \[\leadsto \color{blue}{\frac{\left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \sqrt{\frac{1}{\ell \cdot h}}\right) \cdot \left(D \cdot D\right)}{\ell \cdot d} \cdot -0.125} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 58.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;M\_m \cdot D\_m \leq 2 \cdot 10^{+112}:\\ \;\;\;\;\frac{d\_m}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(D\_m \cdot D\_m\right) \cdot \frac{\left(M\_m \cdot M\_m\right) \cdot h}{\left(\ell \cdot d\_m\right) \cdot t\_0}\right) \cdot -0.125\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<= (* M_m D_m) 2e+112)
     (/ d_m t_0)
     (* (* (* D_m D_m) (/ (* (* M_m M_m) h) (* (* l d_m) t_0))) -0.125))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((l * h));
	double tmp;
	if ((M_m * D_m) <= 2e+112) {
		tmp = d_m / t_0;
	} else {
		tmp = ((D_m * D_m) * (((M_m * M_m) * h) / ((l * d_m) * t_0))) * -0.125;
	}
	return tmp;
}

Fortran

d_m =     private
M_m =     private
D_m =     private
NOTE: d_m, 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_m, h, l, m_m, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l * h))
    if ((m_m * d_m_1) <= 2d+112) then
        tmp = d_m / t_0
    else
        tmp = ((d_m_1 * d_m_1) * (((m_m * m_m) * h) / ((l * d_m) * t_0))) * (-0.125d0)
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
M_m = Math.abs(M);
D_m = Math.abs(D);
assert d_m < h && h < l && l < M_m && M_m < D_m;
public static double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((l * h));
	double tmp;
	if ((M_m * D_m) <= 2e+112) {
		tmp = d_m / t_0;
	} else {
		tmp = ((D_m * D_m) * (((M_m * M_m) * h) / ((l * d_m) * t_0))) * -0.125;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
M_m = math.fabs(M)
D_m = math.fabs(D)
[d_m, h, l, M_m, D_m] = sort([d_m, h, l, M_m, D_m])
def code(d_m, h, l, M_m, D_m):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if (M_m * D_m) <= 2e+112:
		tmp = d_m / t_0
	else:
		tmp = ((D_m * D_m) * (((M_m * M_m) * h) / ((l * d_m) * t_0))) * -0.125
	return tmp

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (Float64(M_m * D_m) <= 2e+112)
		tmp = Float64(d_m / t_0);
	else
		tmp = Float64(Float64(Float64(D_m * D_m) * Float64(Float64(Float64(M_m * M_m) * h) / Float64(Float64(l * d_m) * t_0))) * -0.125);
	end
	return tmp
end

MATLAB

d_m = abs(d);
M_m = abs(M);
D_m = abs(D);
d_m, h, l, M_m, D_m = num2cell(sort([d_m, h, l, M_m, D_m])){:}
function tmp_2 = code(d_m, h, l, M_m, D_m)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if ((M_m * D_m) <= 2e+112)
		tmp = d_m / t_0;
	else
		tmp = ((D_m * D_m) * (((M_m * M_m) * h) / ((l * d_m) * t_0))) * -0.125;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 2e+112], N[(d$95$m / t$95$0), $MachinePrecision], N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] / N[(N[(l * d$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 M D) < 1.9999999999999999e112

   1. Initial program 35.4%

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

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

   4. Applied rewrites 28.1%

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

   5. Taylor expanded in D around inf

      \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right) \cdot {D}^{\color{blue}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right) \cdot {D}^{\color{blue}{2}} \]

   7. Applied rewrites 43.5%

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

   8. Taylor expanded in d around inf

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

      1. sqrt-prod N/A

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

      2. lower-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 42.9

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

   10. Applied rewrites 42.9%

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

   if 1.9999999999999999e112 < (*.f64 M D)

   1. Initial program 35.4%

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

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

   4. Applied rewrites 28.1%

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

   5. Taylor expanded in D around inf

      \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right) \cdot {D}^{\color{blue}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right) \cdot {D}^{\color{blue}{2}} \]

   7. Applied rewrites 43.5%

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

   8. Taylor expanded in d around 0

      \[\leadsto \frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)}} \]

   10. Applied rewrites 30.4%

       \[\leadsto \left(\left(D \cdot D\right) \cdot \frac{\left(M \cdot M\right) \cdot h}{\left(\ell \cdot d\right) \cdot \sqrt{\ell \cdot h}}\right) \cdot \color{blue}{-0.125} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 56.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-109}:\\ \;\;\;\;\frac{\left(M\_m \cdot M\_m\right) \cdot \left(D\_m \cdot D\_m\right)}{\left(\sqrt{\frac{\ell}{h}} \cdot \ell\right) \cdot d\_m} \cdot -0.125\\ \mathbf{elif}\;t\_0 \leq 10^{+126}:\\ \;\;\;\;\frac{\left|d\_m\right|}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d\_m}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d_m h) (/ 1.0 2.0)) (pow (/ d_m l) (/ 1.0 2.0)))
          (-
           1.0
           (*
            (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0))
            (/ h l))))))
   (if (<= t_0 -1e-109)
     (* (/ (* (* M_m M_m) (* D_m D_m)) (* (* (sqrt (/ l h)) l) d_m)) -0.125)
     (if (<= t_0 1e+126)
       (/ (fabs d_m) (* (sqrt l) (sqrt h)))
       (/ d_m (sqrt (* l h)))))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d_m / h), (1.0 / 2.0)) * pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -1e-109) {
		tmp = (((M_m * M_m) * (D_m * D_m)) / ((sqrt((l / h)) * l) * d_m)) * -0.125;
	} else if (t_0 <= 1e+126) {
		tmp = fabs(d_m) / (sqrt(l) * sqrt(h));
	} else {
		tmp = d_m / sqrt((l * h));
	}
	return tmp;
}

Fortran

d_m =     private
M_m =     private
D_m =     private
NOTE: d_m, 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_m, h, l, m_m, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d_m / h) ** (1.0d0 / 2.0d0)) * ((d_m / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m_m * d_m_1) / (2.0d0 * d_m)) ** 2.0d0)) * (h / l)))
    if (t_0 <= (-1d-109)) then
        tmp = (((m_m * m_m) * (d_m_1 * d_m_1)) / ((sqrt((l / h)) * l) * d_m)) * (-0.125d0)
    else if (t_0 <= 1d+126) then
        tmp = abs(d_m) / (sqrt(l) * sqrt(h))
    else
        tmp = d_m / sqrt((l * h))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
M_m = Math.abs(M);
D_m = Math.abs(D);
assert d_m < h && h < l && l < M_m && M_m < D_m;
public static double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = (Math.pow((d_m / h), (1.0 / 2.0)) * Math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -1e-109) {
		tmp = (((M_m * M_m) * (D_m * D_m)) / ((Math.sqrt((l / h)) * l) * d_m)) * -0.125;
	} else if (t_0 <= 1e+126) {
		tmp = Math.abs(d_m) / (Math.sqrt(l) * Math.sqrt(h));
	} else {
		tmp = d_m / Math.sqrt((l * h));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
M_m = math.fabs(M)
D_m = math.fabs(D)
[d_m, h, l, M_m, D_m] = sort([d_m, h, l, M_m, D_m])
def code(d_m, h, l, M_m, D_m):
	t_0 = (math.pow((d_m / h), (1.0 / 2.0)) * math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= -1e-109:
		tmp = (((M_m * M_m) * (D_m * D_m)) / ((math.sqrt((l / h)) * l) * d_m)) * -0.125
	elif t_0 <= 1e+126:
		tmp = math.fabs(d_m) / (math.sqrt(l) * math.sqrt(h))
	else:
		tmp = d_m / math.sqrt((l * h))
	return tmp

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	t_0 = Float64(Float64((Float64(d_m / h) ^ Float64(1.0 / 2.0)) * (Float64(d_m / 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_m)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -1e-109)
		tmp = Float64(Float64(Float64(Float64(M_m * M_m) * Float64(D_m * D_m)) / Float64(Float64(sqrt(Float64(l / h)) * l) * d_m)) * -0.125);
	elseif (t_0 <= 1e+126)
		tmp = Float64(abs(d_m) / Float64(sqrt(l) * sqrt(h)));
	else
		tmp = Float64(d_m / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

d_m = abs(d);
M_m = abs(M);
D_m = abs(D);
d_m, h, l, M_m, D_m = num2cell(sort([d_m, h, l, M_m, D_m])){:}
function tmp_2 = code(d_m, h, l, M_m, D_m)
	t_0 = (((d_m / h) ^ (1.0 / 2.0)) * ((d_m / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M_m * D_m) / (2.0 * d_m)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= -1e-109)
		tmp = (((M_m * M_m) * (D_m * D_m)) / ((sqrt((l / h)) * l) * d_m)) * -0.125;
	elseif (t_0 <= 1e+126)
		tmp = abs(d_m) / (sqrt(l) * sqrt(h));
	else
		tmp = d_m / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d$95$m / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d$95$m / 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$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-109], N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] * d$95$m), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision], If[LessEqual[t$95$0, 1e+126], N[(N[Abs[d$95$m], $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d$95$m / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $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)))) < -9.9999999999999999e-110

   1. Initial program 35.4%

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

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

   4. Applied rewrites 28.1%

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

   5. Taylor expanded in D around inf

      \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right) \cdot {D}^{\color{blue}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right) \cdot {D}^{\color{blue}{2}} \]

   7. Applied rewrites 43.5%

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

   8. Taylor expanded in h around inf

      \[\leadsto \frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{\color{blue}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)} \cdot \frac{-1}{8} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)} \cdot \frac{-1}{8} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot {M}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)} \cdot \frac{-1}{8} \]

      4. *-commutative N/A

         \[\leadsto \frac{{M}^{2} \cdot {D}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)} \cdot \frac{-1}{8} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{{M}^{2} \cdot {D}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)} \cdot \frac{-1}{8} \]

      6. pow2 N/A

         \[\leadsto \frac{\left(M \cdot M\right) \cdot {D}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)} \cdot \frac{-1}{8} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(M \cdot M\right) \cdot {D}^{2}}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)} \cdot \frac{-1}{8} \]

      8. pow2 N/A

         \[\leadsto \frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)} \cdot \frac{-1}{8} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)} \cdot \frac{-1}{8} \]

      10. *-commutative N/A

          \[\leadsto \frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{\left(\ell \cdot \sqrt{\frac{\ell}{h}}\right) \cdot d} \cdot \frac{-1}{8} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{\left(\ell \cdot \sqrt{\frac{\ell}{h}}\right) \cdot d} \cdot \frac{-1}{8} \]

      12. *-commutative N/A

          \[\leadsto \frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{\ell}{h}} \cdot \ell\right) \cdot d} \cdot \frac{-1}{8} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{\ell}{h}} \cdot \ell\right) \cdot d} \cdot \frac{-1}{8} \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{\ell}{h}} \cdot \ell\right) \cdot d} \cdot \frac{-1}{8} \]

      15. lower-/.f64 18.3

          \[\leadsto \frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{\ell}{h}} \cdot \ell\right) \cdot d} \cdot -0.125 \]

   10. Applied rewrites 18.3%

       \[\leadsto \frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{\left(\sqrt{\frac{\ell}{h}} \cdot \ell\right) \cdot d} \cdot -0.125 \]

   if -9.9999999999999999e-110 < (*.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)))) < 9.99999999999999925e125

   1. Initial program 35.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow1/2 N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-neg.f64 0.0

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

   3. Applied rewrites 0.0%

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

   4. Taylor expanded in h around 0

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

   6. Applied rewrites 42.9%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 25.0

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

   8. Applied rewrites 25.0%

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

   if 9.99999999999999925e125 < (*.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 35.4%

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

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

   4. Applied rewrites 28.1%

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

   5. Taylor expanded in D around inf

      \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right) \cdot {D}^{\color{blue}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right) \cdot {D}^{\color{blue}{2}} \]

   7. Applied rewrites 43.5%

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

   8. Taylor expanded in d around inf

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

      1. sqrt-prod N/A

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

      2. lower-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 42.9

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

   10. Applied rewrites 42.9%

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

Alternative 14: 49.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-109}:\\ \;\;\;\;-\sqrt{\frac{\left(-d\_m\right) \cdot \frac{d\_m}{\ell}}{-h}}\\ \mathbf{elif}\;t\_0 \leq 10^{+126}:\\ \;\;\;\;\frac{\left|d\_m\right|}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d\_m}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d_m h) (/ 1.0 2.0)) (pow (/ d_m l) (/ 1.0 2.0)))
          (-
           1.0
           (*
            (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0))
            (/ h l))))))
   (if (<= t_0 -1e-109)
     (- (sqrt (/ (* (- d_m) (/ d_m l)) (- h))))
     (if (<= t_0 1e+126)
       (/ (fabs d_m) (* (sqrt l) (sqrt h)))
       (/ d_m (sqrt (* l h)))))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d_m / h), (1.0 / 2.0)) * pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -1e-109) {
		tmp = -sqrt(((-d_m * (d_m / l)) / -h));
	} else if (t_0 <= 1e+126) {
		tmp = fabs(d_m) / (sqrt(l) * sqrt(h));
	} else {
		tmp = d_m / sqrt((l * h));
	}
	return tmp;
}

Fortran

d_m =     private
M_m =     private
D_m =     private
NOTE: d_m, 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_m, h, l, m_m, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d_m / h) ** (1.0d0 / 2.0d0)) * ((d_m / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m_m * d_m_1) / (2.0d0 * d_m)) ** 2.0d0)) * (h / l)))
    if (t_0 <= (-1d-109)) then
        tmp = -sqrt(((-d_m * (d_m / l)) / -h))
    else if (t_0 <= 1d+126) then
        tmp = abs(d_m) / (sqrt(l) * sqrt(h))
    else
        tmp = d_m / sqrt((l * h))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
M_m = Math.abs(M);
D_m = Math.abs(D);
assert d_m < h && h < l && l < M_m && M_m < D_m;
public static double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = (Math.pow((d_m / h), (1.0 / 2.0)) * Math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -1e-109) {
		tmp = -Math.sqrt(((-d_m * (d_m / l)) / -h));
	} else if (t_0 <= 1e+126) {
		tmp = Math.abs(d_m) / (Math.sqrt(l) * Math.sqrt(h));
	} else {
		tmp = d_m / Math.sqrt((l * h));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
M_m = math.fabs(M)
D_m = math.fabs(D)
[d_m, h, l, M_m, D_m] = sort([d_m, h, l, M_m, D_m])
def code(d_m, h, l, M_m, D_m):
	t_0 = (math.pow((d_m / h), (1.0 / 2.0)) * math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= -1e-109:
		tmp = -math.sqrt(((-d_m * (d_m / l)) / -h))
	elif t_0 <= 1e+126:
		tmp = math.fabs(d_m) / (math.sqrt(l) * math.sqrt(h))
	else:
		tmp = d_m / math.sqrt((l * h))
	return tmp

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	t_0 = Float64(Float64((Float64(d_m / h) ^ Float64(1.0 / 2.0)) * (Float64(d_m / 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_m)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -1e-109)
		tmp = Float64(-sqrt(Float64(Float64(Float64(-d_m) * Float64(d_m / l)) / Float64(-h))));
	elseif (t_0 <= 1e+126)
		tmp = Float64(abs(d_m) / Float64(sqrt(l) * sqrt(h)));
	else
		tmp = Float64(d_m / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

d_m = abs(d);
M_m = abs(M);
D_m = abs(D);
d_m, h, l, M_m, D_m = num2cell(sort([d_m, h, l, M_m, D_m])){:}
function tmp_2 = code(d_m, h, l, M_m, D_m)
	t_0 = (((d_m / h) ^ (1.0 / 2.0)) * ((d_m / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M_m * D_m) / (2.0 * d_m)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= -1e-109)
		tmp = -sqrt(((-d_m * (d_m / l)) / -h));
	elseif (t_0 <= 1e+126)
		tmp = abs(d_m) / (sqrt(l) * sqrt(h));
	else
		tmp = d_m / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d$95$m / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d$95$m / 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$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-109], (-N[Sqrt[N[(N[((-d$95$m) * N[(d$95$m / l), $MachinePrecision]), $MachinePrecision] / (-h)), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t$95$0, 1e+126], N[(N[Abs[d$95$m], $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d$95$m / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $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)))) < -9.9999999999999999e-110

   1. Initial program 35.4%

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

   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. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 21.4

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

   4. Applied rewrites 21.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sqrt-prod N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 18.9

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

   6. Applied rewrites 18.9%

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

   7. Taylor expanded in h around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. sqrt-div N/A

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

      5. sqrt-div N/A

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

      6. frac-times N/A

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

      7. sqrt-prod N/A

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

      8. rem-sqrt-square-rev N/A

         \[\leadsto -\frac{\left|d\right|}{\sqrt{\ell} \cdot \sqrt{h}} \]

      9. sqrt-prod N/A

         \[\leadsto -\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto -\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      11. lift-*.f64 N/A

          \[\leadsto -\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      12. lift-/.f64 N/A

          \[\leadsto -\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      13. lift-fabs.f64 9.6

          \[\leadsto -\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      14. lift-*.f64 N/A

          \[\leadsto -\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      15. *-commutative N/A

          \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      16. lower-*.f64 9.6

          \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

   9. Applied rewrites 9.6%

      \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]
   10. Step-by-step derivation

       1. lift-fabs.f64 N/A

          \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

       2. lift-/.f64 N/A

          \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

       3. rem-sqrt-square-rev N/A

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

       4. sqrt-prod N/A

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

       5. lift-*.f64 N/A

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

       6. lift-sqrt.f64 N/A

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

       7. sqrt-prod N/A

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

       8. frac-times N/A

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

       9. sqrt-div N/A

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

       10. frac-2neg N/A

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

       11. sqrt-undiv N/A

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

       12. sqrt-div N/A

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

       13. associate-*l/ N/A

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

       14. sqrt-unprod N/A

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

       15. sqrt-undiv N/A

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

       16. lower-sqrt.f64 N/A

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

       17. lower-/.f64 N/A

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

       18. lower-*.f64 N/A

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

       19. lower-neg.f64 N/A

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

       20. lift-/.f64 N/A

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

       21. lower-neg.f64 11.5

           \[\leadsto -\sqrt{\frac{\left(-d\right) \cdot \frac{d}{\ell}}{-h}} \]

   11. Applied rewrites 11.5%

       \[\leadsto -\sqrt{\frac{\left(-d\right) \cdot \frac{d}{\ell}}{-h}} \]

   if -9.9999999999999999e-110 < (*.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)))) < 9.99999999999999925e125

   1. Initial program 35.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. frac-2neg N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto \left(\frac{\sqrt{\color{blue}{-d}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\sqrt{-d}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-neg.f64 0.0

          \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{\color{blue}{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 0.0%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-1 \cdot h}}} \]

   6. Applied rewrites 42.9%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      3. sqrt-prod N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell} \cdot \sqrt{\color{blue}{h}}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell} \cdot \sqrt{h}} \]

      6. lift-*.f64 25.0

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

   8. Applied rewrites 25.0%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

   if 9.99999999999999925e125 < (*.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 35.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

   4. Applied rewrites 28.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot h\right) \cdot M\right) \cdot M}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right)} \]

   5. Taylor expanded in D around inf

      \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right) \cdot {D}^{\color{blue}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right) \cdot {D}^{\color{blue}{2}} \]

   7. Applied rewrites 43.5%

      \[\leadsto \mathsf{fma}\left(\frac{\left(M \cdot M\right) \cdot h}{\left(\sqrt{\ell \cdot h} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\ell \cdot h} \cdot \left(D \cdot D\right)}\right) \cdot \color{blue}{\left(D \cdot D\right)} \]

   8. Taylor expanded in d around inf

      \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   9. Step-by-step derivation

      1. sqrt-prod N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

      4. *-commutative N/A

         \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

      5. lift-*.f64 42.9

         \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

   10. Applied rewrites 42.9%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 49.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-109}:\\ \;\;\;\;-\sqrt{\frac{d\_m \cdot d\_m}{\ell \cdot h}}\\ \mathbf{elif}\;t\_0 \leq 10^{+126}:\\ \;\;\;\;\frac{\left|d\_m\right|}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d\_m}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d_m h) (/ 1.0 2.0)) (pow (/ d_m l) (/ 1.0 2.0)))
          (-
           1.0
           (*
            (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0))
            (/ h l))))))
   (if (<= t_0 -1e-109)
     (- (sqrt (/ (* d_m d_m) (* l h))))
     (if (<= t_0 1e+126)
       (/ (fabs d_m) (* (sqrt l) (sqrt h)))
       (/ d_m (sqrt (* l h)))))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d_m / h), (1.0 / 2.0)) * pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -1e-109) {
		tmp = -sqrt(((d_m * d_m) / (l * h)));
	} else if (t_0 <= 1e+126) {
		tmp = fabs(d_m) / (sqrt(l) * sqrt(h));
	} else {
		tmp = d_m / sqrt((l * h));
	}
	return tmp;
}

Fortran

d_m =     private
M_m =     private
D_m =     private
NOTE: d_m, 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_m, h, l, m_m, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d_m / h) ** (1.0d0 / 2.0d0)) * ((d_m / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m_m * d_m_1) / (2.0d0 * d_m)) ** 2.0d0)) * (h / l)))
    if (t_0 <= (-1d-109)) then
        tmp = -sqrt(((d_m * d_m) / (l * h)))
    else if (t_0 <= 1d+126) then
        tmp = abs(d_m) / (sqrt(l) * sqrt(h))
    else
        tmp = d_m / sqrt((l * h))
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
M_m = Math.abs(M);
D_m = Math.abs(D);
assert d_m < h && h < l && l < M_m && M_m < D_m;
public static double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = (Math.pow((d_m / h), (1.0 / 2.0)) * Math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -1e-109) {
		tmp = -Math.sqrt(((d_m * d_m) / (l * h)));
	} else if (t_0 <= 1e+126) {
		tmp = Math.abs(d_m) / (Math.sqrt(l) * Math.sqrt(h));
	} else {
		tmp = d_m / Math.sqrt((l * h));
	}
	return tmp;
}

Python

d_m = math.fabs(d)
M_m = math.fabs(M)
D_m = math.fabs(D)
[d_m, h, l, M_m, D_m] = sort([d_m, h, l, M_m, D_m])
def code(d_m, h, l, M_m, D_m):
	t_0 = (math.pow((d_m / h), (1.0 / 2.0)) * math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= -1e-109:
		tmp = -math.sqrt(((d_m * d_m) / (l * h)))
	elif t_0 <= 1e+126:
		tmp = math.fabs(d_m) / (math.sqrt(l) * math.sqrt(h))
	else:
		tmp = d_m / math.sqrt((l * h))
	return tmp

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	t_0 = Float64(Float64((Float64(d_m / h) ^ Float64(1.0 / 2.0)) * (Float64(d_m / 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_m)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -1e-109)
		tmp = Float64(-sqrt(Float64(Float64(d_m * d_m) / Float64(l * h))));
	elseif (t_0 <= 1e+126)
		tmp = Float64(abs(d_m) / Float64(sqrt(l) * sqrt(h)));
	else
		tmp = Float64(d_m / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

d_m = abs(d);
M_m = abs(M);
D_m = abs(D);
d_m, h, l, M_m, D_m = num2cell(sort([d_m, h, l, M_m, D_m])){:}
function tmp_2 = code(d_m, h, l, M_m, D_m)
	t_0 = (((d_m / h) ^ (1.0 / 2.0)) * ((d_m / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M_m * D_m) / (2.0 * d_m)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= -1e-109)
		tmp = -sqrt(((d_m * d_m) / (l * h)));
	elseif (t_0 <= 1e+126)
		tmp = abs(d_m) / (sqrt(l) * sqrt(h));
	else
		tmp = d_m / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d$95$m / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d$95$m / 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$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-109], (-N[Sqrt[N[(N[(d$95$m * d$95$m), $MachinePrecision] / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t$95$0, 1e+126], N[(N[Abs[d$95$m], $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d$95$m / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $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)))) < -9.9999999999999999e-110

   1. Initial program 35.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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. sqrt-unprod N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

      7. lift-/.f64 21.4

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   4. Applied rewrites 21.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

      4. sqrt-prod N/A

         \[\leadsto \frac{\sqrt{h \cdot d} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h} \]

      7. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{d \cdot h}}{h} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{d \cdot h}}{h} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{d \cdot h}}{h} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{d \cdot h}}{h} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{d \cdot h}}{h} \]

      12. *-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{h \cdot d}}{h} \]

      13. lift-*.f64 18.9

          \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{h \cdot d}}{h} \]

   6. Applied rewrites 18.9%

      \[\leadsto \frac{\sqrt{\frac{d}{\ell}} \cdot \sqrt{h \cdot d}}{h} \]

   7. Taylor expanded in h around -inf

      \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      3. *-commutative N/A

         \[\leadsto -\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}} \]

      4. sqrt-div N/A

         \[\leadsto -\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \sqrt{\frac{d}{h}} \]

      5. sqrt-div N/A

         \[\leadsto -\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \frac{\sqrt{d}}{\sqrt{h}} \]

      6. frac-times N/A

         \[\leadsto -\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}} \]

      7. sqrt-prod N/A

         \[\leadsto -\frac{\sqrt{d \cdot d}}{\sqrt{\ell} \cdot \sqrt{h}} \]

      8. rem-sqrt-square-rev N/A

         \[\leadsto -\frac{\left|d\right|}{\sqrt{\ell} \cdot \sqrt{h}} \]

      9. sqrt-prod N/A

         \[\leadsto -\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto -\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      11. lift-*.f64 N/A

          \[\leadsto -\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      12. lift-/.f64 N/A

          \[\leadsto -\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      13. lift-fabs.f64 9.6

          \[\leadsto -\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      14. lift-*.f64 N/A

          \[\leadsto -\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      15. *-commutative N/A

          \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      16. lower-*.f64 9.6

          \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

   9. Applied rewrites 9.6%

      \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]
   10. Step-by-step derivation

       1. lift-fabs.f64 N/A

          \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

       2. lift-/.f64 N/A

          \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

       3. rem-sqrt-square-rev N/A

          \[\leadsto -\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}} \]

       4. pow2 N/A

          \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

       5. lift-*.f64 N/A

          \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

       6. lift-sqrt.f64 N/A

          \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

       7. sqrt-undiv N/A

          \[\leadsto -\sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

       8. pow2 N/A

          \[\leadsto -\sqrt{\frac{d \cdot d}{h \cdot \ell}} \]

       9. frac-times N/A

          \[\leadsto -\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

       10. rem-square-sqrt N/A

           \[\leadsto -\sqrt{\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

       11. sqrt-unprod N/A

           \[\leadsto -\sqrt{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

       12. frac-2neg N/A

           \[\leadsto -\sqrt{\left(\sqrt{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

       13. sqrt-undiv N/A

           \[\leadsto -\sqrt{\left(\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

       14. associate-*l/ N/A

           \[\leadsto -\sqrt{\frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot \sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}} \]

       15. sqrt-unprod N/A

           \[\leadsto -\sqrt{\frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)} \]

       16. frac-2neg N/A

           \[\leadsto -\sqrt{\frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot \left(\sqrt{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}} \cdot \sqrt{\frac{d}{\ell}}\right)} \]

       17. sqrt-undiv N/A

           \[\leadsto -\sqrt{\frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot \left(\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot \sqrt{\frac{d}{\ell}}\right)} \]

       18. associate-*l/ N/A

           \[\leadsto -\sqrt{\frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot \frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{\mathsf{neg}\left(h\right)}}} \]

   11. Applied rewrites 11.5%

       \[\leadsto -\sqrt{\frac{d \cdot d}{\ell \cdot h}} \]

   if -9.9999999999999999e-110 < (*.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)))) < 9.99999999999999925e125

   1. Initial program 35.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. frac-2neg N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto \left(\frac{\sqrt{\color{blue}{-d}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\sqrt{-d}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-neg.f64 0.0

          \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{\color{blue}{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 0.0%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-1 \cdot h}}} \]

   6. Applied rewrites 42.9%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      3. sqrt-prod N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell} \cdot \sqrt{\color{blue}{h}}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell} \cdot \sqrt{h}} \]

      6. lift-*.f64 25.0

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

   8. Applied rewrites 25.0%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

   if 9.99999999999999925e125 < (*.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 35.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

   4. Applied rewrites 28.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot h\right) \cdot M\right) \cdot M}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right)} \]

   5. Taylor expanded in D around inf

      \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right) \cdot {D}^{\color{blue}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right) \cdot {D}^{\color{blue}{2}} \]

   7. Applied rewrites 43.5%

      \[\leadsto \mathsf{fma}\left(\frac{\left(M \cdot M\right) \cdot h}{\left(\sqrt{\ell \cdot h} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\ell \cdot h} \cdot \left(D \cdot D\right)}\right) \cdot \color{blue}{\left(D \cdot D\right)} \]

   8. Taylor expanded in d around inf

      \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   9. Step-by-step derivation

      1. sqrt-prod N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

      4. *-commutative N/A

         \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

      5. lift-*.f64 42.9

         \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

   10. Applied rewrites 42.9%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 48.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ t_1 := \left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-109}:\\ \;\;\;\;-\frac{\left|d\_m\right|}{t\_0}\\ \mathbf{elif}\;t\_1 \leq 10^{+126}:\\ \;\;\;\;\frac{\left|d\_m\right|}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d\_m}{t\_0}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h)))
        (t_1
         (*
          (* (pow (/ d_m h) (/ 1.0 2.0)) (pow (/ d_m l) (/ 1.0 2.0)))
          (-
           1.0
           (*
            (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0))
            (/ h l))))))
   (if (<= t_1 -1e-109)
     (- (/ (fabs d_m) t_0))
     (if (<= t_1 1e+126) (/ (fabs d_m) (* (sqrt l) (sqrt h))) (/ d_m t_0)))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((l * h));
	double t_1 = (pow((d_m / h), (1.0 / 2.0)) * pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -1e-109) {
		tmp = -(fabs(d_m) / t_0);
	} else if (t_1 <= 1e+126) {
		tmp = fabs(d_m) / (sqrt(l) * sqrt(h));
	} else {
		tmp = d_m / t_0;
	}
	return tmp;
}

Fortran

d_m =     private
M_m =     private
D_m =     private
NOTE: d_m, 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_m, h, l, m_m, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((l * h))
    t_1 = (((d_m / h) ** (1.0d0 / 2.0d0)) * ((d_m / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m_m * d_m_1) / (2.0d0 * d_m)) ** 2.0d0)) * (h / l)))
    if (t_1 <= (-1d-109)) then
        tmp = -(abs(d_m) / t_0)
    else if (t_1 <= 1d+126) then
        tmp = abs(d_m) / (sqrt(l) * sqrt(h))
    else
        tmp = d_m / t_0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
M_m = Math.abs(M);
D_m = Math.abs(D);
assert d_m < h && h < l && l < M_m && M_m < D_m;
public static double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((l * h));
	double t_1 = (Math.pow((d_m / h), (1.0 / 2.0)) * Math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -1e-109) {
		tmp = -(Math.abs(d_m) / t_0);
	} else if (t_1 <= 1e+126) {
		tmp = Math.abs(d_m) / (Math.sqrt(l) * Math.sqrt(h));
	} else {
		tmp = d_m / t_0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
M_m = math.fabs(M)
D_m = math.fabs(D)
[d_m, h, l, M_m, D_m] = sort([d_m, h, l, M_m, D_m])
def code(d_m, h, l, M_m, D_m):
	t_0 = math.sqrt((l * h))
	t_1 = (math.pow((d_m / h), (1.0 / 2.0)) * math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) * (h / l)))
	tmp = 0
	if t_1 <= -1e-109:
		tmp = -(math.fabs(d_m) / t_0)
	elif t_1 <= 1e+126:
		tmp = math.fabs(d_m) / (math.sqrt(l) * math.sqrt(h))
	else:
		tmp = d_m / t_0
	return tmp

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	t_0 = sqrt(Float64(l * h))
	t_1 = Float64(Float64((Float64(d_m / h) ^ Float64(1.0 / 2.0)) * (Float64(d_m / 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_m)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= -1e-109)
		tmp = Float64(-Float64(abs(d_m) / t_0));
	elseif (t_1 <= 1e+126)
		tmp = Float64(abs(d_m) / Float64(sqrt(l) * sqrt(h)));
	else
		tmp = Float64(d_m / t_0);
	end
	return tmp
end

MATLAB

d_m = abs(d);
M_m = abs(M);
D_m = abs(D);
d_m, h, l, M_m, D_m = num2cell(sort([d_m, h, l, M_m, D_m])){:}
function tmp_2 = code(d_m, h, l, M_m, D_m)
	t_0 = sqrt((l * h));
	t_1 = (((d_m / h) ^ (1.0 / 2.0)) * ((d_m / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M_m * D_m) / (2.0 * d_m)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_1 <= -1e-109)
		tmp = -(abs(d_m) / t_0);
	elseif (t_1 <= 1e+126)
		tmp = abs(d_m) / (sqrt(l) * sqrt(h));
	else
		tmp = d_m / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d$95$m / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d$95$m / 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$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-109], (-N[(N[Abs[d$95$m], $MachinePrecision] / t$95$0), $MachinePrecision]), If[LessEqual[t$95$1, 1e+126], N[(N[Abs[d$95$m], $MachinePrecision] / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d$95$m / t$95$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)))) < -9.9999999999999999e-110

   1. Initial program 35.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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. sqrt-unprod N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

      7. lift-/.f64 21.4

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   4. Applied rewrites 21.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around -inf

      \[\leadsto -1 \cdot \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{{d}^{2}}{h \cdot \ell}}\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}}\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right) \]

      5. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \]

      7. lower-neg.f64 N/A

         \[\leadsto -\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      8. lift-/.f64 N/A

         \[\leadsto -\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      9. sqrt-unprod N/A

         \[\leadsto -\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      10. lift-/.f64 N/A

          \[\leadsto -\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      11. times-frac N/A

          \[\leadsto -\sqrt{\frac{d \cdot d}{h \cdot \ell}} \]

      12. pow2 N/A

          \[\leadsto -\sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      13. sqrt-div N/A

          \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      14. pow1/2 N/A

          \[\leadsto -\frac{{\left({d}^{2}\right)}^{\frac{1}{2}}}{\sqrt{h \cdot \ell}} \]

      15. metadata-eval N/A

          \[\leadsto -\frac{{\left({d}^{2}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{h \cdot \ell}} \]

      16. sqrt-unprod N/A

          \[\leadsto -\frac{{\left({d}^{2}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

   7. Applied rewrites 9.6%

      \[\leadsto -\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   if -9.9999999999999999e-110 < (*.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)))) < 9.99999999999999925e125

   1. Initial program 35.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left({\color{blue}{\left(\frac{d}{h}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-pow.f64 N/A

         \[\leadsto \left(\color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. metadata-eval N/A

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. pow1/2 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      6. frac-2neg N/A

         \[\leadsto \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(d\right)}{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      7. sqrt-div N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)}}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \left(\frac{\color{blue}{\sqrt{\mathsf{neg}\left(d\right)}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto \left(\frac{\sqrt{\color{blue}{-d}}}{\sqrt{\mathsf{neg}\left(h\right)}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \left(\frac{\sqrt{-d}}{\color{blue}{\sqrt{\mathsf{neg}\left(h\right)}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. lower-neg.f64 0.0

          \[\leadsto \left(\frac{\sqrt{-d}}{\sqrt{\color{blue}{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 0.0%

      \[\leadsto \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{neg}\left(d\right)} \cdot \sqrt{\frac{d}{\ell}}}{\sqrt{-1 \cdot h}}} \]

   6. Applied rewrites 42.9%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\ell \cdot h}}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      3. sqrt-prod N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell} \cdot \sqrt{\color{blue}{h}}} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell} \cdot \sqrt{h}} \]

      6. lift-*.f64 25.0

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

   8. Applied rewrites 25.0%

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell} \cdot \color{blue}{\sqrt{h}}} \]

   if 9.99999999999999925e125 < (*.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 35.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

   4. Applied rewrites 28.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot h\right) \cdot M\right) \cdot M}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right)} \]

   5. Taylor expanded in D around inf

      \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right) \cdot {D}^{\color{blue}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right) \cdot {D}^{\color{blue}{2}} \]

   7. Applied rewrites 43.5%

      \[\leadsto \mathsf{fma}\left(\frac{\left(M \cdot M\right) \cdot h}{\left(\sqrt{\ell \cdot h} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\ell \cdot h} \cdot \left(D \cdot D\right)}\right) \cdot \color{blue}{\left(D \cdot D\right)} \]

   8. Taylor expanded in d around inf

      \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   9. Step-by-step derivation

      1. sqrt-prod N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

      4. *-commutative N/A

         \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

      5. lift-*.f64 42.9

         \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

   10. Applied rewrites 42.9%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 46.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;\left({\left(\frac{d\_m}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d\_m}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d\_m}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -1 \cdot 10^{-109}:\\ \;\;\;\;-\frac{\left|d\_m\right|}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{d\_m}{t\_0}\\ \end{array} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<=
        (*
         (* (pow (/ d_m h) (/ 1.0 2.0)) (pow (/ d_m l) (/ 1.0 2.0)))
         (-
          1.0
          (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d_m)) 2.0)) (/ h l))))
        -1e-109)
     (- (/ (fabs d_m) t_0))
     (/ d_m t_0))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (((pow((d_m / h), (1.0 / 2.0)) * pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) * (h / l)))) <= -1e-109) {
		tmp = -(fabs(d_m) / t_0);
	} else {
		tmp = d_m / t_0;
	}
	return tmp;
}

Fortran

d_m =     private
M_m =     private
D_m =     private
NOTE: d_m, 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_m, h, l, m_m, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l * h))
    if (((((d_m / h) ** (1.0d0 / 2.0d0)) * ((d_m / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m_m * d_m_1) / (2.0d0 * d_m)) ** 2.0d0)) * (h / l)))) <= (-1d-109)) then
        tmp = -(abs(d_m) / t_0)
    else
        tmp = d_m / t_0
    end if
    code = tmp
end function

Java

d_m = Math.abs(d);
M_m = Math.abs(M);
D_m = Math.abs(D);
assert d_m < h && h < l && l < M_m && M_m < D_m;
public static double code(double d_m, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((l * h));
	double tmp;
	if (((Math.pow((d_m / h), (1.0 / 2.0)) * Math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) * (h / l)))) <= -1e-109) {
		tmp = -(Math.abs(d_m) / t_0);
	} else {
		tmp = d_m / t_0;
	}
	return tmp;
}

Python

d_m = math.fabs(d)
M_m = math.fabs(M)
D_m = math.fabs(D)
[d_m, h, l, M_m, D_m] = sort([d_m, h, l, M_m, D_m])
def code(d_m, h, l, M_m, D_m):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if ((math.pow((d_m / h), (1.0 / 2.0)) * math.pow((d_m / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M_m * D_m) / (2.0 * d_m)), 2.0)) * (h / l)))) <= -1e-109:
		tmp = -(math.fabs(d_m) / t_0)
	else:
		tmp = d_m / t_0
	return tmp

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (Float64(Float64((Float64(d_m / h) ^ Float64(1.0 / 2.0)) * (Float64(d_m / 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_m)) ^ 2.0)) * Float64(h / l)))) <= -1e-109)
		tmp = Float64(-Float64(abs(d_m) / t_0));
	else
		tmp = Float64(d_m / t_0);
	end
	return tmp
end

MATLAB

d_m = abs(d);
M_m = abs(M);
D_m = abs(D);
d_m, h, l, M_m, D_m = num2cell(sort([d_m, h, l, M_m, D_m])){:}
function tmp_2 = code(d_m, h, l, M_m, D_m)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if (((((d_m / h) ^ (1.0 / 2.0)) * ((d_m / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M_m * D_m) / (2.0 * d_m)) ^ 2.0)) * (h / l)))) <= -1e-109)
		tmp = -(abs(d_m) / t_0);
	else
		tmp = d_m / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(d$95$m / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d$95$m / 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$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-109], (-N[(N[Abs[d$95$m], $MachinePrecision] / t$95$0), $MachinePrecision]), N[(d$95$m / t$95$0), $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)))) < -9.9999999999999999e-110

   1. Initial program 35.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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. sqrt-unprod N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(d \cdot h\right) \cdot \frac{d}{\ell}}}{h} \]

      5. *-commutative N/A

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

      7. lift-/.f64 21.4

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   4. Applied rewrites 21.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around -inf

      \[\leadsto -1 \cdot \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{{d}^{2}}{h \cdot \ell}}\right) \]

      2. pow2 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{d \cdot d}{h \cdot \ell}}\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}}\right) \]

      5. sqrt-unprod N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \]

      7. lower-neg.f64 N/A

         \[\leadsto -\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      8. lift-/.f64 N/A

         \[\leadsto -\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}} \]

      9. sqrt-unprod N/A

         \[\leadsto -\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      10. lift-/.f64 N/A

          \[\leadsto -\sqrt{\frac{d}{h} \cdot \frac{d}{\ell}} \]

      11. times-frac N/A

          \[\leadsto -\sqrt{\frac{d \cdot d}{h \cdot \ell}} \]

      12. pow2 N/A

          \[\leadsto -\sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      13. sqrt-div N/A

          \[\leadsto -\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      14. pow1/2 N/A

          \[\leadsto -\frac{{\left({d}^{2}\right)}^{\frac{1}{2}}}{\sqrt{h \cdot \ell}} \]

      15. metadata-eval N/A

          \[\leadsto -\frac{{\left({d}^{2}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{h \cdot \ell}} \]

      16. sqrt-unprod N/A

          \[\leadsto -\frac{{\left({d}^{2}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{h} \cdot \sqrt{\ell}} \]

   7. Applied rewrites 9.6%

      \[\leadsto -\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   if -9.9999999999999999e-110 < (*.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 35.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

   4. Applied rewrites 28.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot h\right) \cdot M\right) \cdot M}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right)} \]

   5. Taylor expanded in D around inf

      \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right) \cdot {D}^{\color{blue}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right) \cdot {D}^{\color{blue}{2}} \]

   7. Applied rewrites 43.5%

      \[\leadsto \mathsf{fma}\left(\frac{\left(M \cdot M\right) \cdot h}{\left(\sqrt{\ell \cdot h} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\ell \cdot h} \cdot \left(D \cdot D\right)}\right) \cdot \color{blue}{\left(D \cdot D\right)} \]

   8. Taylor expanded in d around inf

      \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]
   9. Step-by-step derivation

      1. sqrt-prod N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

      4. *-commutative N/A

         \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

      5. lift-*.f64 42.9

         \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

   10. Applied rewrites 42.9%

       \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 42.9% accurate, 10.2× speedup

Math

\[\begin{array}{l} d_m = \left|d\right| \\ M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d_m, h, l, M_m, D_m] = \mathsf{sort}([d_m, h, l, M_m, D_m])\\ \\ \frac{d\_m}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

d_m = (fabs.f64 d)
M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d_m h l M_m D_m) :precision binary64 (/ d_m (sqrt (* l h))))

C

d_m = fabs(d);
M_m = fabs(M);
D_m = fabs(D);
assert(d_m < h && h < l && l < M_m && M_m < D_m);
double code(double d_m, double h, double l, double M_m, double D_m) {
	return d_m / sqrt((l * h));
}

Fortran

d_m =     private
M_m =     private
D_m =     private
NOTE: d_m, 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_m, h, l, m_m, d_m_1)
use fmin_fmax_functions
    real(8), intent (in) :: d_m
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m_1
    code = d_m / sqrt((l * h))
end function

Java

d_m = Math.abs(d);
M_m = Math.abs(M);
D_m = Math.abs(D);
assert d_m < h && h < l && l < M_m && M_m < D_m;
public static double code(double d_m, double h, double l, double M_m, double D_m) {
	return d_m / Math.sqrt((l * h));
}

Python

d_m = math.fabs(d)
M_m = math.fabs(M)
D_m = math.fabs(D)
[d_m, h, l, M_m, D_m] = sort([d_m, h, l, M_m, D_m])
def code(d_m, h, l, M_m, D_m):
	return d_m / math.sqrt((l * h))

Julia

d_m = abs(d)
M_m = abs(M)
D_m = abs(D)
d_m, h, l, M_m, D_m = sort([d_m, h, l, M_m, D_m])
function code(d_m, h, l, M_m, D_m)
	return Float64(d_m / sqrt(Float64(l * h)))
end

MATLAB

d_m = abs(d);
M_m = abs(M);
D_m = abs(D);
d_m, h, l, M_m, D_m = num2cell(sort([d_m, h, l, M_m, D_m])){:}
function tmp = code(d_m, h, l, M_m, D_m)
	tmp = d_m / sqrt((l * h));
end

Wolfram

d_m = N[Abs[d], $MachinePrecision]
M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d_m, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d$95$m_, h_, l_, M$95$m_, D$95$m_] := N[(d$95$m / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.4%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

2. Taylor expanded in l around inf

   \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}} \]

4. Applied rewrites 28.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{\left(\left(\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot h\right) \cdot M\right) \cdot M}{\left(d \cdot d\right) \cdot \ell} \cdot -0.125\right) \cdot D, D, \frac{d}{\sqrt{\ell} \cdot \sqrt{h}}\right)} \]

5. Taylor expanded in D around inf

   \[\leadsto {D}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right) \cdot {D}^{\color{blue}{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot h}{d \cdot \left(\ell \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)\right)} + \frac{d}{{D}^{2} \cdot \left(\sqrt{h} \cdot \sqrt{\ell}\right)}\right) \cdot {D}^{\color{blue}{2}} \]

7. Applied rewrites 43.5%

   \[\leadsto \mathsf{fma}\left(\frac{\left(M \cdot M\right) \cdot h}{\left(\sqrt{\ell \cdot h} \cdot \ell\right) \cdot d}, -0.125, \frac{d}{\sqrt{\ell \cdot h} \cdot \left(D \cdot D\right)}\right) \cdot \color{blue}{\left(D \cdot D\right)} \]

8. Taylor expanded in d around inf

   \[\leadsto \frac{d}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]
9. Step-by-step derivation

   1. sqrt-prod N/A

      \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \]

   4. *-commutative N/A

      \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

   5. lift-*.f64 42.9

      \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

10. Applied rewrites 42.9%

    \[\leadsto \frac{d}{\color{blue}{\sqrt{\ell \cdot h}}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(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)))))