Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.9% → 73.7%

Time: 11.6s

Alternatives: 23

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.9% 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: 73.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{D\_m \cdot M\_m}{d}\\ t_1 := \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\left|d\right|}{\left(\sqrt{\frac{\ell}{h}} \cdot \ell\right) \cdot d} \cdot \frac{\left(D\_m \cdot M\_m\right) \cdot M\_m}{d}\right) \cdot 0.125, D\_m, t\_1\right)\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{+122}:\\ \;\;\;\;t\_1 \cdot \left(1 - \left(\left(t\_0 \cdot t\_0\right) \cdot 0.125\right) \cdot \left(h \cdot \frac{1}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot \frac{\left(\frac{D\_m}{d + d} \cdot M\_m\right) \cdot \left(D\_m \cdot M\_m\right)}{d + d}\right) \cdot \frac{h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ (* D_m M_m) d)) (t_1 (/ (fabs d) (sqrt (* h l)))))
   (if (<= l -5e-31)
     (fma
      (*
       (* (/ (fabs d) (* (* (sqrt (/ l h)) l) d)) (/ (* (* D_m M_m) M_m) d))
       0.125)
      D_m
      t_1)
     (if (<= l 3.4e+122)
       (* t_1 (- 1.0 (* (* (* t_0 t_0) 0.125) (* h (/ 1.0 l)))))
       (*
        (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
        (-
         1.0
         (*
          (* (/ 1.0 2.0) (/ (* (* (/ D_m (+ d d)) M_m) (* D_m M_m)) (+ d d)))
          (/ h l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (D_m * M_m) / d;
	double t_1 = fabs(d) / sqrt((h * l));
	double tmp;
	if (l <= -5e-31) {
		tmp = fma((((fabs(d) / ((sqrt((l / h)) * l) * d)) * (((D_m * M_m) * M_m) / d)) * 0.125), D_m, t_1);
	} else if (l <= 3.4e+122) {
		tmp = t_1 * (1.0 - (((t_0 * t_0) * 0.125) * (h * (1.0 / l))));
	} else {
		tmp = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * ((((D_m / (d + d)) * M_m) * (D_m * M_m)) / (d + d))) * (h / l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(D_m * M_m) / d)
	t_1 = Float64(abs(d) / sqrt(Float64(h * l)))
	tmp = 0.0
	if (l <= -5e-31)
		tmp = fma(Float64(Float64(Float64(abs(d) / Float64(Float64(sqrt(Float64(l / h)) * l) * d)) * Float64(Float64(Float64(D_m * M_m) * M_m) / d)) * 0.125), D_m, t_1);
	elseif (l <= 3.4e+122)
		tmp = Float64(t_1 * Float64(1.0 - Float64(Float64(Float64(t_0 * t_0) * 0.125) * Float64(h * Float64(1.0 / l)))));
	else
		tmp = 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(Float64(Float64(D_m / Float64(d + d)) * M_m) * Float64(D_m * M_m)) / Float64(d + d))) * Float64(h / l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5e-31], N[(N[(N[(N[(N[Abs[d], $MachinePrecision] / N[(N[(N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * D$95$m + t$95$1), $MachinePrecision], If[LessEqual[l, 3.4e+122], N[(t$95$1 * N[(1.0 - N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.125), $MachinePrecision] * N[(h * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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[(N[(N[(N[(D$95$m / N[(d + d), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(D$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(d + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5e-31

   1. Initial program 66.9%

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

   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 48.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 50.1

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

   6. Applied rewrites 50.1%

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

   7. Taylor expanded in h around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 40.1%

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

       1. lift-/.f64 N/A

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

       2. lift-fabs.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. lift-sqrt.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. associate-*r* N/A

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

       11. times-frac N/A

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

       12. lower-*.f64 N/A

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

   11. Applied rewrites 51.1%

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

   if -5e-31 < l < 3.4e122

   1. Initial program 66.9%

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

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

      11. unpow2 N/A

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

      12. fabs-pow2-rev N/A

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

      13. sqrt-div-sound-left N/A

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

      14. fabs-pow2-rev N/A

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

      15. unpow2 N/A

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

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

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

      17. lower-/.f64 N/A

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

      18. lower-fabs.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-*.f64 69.8

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

   3. Applied rewrites 69.8%

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

   4. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 46.5

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

   6. Applied rewrites 46.5%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.125\right)} \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. unswap-sqr N/A

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

      7. times-frac N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 69.8

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

   8. Applied rewrites 69.8%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 69.8

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

   10. Applied rewrites 69.8%

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

   if 3.4e122 < l

   1. Initial program 66.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. count-2-rev N/A

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

      18. lower-+.f64 65.0

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

   3. Applied rewrites 65.0%

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

Alternative 2: 73.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{D\_m \cdot M\_m}{d}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot 0.125\\ t_2 := \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\left|d\right|}{\left(\sqrt{\frac{\ell}{h}} \cdot \ell\right) \cdot d} \cdot \frac{\left(D\_m \cdot M\_m\right) \cdot M\_m}{d}\right) \cdot 0.125, D\_m, t\_2\right)\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{+122}:\\ \;\;\;\;t\_2 \cdot \left(1 - t\_1 \cdot \left(h \cdot \frac{1}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right) \cdot \left(1 - t\_1 \cdot \frac{h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ (* D_m M_m) d))
        (t_1 (* (* t_0 t_0) 0.125))
        (t_2 (/ (fabs d) (sqrt (* h l)))))
   (if (<= l -5e-31)
     (fma
      (*
       (* (/ (fabs d) (* (* (sqrt (/ l h)) l) d)) (/ (* (* D_m M_m) M_m) d))
       0.125)
      D_m
      t_2)
     (if (<= l 3.4e+122)
       (* t_2 (- 1.0 (* t_1 (* h (/ 1.0 l)))))
       (* (* (sqrt (/ d l)) (sqrt (/ d h))) (- 1.0 (* t_1 (/ h l))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (D_m * M_m) / d;
	double t_1 = (t_0 * t_0) * 0.125;
	double t_2 = fabs(d) / sqrt((h * l));
	double tmp;
	if (l <= -5e-31) {
		tmp = fma((((fabs(d) / ((sqrt((l / h)) * l) * d)) * (((D_m * M_m) * M_m) / d)) * 0.125), D_m, t_2);
	} else if (l <= 3.4e+122) {
		tmp = t_2 * (1.0 - (t_1 * (h * (1.0 / l))));
	} else {
		tmp = (sqrt((d / l)) * sqrt((d / h))) * (1.0 - (t_1 * (h / l)));
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(D_m * M_m) / d)
	t_1 = Float64(Float64(t_0 * t_0) * 0.125)
	t_2 = Float64(abs(d) / sqrt(Float64(h * l)))
	tmp = 0.0
	if (l <= -5e-31)
		tmp = fma(Float64(Float64(Float64(abs(d) / Float64(Float64(sqrt(Float64(l / h)) * l) * d)) * Float64(Float64(Float64(D_m * M_m) * M_m) / d)) * 0.125), D_m, t_2);
	elseif (l <= 3.4e+122)
		tmp = Float64(t_2 * Float64(1.0 - Float64(t_1 * Float64(h * Float64(1.0 / l)))));
	else
		tmp = Float64(Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h))) * Float64(1.0 - Float64(t_1 * Float64(h / l))));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.125), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5e-31], N[(N[(N[(N[(N[Abs[d], $MachinePrecision] / N[(N[(N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * D$95$m + t$95$2), $MachinePrecision], If[LessEqual[l, 3.4e+122], N[(t$95$2 * N[(1.0 - N[(t$95$1 * N[(h * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(t$95$1 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5e-31

   1. Initial program 66.9%

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

   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 48.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 50.1

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

   6. Applied rewrites 50.1%

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

   7. Taylor expanded in h around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 40.1%

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

       1. lift-/.f64 N/A

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

       2. lift-fabs.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. lift-sqrt.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. associate-*r* N/A

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

       11. times-frac N/A

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

       12. lower-*.f64 N/A

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

   11. Applied rewrites 51.1%

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

   if -5e-31 < l < 3.4e122

   1. Initial program 66.9%

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

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

      11. unpow2 N/A

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

      12. fabs-pow2-rev N/A

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

      13. sqrt-div-sound-left N/A

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

      14. fabs-pow2-rev N/A

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

      15. unpow2 N/A

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

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

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

      17. lower-/.f64 N/A

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

      18. lower-fabs.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-*.f64 69.8

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

   3. Applied rewrites 69.8%

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

   4. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 46.5

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

   6. Applied rewrites 46.5%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.125\right)} \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. unswap-sqr N/A

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

      7. times-frac N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 69.8

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

   8. Applied rewrites 69.8%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 69.8

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

   10. Applied rewrites 69.8%

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

   if 3.4e122 < l

   1. Initial program 66.9%

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

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

      11. unpow2 N/A

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

      12. fabs-pow2-rev N/A

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

      13. sqrt-div-sound-left N/A

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

      14. fabs-pow2-rev N/A

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

      15. unpow2 N/A

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

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

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

      17. lower-/.f64 N/A

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

      18. lower-fabs.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-*.f64 69.8

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

   3. Applied rewrites 69.8%

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

   4. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 46.5

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

   6. Applied rewrites 46.5%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.125\right)} \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. unswap-sqr N/A

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

      7. times-frac N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 69.8

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

   8. Applied rewrites 69.8%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

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

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

      6. pow2 N/A

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

      7. sqrt-undiv N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. times-frac N/A

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

      11. sqrt-unprod N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-/.f64 66.9

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

   10. Applied rewrites 66.9%

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

Alternative 3: 72.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{D\_m \cdot M\_m}{d}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot 0.125\\ \mathbf{if}\;h \leq 20000000000:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - t\_1 \cdot \left(h \cdot \frac{1}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\frac{\ell}{h}} \cdot h} \cdot \left(1 - t\_1 \cdot \frac{h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ (* D_m M_m) d)) (t_1 (* (* t_0 t_0) 0.125)))
   (if (<= h 20000000000.0)
     (* (/ (fabs d) (sqrt (* h l))) (- 1.0 (* t_1 (* h (/ 1.0 l)))))
     (* (/ (fabs d) (* (sqrt (/ l h)) h)) (- 1.0 (* t_1 (/ h l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (D_m * M_m) / d;
	double t_1 = (t_0 * t_0) * 0.125;
	double tmp;
	if (h <= 20000000000.0) {
		tmp = (fabs(d) / sqrt((h * l))) * (1.0 - (t_1 * (h * (1.0 / l))));
	} else {
		tmp = (fabs(d) / (sqrt((l / h)) * h)) * (1.0 - (t_1 * (h / l)));
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (d_m * m_m) / d
    t_1 = (t_0 * t_0) * 0.125d0
    if (h <= 20000000000.0d0) then
        tmp = (abs(d) / sqrt((h * l))) * (1.0d0 - (t_1 * (h * (1.0d0 / l))))
    else
        tmp = (abs(d) / (sqrt((l / h)) * h)) * (1.0d0 - (t_1 * (h / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (D_m * M_m) / d;
	double t_1 = (t_0 * t_0) * 0.125;
	double tmp;
	if (h <= 20000000000.0) {
		tmp = (Math.abs(d) / Math.sqrt((h * l))) * (1.0 - (t_1 * (h * (1.0 / l))));
	} else {
		tmp = (Math.abs(d) / (Math.sqrt((l / h)) * h)) * (1.0 - (t_1 * (h / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (D_m * M_m) / d
	t_1 = (t_0 * t_0) * 0.125
	tmp = 0
	if h <= 20000000000.0:
		tmp = (math.fabs(d) / math.sqrt((h * l))) * (1.0 - (t_1 * (h * (1.0 / l))))
	else:
		tmp = (math.fabs(d) / (math.sqrt((l / h)) * h)) * (1.0 - (t_1 * (h / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64(D_m * M_m) / d)
	t_1 = Float64(Float64(t_0 * t_0) * 0.125)
	tmp = 0.0
	if (h <= 20000000000.0)
		tmp = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * Float64(1.0 - Float64(t_1 * Float64(h * Float64(1.0 / l)))));
	else
		tmp = Float64(Float64(abs(d) / Float64(sqrt(Float64(l / h)) * h)) * Float64(1.0 - Float64(t_1 * Float64(h / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (D_m * M_m) / d;
	t_1 = (t_0 * t_0) * 0.125;
	tmp = 0.0;
	if (h <= 20000000000.0)
		tmp = (abs(d) / sqrt((h * l))) * (1.0 - (t_1 * (h * (1.0 / l))));
	else
		tmp = (abs(d) / (sqrt((l / h)) * h)) * (1.0 - (t_1 * (h / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 0.125), $MachinePrecision]}, If[LessEqual[h, 20000000000.0], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(t$95$1 * N[(h * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[(N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(t$95$1 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if h < 2e10

   1. Initial program 66.9%

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

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

      11. unpow2 N/A

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

      12. fabs-pow2-rev N/A

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

      13. sqrt-div-sound-left N/A

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

      14. fabs-pow2-rev N/A

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

      15. unpow2 N/A

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

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

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

      17. lower-/.f64 N/A

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

      18. lower-fabs.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-*.f64 69.8

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

   3. Applied rewrites 69.8%

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

   4. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 46.5

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

   6. Applied rewrites 46.5%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.125\right)} \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. unswap-sqr N/A

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

      7. times-frac N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 69.8

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

   8. Applied rewrites 69.8%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 69.8

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

   10. Applied rewrites 69.8%

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

   if 2e10 < h

   1. Initial program 66.9%

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

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

      11. unpow2 N/A

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

      12. fabs-pow2-rev N/A

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

      13. sqrt-div-sound-left N/A

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

      14. fabs-pow2-rev N/A

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

      15. unpow2 N/A

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

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

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

      17. lower-/.f64 N/A

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

      18. lower-fabs.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-*.f64 69.8

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

   3. Applied rewrites 69.8%

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

   4. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 46.5

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

   6. Applied rewrites 46.5%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.125\right)} \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. unswap-sqr N/A

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

      7. times-frac N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 69.8

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

   8. Applied rewrites 69.8%

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

   9. Taylor expanded in h around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lift-sqrt.f64 N/A

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

       4. lift-/.f64 38.2

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

   11. Applied rewrites 38.2%

       \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\frac{\ell}{h}} \cdot h}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot \frac{D \cdot M}{d}\right) \cdot 0.125\right) \cdot \frac{h}{\ell}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 70.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_2 := \frac{D\_m \cdot M\_m}{d}\\ t_3 := \left(t\_2 \cdot t\_2\right) \cdot 0.125\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\left(\left(\left(t\_0 \cdot h\right) \cdot \left(\frac{M\_m}{d} \cdot \frac{M\_m}{\ell \cdot d}\right)\right) \cdot -0.125\right) \cdot D\_m\right) \cdot D\_m\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-79}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\frac{h}{\ell}} \cdot \ell} \cdot \left(1 - t\_3 \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 - t\_3 \cdot \left(h \cdot \frac{1}{\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ (fabs d) (sqrt (* h l))))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l)))))
        (t_2 (/ (* D_m M_m) d))
        (t_3 (* (* t_2 t_2) 0.125)))
   (if (<= t_1 (- INFINITY))
     (* (* (* (* (* t_0 h) (* (/ M_m d) (/ M_m (* l d)))) -0.125) D_m) D_m)
     (if (<= t_1 5e-79)
       (* (/ (fabs d) (* (sqrt (/ h l)) l)) (- 1.0 (* t_3 (/ h l))))
       (* t_0 (- 1.0 (* t_3 (* h (/ 1.0 l)))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = fabs(d) / sqrt((h * l));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = (D_m * M_m) / d;
	double t_3 = (t_2 * t_2) * 0.125;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((((t_0 * h) * ((M_m / d) * (M_m / (l * d)))) * -0.125) * D_m) * D_m;
	} else if (t_1 <= 5e-79) {
		tmp = (fabs(d) / (sqrt((h / l)) * l)) * (1.0 - (t_3 * (h / l)));
	} else {
		tmp = t_0 * (1.0 - (t_3 * (h * (1.0 / l))));
	}
	return tmp;
}

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.abs(d) / Math.sqrt((h * l));
	double t_1 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double t_2 = (D_m * M_m) / d;
	double t_3 = (t_2 * t_2) * 0.125;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = ((((t_0 * h) * ((M_m / d) * (M_m / (l * d)))) * -0.125) * D_m) * D_m;
	} else if (t_1 <= 5e-79) {
		tmp = (Math.abs(d) / (Math.sqrt((h / l)) * l)) * (1.0 - (t_3 * (h / l)));
	} else {
		tmp = t_0 * (1.0 - (t_3 * (h * (1.0 / l))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.fabs(d) / math.sqrt((h * l))
	t_1 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	t_2 = (D_m * M_m) / d
	t_3 = (t_2 * t_2) * 0.125
	tmp = 0
	if t_1 <= -math.inf:
		tmp = ((((t_0 * h) * ((M_m / d) * (M_m / (l * d)))) * -0.125) * D_m) * D_m
	elif t_1 <= 5e-79:
		tmp = (math.fabs(d) / (math.sqrt((h / l)) * l)) * (1.0 - (t_3 * (h / l)))
	else:
		tmp = t_0 * (1.0 - (t_3 * (h * (1.0 / l))))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(abs(d) / sqrt(Float64(h * l)))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	t_2 = Float64(Float64(D_m * M_m) / d)
	t_3 = Float64(Float64(t_2 * t_2) * 0.125)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(Float64(t_0 * h) * Float64(Float64(M_m / d) * Float64(M_m / Float64(l * d)))) * -0.125) * D_m) * D_m);
	elseif (t_1 <= 5e-79)
		tmp = Float64(Float64(abs(d) / Float64(sqrt(Float64(h / l)) * l)) * Float64(1.0 - Float64(t_3 * Float64(h / l))));
	else
		tmp = Float64(t_0 * Float64(1.0 - Float64(t_3 * Float64(h * Float64(1.0 / l)))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = abs(d) / sqrt((h * l));
	t_1 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	t_2 = (D_m * M_m) / d;
	t_3 = (t_2 * t_2) * 0.125;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = ((((t_0 * h) * ((M_m / d) * (M_m / (l * d)))) * -0.125) * D_m) * D_m;
	elseif (t_1 <= 5e-79)
		tmp = (abs(d) / (sqrt((h / l)) * l)) * (1.0 - (t_3 * (h / l)));
	else
		tmp = t_0 * (1.0 - (t_3 * (h * (1.0 / l))));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(D$95$m * M$95$m), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 * t$95$2), $MachinePrecision] * 0.125), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(t$95$0 * h), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * N[(M$95$m / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision], If[LessEqual[t$95$1, 5e-79], N[(N[(N[Abs[d], $MachinePrecision] / N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(t$95$3 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 - N[(t$95$3 * N[(h * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $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)))) < -inf.0

   1. Initial program 66.9%

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

   2. Taylor expanded in d around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 26.7%

      \[\leadsto \color{blue}{\left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot -0.125\right) \cdot D\right) \cdot D} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 33.9

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

   6. Applied rewrites 33.9%

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

   if -inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < 4.99999999999999999e-79

   1. Initial program 66.9%

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

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

      11. unpow2 N/A

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

      12. fabs-pow2-rev N/A

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

      13. sqrt-div-sound-left N/A

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

      14. fabs-pow2-rev N/A

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

      15. unpow2 N/A

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

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

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

      17. lower-/.f64 N/A

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

      18. lower-fabs.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-*.f64 69.8

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

   3. Applied rewrites 69.8%

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

   4. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 46.5

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

   6. Applied rewrites 46.5%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.125\right)} \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. unswap-sqr N/A

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

      7. times-frac N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 69.8

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

   8. Applied rewrites 69.8%

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

   9. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lift-/.f64 34.9

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

   11. Applied rewrites 34.9%

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

   if 4.99999999999999999e-79 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.9%

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

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

      11. unpow2 N/A

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

      12. fabs-pow2-rev N/A

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

      13. sqrt-div-sound-left N/A

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

      14. fabs-pow2-rev N/A

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

      15. unpow2 N/A

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

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

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

      17. lower-/.f64 N/A

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

      18. lower-fabs.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-*.f64 69.8

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

   3. Applied rewrites 69.8%

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

   4. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 46.5

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

   6. Applied rewrites 46.5%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.125\right)} \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. unswap-sqr N/A

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

      7. times-frac N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 69.8

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

   8. Applied rewrites 69.8%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 69.8

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

   10. Applied rewrites 69.8%

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

Alternative 5: 69.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (D_m * M_m) / d;
	double t_1 = Math.abs(d) / Math.sqrt((h * l));
	double tmp;
	if (((Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -Double.POSITIVE_INFINITY) {
		tmp = ((((t_1 * h) * ((M_m / d) * (M_m / (l * d)))) * -0.125) * D_m) * D_m;
	} else {
		tmp = t_1 * (1.0 - (((t_0 * t_0) * 0.125) * (h * (1.0 / l))));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (D_m * M_m) / d
	t_1 = math.fabs(d) / math.sqrt((h * l))
	tmp = 0
	if ((math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -math.inf:
		tmp = ((((t_1 * h) * ((M_m / d) * (M_m / (l * d)))) * -0.125) * D_m) * D_m
	else:
		tmp = t_1 * (1.0 - (((t_0 * t_0) * 0.125) * (h * (1.0 / l))))
	return tmp

Julia

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

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (D_m * M_m) / d;
	t_1 = abs(d) / sqrt((h * l));
	tmp = 0.0;
	if (((((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -Inf)
		tmp = ((((t_1 * h) * ((M_m / d) * (M_m / (l * d)))) * -0.125) * D_m) * D_m;
	else
		tmp = t_1 * (1.0 - (((t_0 * t_0) * 0.125) * (h * (1.0 / l))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 66.9%

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

   2. Taylor expanded in d around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 26.7%

      \[\leadsto \color{blue}{\left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot -0.125\right) \cdot D\right) \cdot D} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 33.9

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

   6. Applied rewrites 33.9%

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

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

   1. Initial program 66.9%

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

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

      11. unpow2 N/A

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

      12. fabs-pow2-rev N/A

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

      13. sqrt-div-sound-left N/A

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

      14. fabs-pow2-rev N/A

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

      15. unpow2 N/A

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

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

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

      17. lower-/.f64 N/A

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

      18. lower-fabs.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-*.f64 69.8

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

   3. Applied rewrites 69.8%

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

   4. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 46.5

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

   6. Applied rewrites 46.5%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.125\right)} \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. unswap-sqr N/A

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

      7. times-frac N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 69.8

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

   8. Applied rewrites 69.8%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 69.8

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

   10. Applied rewrites 69.8%

       \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot \frac{D \cdot M}{d}\right) \cdot 0.125\right) \cdot \color{blue}{\left(h \cdot \frac{1}{\ell}\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 69.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (D_m * M_m) / d;
	double t_1 = Math.abs(d) / Math.sqrt((h * l));
	double tmp;
	if (((Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -Double.POSITIVE_INFINITY) {
		tmp = ((((t_1 * h) * ((M_m / d) * (M_m / (l * d)))) * -0.125) * D_m) * D_m;
	} else {
		tmp = t_1 * (1.0 - (((t_0 * t_0) * 0.125) * (h / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (D_m * M_m) / d
	t_1 = math.fabs(d) / math.sqrt((h * l))
	tmp = 0
	if ((math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -math.inf:
		tmp = ((((t_1 * h) * ((M_m / d) * (M_m / (l * d)))) * -0.125) * D_m) * D_m
	else:
		tmp = t_1 * (1.0 - (((t_0 * t_0) * 0.125) * (h / l)))
	return tmp

Julia

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

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (D_m * M_m) / d;
	t_1 = abs(d) / sqrt((h * l));
	tmp = 0.0;
	if (((((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -Inf)
		tmp = ((((t_1 * h) * ((M_m / d) * (M_m / (l * d)))) * -0.125) * D_m) * D_m;
	else
		tmp = t_1 * (1.0 - (((t_0 * t_0) * 0.125) * (h / l)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 66.9%

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

   2. Taylor expanded in d around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 26.7%

      \[\leadsto \color{blue}{\left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot -0.125\right) \cdot D\right) \cdot D} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 33.9

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

   6. Applied rewrites 33.9%

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

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

   1. Initial program 66.9%

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

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

      11. unpow2 N/A

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

      12. fabs-pow2-rev N/A

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

      13. sqrt-div-sound-left N/A

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

      14. fabs-pow2-rev N/A

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

      15. unpow2 N/A

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

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

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

      17. lower-/.f64 N/A

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

      18. lower-fabs.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-*.f64 69.8

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

   3. Applied rewrites 69.8%

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

   4. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 46.5

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

   6. Applied rewrites 46.5%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.125\right)} \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. unswap-sqr N/A

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

      7. times-frac N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 69.8

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

   8. Applied rewrites 69.8%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\frac{D \cdot M}{d} \cdot \frac{D \cdot M}{d}\right) \cdot 0.125\right) \cdot \frac{h}{\ell}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 66.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;M\_m \cdot D\_m \leq 5 \cdot 10^{-163}:\\ \;\;\;\;\frac{\left|d\right|}{t\_0}\\ \mathbf{elif}\;M\_m \cdot D\_m \leq 5 \cdot 10^{+130}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\left(D\_m \cdot M\_m\right) \cdot \frac{D\_m \cdot M\_m}{d \cdot d}\right) \cdot 0.125\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|d\right| \cdot \frac{1 - \left(\left(\left(\left(M\_m \cdot M\_m\right) \cdot D\_m\right) \cdot \frac{D\_m}{d \cdot d}\right) \cdot 0.125\right) \cdot \frac{h}{\ell}}{t\_0}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<= (* M_m D_m) 5e-163)
     (/ (fabs d) t_0)
     (if (<= (* M_m D_m) 5e+130)
       (*
        (/ (fabs d) (sqrt (* h l)))
        (- 1.0 (* (* (* (* D_m M_m) (/ (* D_m M_m) (* d d))) 0.125) (/ h l))))
       (*
        (fabs d)
        (/
         (- 1.0 (* (* (* (* (* M_m M_m) D_m) (/ D_m (* d d))) 0.125) (/ h l)))
         t_0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((l * h));
	double tmp;
	if ((M_m * D_m) <= 5e-163) {
		tmp = fabs(d) / t_0;
	} else if ((M_m * D_m) <= 5e+130) {
		tmp = (fabs(d) / sqrt((h * l))) * (1.0 - ((((D_m * M_m) * ((D_m * M_m) / (d * d))) * 0.125) * (h / l)));
	} else {
		tmp = fabs(d) * ((1.0 - (((((M_m * M_m) * D_m) * (D_m / (d * d))) * 0.125) * (h / l))) / t_0);
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l * h))
    if ((m_m * d_m) <= 5d-163) then
        tmp = abs(d) / t_0
    else if ((m_m * d_m) <= 5d+130) then
        tmp = (abs(d) / sqrt((h * l))) * (1.0d0 - ((((d_m * m_m) * ((d_m * m_m) / (d * d))) * 0.125d0) * (h / l)))
    else
        tmp = abs(d) * ((1.0d0 - (((((m_m * m_m) * d_m) * (d_m / (d * d))) * 0.125d0) * (h / l))) / t_0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((l * h));
	double tmp;
	if ((M_m * D_m) <= 5e-163) {
		tmp = Math.abs(d) / t_0;
	} else if ((M_m * D_m) <= 5e+130) {
		tmp = (Math.abs(d) / Math.sqrt((h * l))) * (1.0 - ((((D_m * M_m) * ((D_m * M_m) / (d * d))) * 0.125) * (h / l)));
	} else {
		tmp = Math.abs(d) * ((1.0 - (((((M_m * M_m) * D_m) * (D_m / (d * d))) * 0.125) * (h / l))) / t_0);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if (M_m * D_m) <= 5e-163:
		tmp = math.fabs(d) / t_0
	elif (M_m * D_m) <= 5e+130:
		tmp = (math.fabs(d) / math.sqrt((h * l))) * (1.0 - ((((D_m * M_m) * ((D_m * M_m) / (d * d))) * 0.125) * (h / l)))
	else:
		tmp = math.fabs(d) * ((1.0 - (((((M_m * M_m) * D_m) * (D_m / (d * d))) * 0.125) * (h / l))) / t_0)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (Float64(M_m * D_m) <= 5e-163)
		tmp = Float64(abs(d) / t_0);
	elseif (Float64(M_m * D_m) <= 5e+130)
		tmp = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * Float64(1.0 - Float64(Float64(Float64(Float64(D_m * M_m) * Float64(Float64(D_m * M_m) / Float64(d * d))) * 0.125) * Float64(h / l))));
	else
		tmp = Float64(abs(d) * Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(M_m * M_m) * D_m) * Float64(D_m / Float64(d * d))) * 0.125) * Float64(h / l))) / t_0));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if ((M_m * D_m) <= 5e-163)
		tmp = abs(d) / t_0;
	elseif ((M_m * D_m) <= 5e+130)
		tmp = (abs(d) / sqrt((h * l))) * (1.0 - ((((D_m * M_m) * ((D_m * M_m) / (d * d))) * 0.125) * (h / l)));
	else
		tmp = abs(d) * ((1.0 - (((((M_m * M_m) * D_m) * (D_m / (d * d))) * 0.125) * (h / l))) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 5e-163], N[(N[Abs[d], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[N[(M$95$m * D$95$m), $MachinePrecision], 5e+130], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[d], $MachinePrecision] * N[(N[(1.0 - N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(D$95$m / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 M D) < 4.99999999999999977e-163

   1. Initial program 66.9%

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

   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 21.1

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

   4. Applied rewrites 21.1%

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

   5. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. sqrt-undiv N/A

         \[\leadsto \frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      2. pow2 N/A

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

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

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-fabs.f64 42.7

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 42.7

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

   7. Applied rewrites 42.7%

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

   if 4.99999999999999977e-163 < (*.f64 M D) < 4.9999999999999996e130

   1. Initial program 66.9%

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

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

      11. unpow2 N/A

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

      12. fabs-pow2-rev N/A

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

      13. sqrt-div-sound-left N/A

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

      14. fabs-pow2-rev N/A

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

      15. unpow2 N/A

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

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

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

      17. lower-/.f64 N/A

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

      18. lower-fabs.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-*.f64 69.8

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

   3. Applied rewrites 69.8%

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

   4. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 46.5

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

   6. Applied rewrites 46.5%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.125\right)} \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. unswap-sqr N/A

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

      7. pow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 59.5

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

   8. Applied rewrites 59.5%

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

   if 4.9999999999999996e130 < (*.f64 M D)

   1. Initial program 66.9%

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

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

      11. unpow2 N/A

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

      12. fabs-pow2-rev N/A

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

      13. sqrt-div-sound-left N/A

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

      14. fabs-pow2-rev N/A

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

      15. unpow2 N/A

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

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

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

      17. lower-/.f64 N/A

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

      18. lower-fabs.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-*.f64 69.8

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

   3. Applied rewrites 69.8%

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

   4. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 46.5

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

   6. Applied rewrites 46.5%

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

   8. Applied rewrites 54.9%

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

Alternative 8: 66.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ \mathbf{if}\;D\_m \leq 1.1 \cdot 10^{+144}:\\ \;\;\;\;t\_0 \cdot \left(1 - \left(\left(\frac{M\_m}{d} \cdot \frac{M\_m \cdot \left(D\_m \cdot D\_m\right)}{d}\right) \cdot 0.125\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(1 - \left(\left(\left(D\_m \cdot M\_m\right) \cdot \frac{D\_m \cdot M\_m}{d \cdot d}\right) \cdot 0.125\right) \cdot \frac{h}{\ell}\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ (fabs d) (sqrt (* h l)))))
   (if (<= D_m 1.1e+144)
     (*
      t_0
      (- 1.0 (* (* (* (/ M_m d) (/ (* M_m (* D_m D_m)) d)) 0.125) (/ h l))))
     (*
      t_0
      (- 1.0 (* (* (* (* D_m M_m) (/ (* D_m M_m) (* d d))) 0.125) (/ h l)))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = fabs(d) / sqrt((h * l));
	double tmp;
	if (D_m <= 1.1e+144) {
		tmp = t_0 * (1.0 - ((((M_m / d) * ((M_m * (D_m * D_m)) / d)) * 0.125) * (h / l)));
	} else {
		tmp = t_0 * (1.0 - ((((D_m * M_m) * ((D_m * M_m) / (d * d))) * 0.125) * (h / l)));
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(d) / sqrt((h * l))
    if (d_m <= 1.1d+144) then
        tmp = t_0 * (1.0d0 - ((((m_m / d) * ((m_m * (d_m * d_m)) / d)) * 0.125d0) * (h / l)))
    else
        tmp = t_0 * (1.0d0 - ((((d_m * m_m) * ((d_m * m_m) / (d * d))) * 0.125d0) * (h / l)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.abs(d) / Math.sqrt((h * l));
	double tmp;
	if (D_m <= 1.1e+144) {
		tmp = t_0 * (1.0 - ((((M_m / d) * ((M_m * (D_m * D_m)) / d)) * 0.125) * (h / l)));
	} else {
		tmp = t_0 * (1.0 - ((((D_m * M_m) * ((D_m * M_m) / (d * d))) * 0.125) * (h / l)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.fabs(d) / math.sqrt((h * l))
	tmp = 0
	if D_m <= 1.1e+144:
		tmp = t_0 * (1.0 - ((((M_m / d) * ((M_m * (D_m * D_m)) / d)) * 0.125) * (h / l)))
	else:
		tmp = t_0 * (1.0 - ((((D_m * M_m) * ((D_m * M_m) / (d * d))) * 0.125) * (h / l)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(abs(d) / sqrt(Float64(h * l)))
	tmp = 0.0
	if (D_m <= 1.1e+144)
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(Float64(Float64(M_m / d) * Float64(Float64(M_m * Float64(D_m * D_m)) / d)) * 0.125) * Float64(h / l))));
	else
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(Float64(Float64(D_m * M_m) * Float64(Float64(D_m * M_m) / Float64(d * d))) * 0.125) * Float64(h / l))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = abs(d) / sqrt((h * l));
	tmp = 0.0;
	if (D_m <= 1.1e+144)
		tmp = t_0 * (1.0 - ((((M_m / d) * ((M_m * (D_m * D_m)) / d)) * 0.125) * (h / l)));
	else
		tmp = t_0 * (1.0 - ((((D_m * M_m) * ((D_m * M_m) / (d * d))) * 0.125) * (h / l)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D$95$m, 1.1e+144], N[(t$95$0 * N[(1.0 - N[(N[(N[(N[(M$95$m / d), $MachinePrecision] * N[(N[(M$95$m * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 - N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * N[(N[(D$95$m * M$95$m), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if D < 1.09999999999999994e144

   1. Initial program 66.9%

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

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

      11. unpow2 N/A

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

      12. fabs-pow2-rev N/A

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

      13. sqrt-div-sound-left N/A

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

      14. fabs-pow2-rev N/A

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

      15. unpow2 N/A

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

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

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

      17. lower-/.f64 N/A

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

      18. lower-fabs.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-*.f64 69.8

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

   3. Applied rewrites 69.8%

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

   4. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 46.5

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

   6. Applied rewrites 46.5%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.125\right)} \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 59.1

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

   8. Applied rewrites 59.1%

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

   if 1.09999999999999994e144 < D

   1. Initial program 66.9%

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

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

      11. unpow2 N/A

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

      12. fabs-pow2-rev N/A

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

      13. sqrt-div-sound-left N/A

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

      14. fabs-pow2-rev N/A

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

      15. unpow2 N/A

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

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

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

      17. lower-/.f64 N/A

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

      18. lower-fabs.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-*.f64 69.8

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

   3. Applied rewrites 69.8%

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

   4. Taylor expanded in d around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 46.5

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

   6. Applied rewrites 46.5%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.125\right)} \cdot \frac{h}{\ell}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. unswap-sqr N/A

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

      7. pow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 59.5

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

   8. Applied rewrites 59.5%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\left(\left(D \cdot M\right) \cdot \frac{D \cdot M}{d \cdot d}\right) \cdot 0.125\right) \cdot \frac{h}{\ell}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 65.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-214}:\\ \;\;\;\;\left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(\frac{M\_m}{d} \cdot \frac{M\_m}{\ell \cdot d}\right)\right) \cdot -0.125\right) \cdot D\_m\right) \cdot D\_m\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-79}:\\ \;\;\;\;\sqrt{\frac{h}{\ell}} \cdot \frac{d}{h}\\ \mathbf{else}:\\ \;\;\;\;\left|d\right| \cdot \frac{1 - \left(\left(\left(\left(M\_m \cdot M\_m\right) \cdot D\_m\right) \cdot \frac{D\_m}{d \cdot d}\right) \cdot 0.125\right) \cdot \frac{h}{\ell}}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_0 -2e-214)
     (*
      (*
       (*
        (* (* (/ (fabs d) (sqrt (* h l))) h) (* (/ M_m d) (/ M_m (* l d))))
        -0.125)
       D_m)
      D_m)
     (if (<= t_0 5e-79)
       (* (sqrt (/ h l)) (/ d h))
       (*
        (fabs d)
        (/
         (- 1.0 (* (* (* (* (* M_m M_m) D_m) (/ D_m (* d d))) 0.125) (/ h l)))
         (sqrt (* l h))))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -2e-214) {
		tmp = (((((fabs(d) / sqrt((h * l))) * h) * ((M_m / d) * (M_m / (l * d)))) * -0.125) * D_m) * D_m;
	} else if (t_0 <= 5e-79) {
		tmp = sqrt((h / l)) * (d / h);
	} else {
		tmp = fabs(d) * ((1.0 - (((((M_m * M_m) * D_m) * (D_m / (d * d))) * 0.125) * (h / l))) / sqrt((l * h)));
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m_m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
    if (t_0 <= (-2d-214)) then
        tmp = (((((abs(d) / sqrt((h * l))) * h) * ((m_m / d) * (m_m / (l * d)))) * (-0.125d0)) * d_m) * d_m
    else if (t_0 <= 5d-79) then
        tmp = sqrt((h / l)) * (d / h)
    else
        tmp = abs(d) * ((1.0d0 - (((((m_m * m_m) * d_m) * (d_m / (d * d))) * 0.125d0) * (h / l))) / sqrt((l * h)))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_0 <= -2e-214) {
		tmp = (((((Math.abs(d) / Math.sqrt((h * l))) * h) * ((M_m / d) * (M_m / (l * d)))) * -0.125) * D_m) * D_m;
	} else if (t_0 <= 5e-79) {
		tmp = Math.sqrt((h / l)) * (d / h);
	} else {
		tmp = Math.abs(d) * ((1.0 - (((((M_m * M_m) * D_m) * (D_m / (d * d))) * 0.125) * (h / l))) / Math.sqrt((l * h)));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))
	tmp = 0
	if t_0 <= -2e-214:
		tmp = (((((math.fabs(d) / math.sqrt((h * l))) * h) * ((M_m / d) * (M_m / (l * d)))) * -0.125) * D_m) * D_m
	elif t_0 <= 5e-79:
		tmp = math.sqrt((h / l)) * (d / h)
	else:
		tmp = math.fabs(d) * ((1.0 - (((((M_m * M_m) * D_m) * (D_m / (d * d))) * 0.125) * (h / l))) / math.sqrt((l * h)))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_0 <= -2e-214)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(abs(d) / sqrt(Float64(h * l))) * h) * Float64(Float64(M_m / d) * Float64(M_m / Float64(l * d)))) * -0.125) * D_m) * D_m);
	elseif (t_0 <= 5e-79)
		tmp = Float64(sqrt(Float64(h / l)) * Float64(d / h));
	else
		tmp = Float64(abs(d) * Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(M_m * M_m) * D_m) * Float64(D_m / Float64(d * d))) * 0.125) * Float64(h / l))) / sqrt(Float64(l * h))));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)));
	tmp = 0.0;
	if (t_0 <= -2e-214)
		tmp = (((((abs(d) / sqrt((h * l))) * h) * ((M_m / d) * (M_m / (l * d)))) * -0.125) * D_m) * D_m;
	elseif (t_0 <= 5e-79)
		tmp = sqrt((h / l)) * (d / h);
	else
		tmp = abs(d) * ((1.0 - (((((M_m * M_m) * D_m) * (D_m / (d * d))) * 0.125) * (h / l))) / sqrt((l * h)));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-214], N[(N[(N[(N[(N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * N[(M$95$m / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision], If[LessEqual[t$95$0, 5e-79], N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision], N[(N[Abs[d], $MachinePrecision] * N[(N[(1.0 - N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(D$95$m / N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $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)))) < -1.99999999999999983e-214

   1. Initial program 66.9%

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

   2. Taylor expanded in d around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}\right) \cdot D\right) \cdot \color{blue}{D} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}\right) \cdot D\right) \cdot \color{blue}{D} \]

   4. Applied rewrites 26.7%

      \[\leadsto \color{blue}{\left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot -0.125\right) \cdot D\right) \cdot D} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      3. associate-*r/ N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      6. associate-*l* N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \frac{M \cdot M}{d \cdot \left(d \cdot \ell\right)}\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      7. times-frac N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{d \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{d \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{d \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{d \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      11. *-commutative N/A

          \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{\ell \cdot d}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      12. lower-*.f64 33.9

          \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{\ell \cdot d}\right)\right) \cdot -0.125\right) \cdot D\right) \cdot D \]

   6. Applied rewrites 33.9%

      \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{\ell \cdot d}\right)\right) \cdot -0.125\right) \cdot D\right) \cdot D \]

   if -1.99999999999999983e-214 < (*.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)))) < 4.99999999999999999e-79

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 21.1

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   4. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. sqrt-undiv N/A

         \[\leadsto \frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      2. pow2 N/A

         \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}} \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      7. lift-fabs.f64 42.7

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      10. lower-*.f64 42.7

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   7. Applied rewrites 42.7%

      \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   8. Taylor expanded in d around 0

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{\frac{h}{\ell}} \cdot \frac{d}{\color{blue}{h}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{h}{\ell}} \cdot \frac{d}{\color{blue}{h}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{h}{\ell}} \cdot \frac{d}{h} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{h}{\ell}} \cdot \frac{d}{h} \]

      6. lower-/.f64 36.9

         \[\leadsto \sqrt{\frac{h}{\ell}} \cdot \frac{d}{h} \]

   10. Applied rewrites 36.9%

       \[\leadsto \sqrt{\frac{h}{\ell}} \cdot \color{blue}{\frac{d}{h}} \]

   if 4.99999999999999999e-79 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{{d}^{2}}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. fabs-pow2-rev N/A

          \[\leadsto \sqrt{\frac{\color{blue}{\left|{d}^{2}\right|}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. sqrt-div-sound-left N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\left|{d}^{2}\right|}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. fabs-pow2-rev N/A

          \[\leadsto \frac{\sqrt{\color{blue}{{d}^{2}}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-*.f64 69.8

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in d around 0

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}}\right)} \cdot \frac{h}{\ell}\right) \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \color{blue}{\frac{1}{8}}\right) \cdot \frac{h}{\ell}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \color{blue}{\frac{1}{8}}\right) \cdot \frac{h}{\ell}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{{D}^{2} \cdot {M}^{2}}{{d}^{2}} \cdot \frac{1}{8}\right) \cdot \frac{h}{\ell}\right) \]

      4. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{{M}^{2} \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{8}\right) \cdot \frac{h}{\ell}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{{M}^{2} \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{8}\right) \cdot \frac{h}{\ell}\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{\left(M \cdot M\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{8}\right) \cdot \frac{h}{\ell}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{\left(M \cdot M\right) \cdot {D}^{2}}{{d}^{2}} \cdot \frac{1}{8}\right) \cdot \frac{h}{\ell}\right) \]

      8. unpow2 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{8}\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{{d}^{2}} \cdot \frac{1}{8}\right) \cdot \frac{h}{\ell}\right) \]

      10. pow2 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot \frac{1}{8}\right) \cdot \frac{h}{\ell}\right) \]

      11. lift-*.f64 46.5

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.125\right) \cdot \frac{h}{\ell}\right) \]

   6. Applied rewrites 46.5%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\frac{\left(M \cdot M\right) \cdot \left(D \cdot D\right)}{d \cdot d} \cdot 0.125\right)} \cdot \frac{h}{\ell}\right) \]

   8. Applied rewrites 54.9%

      \[\leadsto \color{blue}{\left|d\right| \cdot \frac{1 - \left(\left(\left(\left(M \cdot M\right) \cdot D\right) \cdot \frac{D}{d \cdot d}\right) \cdot 0.125\right) \cdot \frac{h}{\ell}}{\sqrt{\ell \cdot h}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 65.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\left|d\right|}{\sqrt{h \cdot \ell}}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-214}:\\ \;\;\;\;\left(\left(\left(\left(t\_0 \cdot h\right) \cdot \left(\frac{M\_m}{d} \cdot \frac{M\_m}{\ell \cdot d}\right)\right) \cdot -0.125\right) \cdot D\_m\right) \cdot D\_m\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\left(M\_m \cdot \frac{M\_m \cdot \left(D\_m \cdot D\_m\right)}{\left(d \cdot d\right) \cdot \ell}\right) \cdot -0.125, h, 1\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ (fabs d) (sqrt (* h l))))
        (t_1
         (*
          (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
          (-
           1.0
           (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l))))))
   (if (<= t_1 -2e-214)
     (* (* (* (* (* t_0 h) (* (/ M_m d) (/ M_m (* l d)))) -0.125) D_m) D_m)
     (if (<= t_1 INFINITY)
       (/ (fabs d) (sqrt (* l h)))
       (*
        t_0
        (fma
         (* (* M_m (/ (* M_m (* D_m D_m)) (* (* d d) l))) -0.125)
         h
         1.0))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = fabs(d) / sqrt((h * l));
	double t_1 = (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)));
	double tmp;
	if (t_1 <= -2e-214) {
		tmp = ((((t_0 * h) * ((M_m / d) * (M_m / (l * d)))) * -0.125) * D_m) * D_m;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fabs(d) / sqrt((l * h));
	} else {
		tmp = t_0 * fma(((M_m * ((M_m * (D_m * D_m)) / ((d * d) * l))) * -0.125), h, 1.0);
	}
	return tmp;
}

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(abs(d) / sqrt(Float64(h * l)))
	t_1 = Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
	tmp = 0.0
	if (t_1 <= -2e-214)
		tmp = Float64(Float64(Float64(Float64(Float64(t_0 * h) * Float64(Float64(M_m / d) * Float64(M_m / Float64(l * d)))) * -0.125) * D_m) * D_m);
	elseif (t_1 <= Inf)
		tmp = Float64(abs(d) / sqrt(Float64(l * h)));
	else
		tmp = Float64(t_0 * fma(Float64(Float64(M_m * Float64(Float64(M_m * Float64(D_m * D_m)) / Float64(Float64(d * d) * l))) * -0.125), h, 1.0));
	end
	return tmp
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-214], N[(N[(N[(N[(N[(t$95$0 * h), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * N[(M$95$m / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[(M$95$m * N[(N[(M$95$m * N[(D$95$m * D$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * h + 1.0), $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)))) < -1.99999999999999983e-214

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left({D}^{2} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{{D}^{2}}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}\right) \cdot \color{blue}{{D}^{2}} \]

      4. unpow2 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}\right) \cdot \left(D \cdot \color{blue}{D}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}\right) \cdot D\right) \cdot \color{blue}{D} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}\right) \cdot D\right) \cdot \color{blue}{D} \]

   4. Applied rewrites 26.7%

      \[\leadsto \color{blue}{\left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot -0.125\right) \cdot D\right) \cdot D} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      3. associate-*r/ N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      6. associate-*l* N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \frac{M \cdot M}{d \cdot \left(d \cdot \ell\right)}\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      7. times-frac N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{d \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{d \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{d \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{d \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      11. *-commutative N/A

          \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{\ell \cdot d}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      12. lower-*.f64 33.9

          \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{\ell \cdot d}\right)\right) \cdot -0.125\right) \cdot D\right) \cdot D \]

   6. Applied rewrites 33.9%

      \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{\ell \cdot d}\right)\right) \cdot -0.125\right) \cdot D\right) \cdot D \]

   if -1.99999999999999983e-214 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < +inf.0

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 21.1

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   4. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. sqrt-undiv N/A

         \[\leadsto \frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      2. pow2 N/A

         \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}} \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      7. lift-fabs.f64 42.7

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      10. lower-*.f64 42.7

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   7. Applied rewrites 42.7%

      \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{{d}^{2}}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. fabs-pow2-rev N/A

          \[\leadsto \sqrt{\frac{\color{blue}{\left|{d}^{2}\right|}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. sqrt-div-sound-left N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\left|{d}^{2}\right|}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. fabs-pow2-rev N/A

          \[\leadsto \frac{\sqrt{\color{blue}{{d}^{2}}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-*.f64 69.8

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      7. associate-/l* N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. count-2-rev N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(M \cdot \frac{D}{\color{blue}{d + d}}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(M \cdot \frac{D}{\color{blue}{d + d}}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. associate-/l* N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. count-2-rev N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{\color{blue}{d + d}}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-+.f64 68.9

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{\color{blue}{d + d}}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 68.9%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]

   6. Taylor expanded in h around inf

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(h \cdot \left(\frac{1}{h} - \frac{1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right)\right)} \]
   7. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(h \cdot \left(\frac{1}{h} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(h \cdot \left(\frac{1}{h} + \frac{-1}{8} \cdot \frac{\color{blue}{{D}^{2} \cdot {M}^{2}}}{{d}^{2} \cdot \ell}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(h \cdot \left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell} + \color{blue}{\frac{1}{h}}\right)\right) \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + \color{blue}{\frac{1}{h} \cdot h}\right) \]

      5. lft-mult-inverse N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}\right) \cdot h + 1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot {M}^{2}}{{d}^{2} \cdot \ell}, \color{blue}{h}, 1\right) \]

   8. Applied rewrites 52.7%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(\left(M \cdot \frac{M \cdot \left(D \cdot D\right)}{\left(d \cdot d\right) \cdot \ell}\right) \cdot -0.125, h, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 64.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 1.85 \cdot 10^{-24}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(\frac{M\_m}{d} \cdot \frac{M\_m}{\ell \cdot d}\right)\right) \cdot -0.125\right) \cdot D\_m\right) \cdot D\_m\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= M_m 1.85e-24)
   (/ (fabs d) (sqrt (* l h)))
   (*
    (*
     (*
      (* (* (/ (fabs d) (sqrt (* h l))) h) (* (/ M_m d) (/ M_m (* l d))))
      -0.125)
     D_m)
    D_m)))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (M_m <= 1.85e-24) {
		tmp = fabs(d) / sqrt((l * h));
	} else {
		tmp = (((((fabs(d) / sqrt((h * l))) * h) * ((M_m / d) * (M_m / (l * d)))) * -0.125) * D_m) * D_m;
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (m_m <= 1.85d-24) then
        tmp = abs(d) / sqrt((l * h))
    else
        tmp = (((((abs(d) / sqrt((h * l))) * h) * ((m_m / d) * (m_m / (l * d)))) * (-0.125d0)) * d_m) * d_m
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (M_m <= 1.85e-24) {
		tmp = Math.abs(d) / Math.sqrt((l * h));
	} else {
		tmp = (((((Math.abs(d) / Math.sqrt((h * l))) * h) * ((M_m / d) * (M_m / (l * d)))) * -0.125) * D_m) * D_m;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if M_m <= 1.85e-24:
		tmp = math.fabs(d) / math.sqrt((l * h))
	else:
		tmp = (((((math.fabs(d) / math.sqrt((h * l))) * h) * ((M_m / d) * (M_m / (l * d)))) * -0.125) * D_m) * D_m
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (M_m <= 1.85e-24)
		tmp = Float64(abs(d) / sqrt(Float64(l * h)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(abs(d) / sqrt(Float64(h * l))) * h) * Float64(Float64(M_m / d) * Float64(M_m / Float64(l * d)))) * -0.125) * D_m) * D_m);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (M_m <= 1.85e-24)
		tmp = abs(d) / sqrt((l * h));
	else
		tmp = (((((abs(d) / sqrt((h * l))) * h) * ((M_m / d) * (M_m / (l * d)))) * -0.125) * D_m) * D_m;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[M$95$m, 1.85e-24], N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * N[(N[(M$95$m / d), $MachinePrecision] * N[(M$95$m / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.8499999999999999e-24

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 21.1

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   4. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. sqrt-undiv N/A

         \[\leadsto \frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      2. pow2 N/A

         \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}} \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      7. lift-fabs.f64 42.7

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      10. lower-*.f64 42.7

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   7. Applied rewrites 42.7%

      \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   if 1.8499999999999999e-24 < M

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left({D}^{2} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{{D}^{2}}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}\right) \cdot \color{blue}{{D}^{2}} \]

      4. unpow2 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}\right) \cdot \left(D \cdot \color{blue}{D}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}\right) \cdot D\right) \cdot \color{blue}{D} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}\right) \cdot D\right) \cdot \color{blue}{D} \]

   4. Applied rewrites 26.7%

      \[\leadsto \color{blue}{\left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot -0.125\right) \cdot D\right) \cdot D} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      3. associate-*r/ N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \frac{M \cdot M}{\left(d \cdot d\right) \cdot \ell}\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      6. associate-*l* N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \frac{M \cdot M}{d \cdot \left(d \cdot \ell\right)}\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      7. times-frac N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{d \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{d \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      9. lower-/.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{d \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{d \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      11. *-commutative N/A

          \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{\ell \cdot d}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      12. lower-*.f64 33.9

          \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{\ell \cdot d}\right)\right) \cdot -0.125\right) \cdot D\right) \cdot D \]

   6. Applied rewrites 33.9%

      \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(\frac{M}{d} \cdot \frac{M}{\ell \cdot d}\right)\right) \cdot -0.125\right) \cdot D\right) \cdot D \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 56.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;M\_m \leq 1.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(\left(\left(D\_m \cdot D\_m\right) \cdot \left(h \cdot \left(\frac{M\_m}{\left(d \cdot d\right) \cdot \ell} \cdot M\_m\right)\right)\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= M_m 1.5e-30)
   (/ (fabs d) (sqrt (* l h)))
   (*
    (/ (fabs d) (sqrt (* h l)))
    (* (* (* D_m D_m) (* h (* (/ M_m (* (* d d) l)) M_m))) -0.125))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (M_m <= 1.5e-30) {
		tmp = fabs(d) / sqrt((l * h));
	} else {
		tmp = (fabs(d) / sqrt((h * l))) * (((D_m * D_m) * (h * ((M_m / ((d * d) * l)) * M_m))) * -0.125);
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (m_m <= 1.5d-30) then
        tmp = abs(d) / sqrt((l * h))
    else
        tmp = (abs(d) / sqrt((h * l))) * (((d_m * d_m) * (h * ((m_m / ((d * d) * l)) * m_m))) * (-0.125d0))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (M_m <= 1.5e-30) {
		tmp = Math.abs(d) / Math.sqrt((l * h));
	} else {
		tmp = (Math.abs(d) / Math.sqrt((h * l))) * (((D_m * D_m) * (h * ((M_m / ((d * d) * l)) * M_m))) * -0.125);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if M_m <= 1.5e-30:
		tmp = math.fabs(d) / math.sqrt((l * h))
	else:
		tmp = (math.fabs(d) / math.sqrt((h * l))) * (((D_m * D_m) * (h * ((M_m / ((d * d) * l)) * M_m))) * -0.125)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (M_m <= 1.5e-30)
		tmp = Float64(abs(d) / sqrt(Float64(l * h)));
	else
		tmp = Float64(Float64(abs(d) / sqrt(Float64(h * l))) * Float64(Float64(Float64(D_m * D_m) * Float64(h * Float64(Float64(M_m / Float64(Float64(d * d) * l)) * M_m))) * -0.125));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (M_m <= 1.5e-30)
		tmp = abs(d) / sqrt((l * h));
	else
		tmp = (abs(d) / sqrt((h * l))) * (((D_m * D_m) * (h * ((M_m / ((d * d) * l)) * M_m))) * -0.125);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[M$95$m, 1.5e-30], N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(h * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(h * N[(N[(M$95$m / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * M$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 1.49999999999999995e-30

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 21.1

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   4. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. sqrt-undiv N/A

         \[\leadsto \frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      2. pow2 N/A

         \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}} \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      7. lift-fabs.f64 42.7

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      10. lower-*.f64 42.7

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   7. Applied rewrites 42.7%

      \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   if 1.49999999999999995e-30 < M

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{{d}^{2}}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. fabs-pow2-rev N/A

          \[\leadsto \sqrt{\frac{\color{blue}{\left|{d}^{2}\right|}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. sqrt-div-sound-left N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\left|{d}^{2}\right|}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. fabs-pow2-rev N/A

          \[\leadsto \frac{\sqrt{\color{blue}{{d}^{2}}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-*.f64 69.8

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      7. associate-/l* N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. count-2-rev N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(M \cdot \frac{D}{\color{blue}{d + d}}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(M \cdot \frac{D}{\color{blue}{d + d}}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. associate-/l* N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. count-2-rev N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{\color{blue}{d + d}}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-+.f64 68.9

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{\color{blue}{d + d}}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 68.9%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]

   6. Taylor expanded in d around 0

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell}\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(\frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{\frac{-1}{8}}\right) \]

   8. Applied rewrites 26.2%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(\left(\left(D \cdot D\right) \cdot \left(h \cdot \left(\frac{M}{\left(d \cdot d\right) \cdot \ell} \cdot M\right)\right)\right) \cdot -0.125\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 54.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;M\_m \leq 2.8 \cdot 10^{-31}:\\ \;\;\;\;\frac{\left|d\right|}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\frac{\left|d\right| \cdot h}{t\_0} \cdot M\_m\right) \cdot \frac{M\_m}{\left(d \cdot d\right) \cdot \ell}\right) \cdot -0.125\right) \cdot D\_m\right) \cdot D\_m\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<= M_m 2.8e-31)
     (/ (fabs d) t_0)
     (*
      (*
       (* (* (* (/ (* (fabs d) h) t_0) M_m) (/ M_m (* (* d d) l))) -0.125)
       D_m)
      D_m))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (M_m <= 2.8e-31) {
		tmp = fabs(d) / t_0;
	} else {
		tmp = ((((((fabs(d) * h) / t_0) * M_m) * (M_m / ((d * d) * l))) * -0.125) * D_m) * D_m;
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l * h))
    if (m_m <= 2.8d-31) then
        tmp = abs(d) / t_0
    else
        tmp = ((((((abs(d) * h) / t_0) * m_m) * (m_m / ((d * d) * l))) * (-0.125d0)) * d_m) * d_m
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((l * h));
	double tmp;
	if (M_m <= 2.8e-31) {
		tmp = Math.abs(d) / t_0;
	} else {
		tmp = ((((((Math.abs(d) * h) / t_0) * M_m) * (M_m / ((d * d) * l))) * -0.125) * D_m) * D_m;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if M_m <= 2.8e-31:
		tmp = math.fabs(d) / t_0
	else:
		tmp = ((((((math.fabs(d) * h) / t_0) * M_m) * (M_m / ((d * d) * l))) * -0.125) * D_m) * D_m
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (M_m <= 2.8e-31)
		tmp = Float64(abs(d) / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(abs(d) * h) / t_0) * M_m) * Float64(M_m / Float64(Float64(d * d) * l))) * -0.125) * D_m) * D_m);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if (M_m <= 2.8e-31)
		tmp = abs(d) / t_0;
	else
		tmp = ((((((abs(d) * h) / t_0) * M_m) * (M_m / ((d * d) * l))) * -0.125) * D_m) * D_m;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M$95$m, 2.8e-31], N[(N[Abs[d], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[Abs[d], $MachinePrecision] * h), $MachinePrecision] / t$95$0), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(M$95$m / N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 2.7999999999999999e-31

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 21.1

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   4. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. sqrt-undiv N/A

         \[\leadsto \frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      2. pow2 N/A

         \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}} \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      7. lift-fabs.f64 42.7

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      10. lower-*.f64 42.7

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   7. Applied rewrites 42.7%

      \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   if 2.7999999999999999e-31 < M

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left({D}^{2} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{{D}^{2}}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}\right) \cdot \color{blue}{{D}^{2}} \]

      4. unpow2 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}\right) \cdot \left(D \cdot \color{blue}{D}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}\right) \cdot D\right) \cdot \color{blue}{D} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}\right) \cdot D\right) \cdot \color{blue}{D} \]

   4. Applied rewrites 26.7%

      \[\leadsto \color{blue}{\left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot -0.125\right) \cdot D\right) \cdot D} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      3. lift-fabs.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      8. associate-*r* N/A

         \[\leadsto \left(\left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

   6. Applied rewrites 24.1%

      \[\leadsto \left(\left(\left(\left(\frac{\left|d\right| \cdot h}{\sqrt{\ell \cdot h}} \cdot M\right) \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right) \cdot -0.125\right) \cdot D\right) \cdot D \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 53.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;M\_m \leq 2.8 \cdot 10^{-31}:\\ \;\;\;\;\frac{\left|d\right|}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(M\_m \cdot M\_m\right) \cdot D\_m\right) \cdot D\_m\right) \cdot \left(h \cdot \frac{\left|d\right|}{\left(\left(d \cdot d\right) \cdot \ell\right) \cdot t\_0}\right)\right) \cdot -0.125\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<= M_m 2.8e-31)
     (/ (fabs d) t_0)
     (*
      (* (* (* (* M_m M_m) D_m) D_m) (* h (/ (fabs d) (* (* (* d d) l) t_0))))
      -0.125))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (M_m <= 2.8e-31) {
		tmp = fabs(d) / t_0;
	} else {
		tmp = ((((M_m * M_m) * D_m) * D_m) * (h * (fabs(d) / (((d * d) * l) * t_0)))) * -0.125;
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l * h))
    if (m_m <= 2.8d-31) then
        tmp = abs(d) / t_0
    else
        tmp = ((((m_m * m_m) * d_m) * d_m) * (h * (abs(d) / (((d * d) * l) * t_0)))) * (-0.125d0)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((l * h));
	double tmp;
	if (M_m <= 2.8e-31) {
		tmp = Math.abs(d) / t_0;
	} else {
		tmp = ((((M_m * M_m) * D_m) * D_m) * (h * (Math.abs(d) / (((d * d) * l) * t_0)))) * -0.125;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if M_m <= 2.8e-31:
		tmp = math.fabs(d) / t_0
	else:
		tmp = ((((M_m * M_m) * D_m) * D_m) * (h * (math.fabs(d) / (((d * d) * l) * t_0)))) * -0.125
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (M_m <= 2.8e-31)
		tmp = Float64(abs(d) / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(M_m * M_m) * D_m) * D_m) * Float64(h * Float64(abs(d) / Float64(Float64(Float64(d * d) * l) * t_0)))) * -0.125);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if (M_m <= 2.8e-31)
		tmp = abs(d) / t_0;
	else
		tmp = ((((M_m * M_m) * D_m) * D_m) * (h * (abs(d) / (((d * d) * l) * t_0)))) * -0.125;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M$95$m, 2.8e-31], N[(N[Abs[d], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision] * N[(h * N[(N[Abs[d], $MachinePrecision] / N[(N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 2.7999999999999999e-31

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 21.1

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   4. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. sqrt-undiv N/A

         \[\leadsto \frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      2. pow2 N/A

         \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}} \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      7. lift-fabs.f64 42.7

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      10. lower-*.f64 42.7

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   7. Applied rewrites 42.7%

      \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   if 2.7999999999999999e-31 < M

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{{d}^{2}}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. fabs-pow2-rev N/A

          \[\leadsto \sqrt{\frac{\color{blue}{\left|{d}^{2}\right|}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. sqrt-div-sound-left N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\left|{d}^{2}\right|}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. fabs-pow2-rev N/A

          \[\leadsto \frac{\sqrt{\color{blue}{{d}^{2}}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-*.f64 69.8

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\right) \cdot \frac{h}{\ell}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{\color{blue}{M \cdot D}}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{\color{blue}{2 \cdot d}}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      5. unpow2 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\right) \cdot \frac{h}{\ell}\right) \]

      7. associate-/l* N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)} \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      10. count-2-rev N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(M \cdot \frac{D}{\color{blue}{d + d}}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(M \cdot \frac{D}{\color{blue}{d + d}}\right) \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{h}{\ell}\right) \]

      12. associate-/l* N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot \color{blue}{\left(M \cdot \frac{D}{2 \cdot d}\right)}\right)\right) \cdot \frac{h}{\ell}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \color{blue}{\frac{D}{2 \cdot d}}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      15. count-2-rev N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{\color{blue}{d + d}}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

      16. lower-+.f64 68.9

          \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{\color{blue}{d + d}}\right)\right)\right) \cdot \frac{h}{\ell}\right) \]

   5. Applied rewrites 68.9%

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot \color{blue}{\left(\left(M \cdot \frac{D}{d + d}\right) \cdot \left(M \cdot \frac{D}{d + d}\right)\right)}\right) \cdot \frac{h}{\ell}\right) \]

   6. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left|d\right|\right)\right)}{{d}^{2} \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)}} \]

   8. Applied rewrites 25.1%

      \[\leadsto \color{blue}{\left(\left(\left(\left(M \cdot M\right) \cdot D\right) \cdot D\right) \cdot \left(h \cdot \frac{\left|d\right|}{\left(\left(d \cdot d\right) \cdot \ell\right) \cdot \sqrt{\ell \cdot h}}\right)\right) \cdot -0.125} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 53.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;M\_m \leq 2.8 \cdot 10^{-31}:\\ \;\;\;\;\frac{\left|d\right|}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-0.125 \cdot \left(\left(\left(M\_m \cdot M\_m\right) \cdot h\right) \cdot \left|d\right|\right)}{\left(\left(d \cdot d\right) \cdot \ell\right) \cdot t\_0} \cdot D\_m\right) \cdot D\_m\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<= M_m 2.8e-31)
     (/ (fabs d) t_0)
     (*
      (*
       (/ (* -0.125 (* (* (* M_m M_m) h) (fabs d))) (* (* (* d d) l) t_0))
       D_m)
      D_m))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (M_m <= 2.8e-31) {
		tmp = fabs(d) / t_0;
	} else {
		tmp = (((-0.125 * (((M_m * M_m) * h) * fabs(d))) / (((d * d) * l) * t_0)) * D_m) * D_m;
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l * h))
    if (m_m <= 2.8d-31) then
        tmp = abs(d) / t_0
    else
        tmp = ((((-0.125d0) * (((m_m * m_m) * h) * abs(d))) / (((d * d) * l) * t_0)) * d_m) * d_m
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((l * h));
	double tmp;
	if (M_m <= 2.8e-31) {
		tmp = Math.abs(d) / t_0;
	} else {
		tmp = (((-0.125 * (((M_m * M_m) * h) * Math.abs(d))) / (((d * d) * l) * t_0)) * D_m) * D_m;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if M_m <= 2.8e-31:
		tmp = math.fabs(d) / t_0
	else:
		tmp = (((-0.125 * (((M_m * M_m) * h) * math.fabs(d))) / (((d * d) * l) * t_0)) * D_m) * D_m
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (M_m <= 2.8e-31)
		tmp = Float64(abs(d) / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(-0.125 * Float64(Float64(Float64(M_m * M_m) * h) * abs(d))) / Float64(Float64(Float64(d * d) * l) * t_0)) * D_m) * D_m);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if (M_m <= 2.8e-31)
		tmp = abs(d) / t_0;
	else
		tmp = (((-0.125 * (((M_m * M_m) * h) * abs(d))) / (((d * d) * l) * t_0)) * D_m) * D_m;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M$95$m, 2.8e-31], N[(N[Abs[d], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(-0.125 * N[(N[(N[(M$95$m * M$95$m), $MachinePrecision] * h), $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * D$95$m), $MachinePrecision] * D$95$m), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 2.7999999999999999e-31

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 21.1

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   4. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. sqrt-undiv N/A

         \[\leadsto \frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      2. pow2 N/A

         \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}} \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      7. lift-fabs.f64 42.7

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      10. lower-*.f64 42.7

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   7. Applied rewrites 42.7%

      \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   if 2.7999999999999999e-31 < M

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left({D}^{2} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{{D}^{2}}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}\right) \cdot \color{blue}{{D}^{2}} \]

      4. unpow2 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}\right) \cdot \left(D \cdot \color{blue}{D}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}\right) \cdot D\right) \cdot \color{blue}{D} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}\right) \cdot D\right) \cdot \color{blue}{D} \]

   4. Applied rewrites 26.7%

      \[\leadsto \color{blue}{\left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot -0.125\right) \cdot D\right) \cdot D} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      4. lift-fabs.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot \frac{-1}{8}\right) \cdot D\right) \cdot D \]

   6. Applied rewrites 21.7%

      \[\leadsto \left(\frac{-0.125 \cdot \left(\left(\left(M \cdot M\right) \cdot h\right) \cdot \left|d\right|\right)}{\left(\left(d \cdot d\right) \cdot \ell\right) \cdot \sqrt{\ell \cdot h}} \cdot D\right) \cdot D \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 52.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;M\_m \leq 2.8 \cdot 10^{-31}:\\ \;\;\;\;\frac{\left|d\right|}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\left(\left(D\_m \cdot M\_m\right) \cdot M\_m\right) \cdot \left(\left|d\right| \cdot h\right)}{\left(\left(d \cdot d\right) \cdot \ell\right) \cdot t\_0} \cdot -0.125\right) \cdot D\_m\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))))
   (if (<= M_m 2.8e-31)
     (/ (fabs d) t_0)
     (*
      (*
       (/ (* (* (* D_m M_m) M_m) (* (fabs d) h)) (* (* (* d d) l) t_0))
       -0.125)
      D_m))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((l * h));
	double tmp;
	if (M_m <= 2.8e-31) {
		tmp = fabs(d) / t_0;
	} else {
		tmp = (((((D_m * M_m) * M_m) * (fabs(d) * h)) / (((d * d) * l) * t_0)) * -0.125) * D_m;
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((l * h))
    if (m_m <= 2.8d-31) then
        tmp = abs(d) / t_0
    else
        tmp = (((((d_m * m_m) * m_m) * (abs(d) * h)) / (((d * d) * l) * t_0)) * (-0.125d0)) * d_m
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((l * h));
	double tmp;
	if (M_m <= 2.8e-31) {
		tmp = Math.abs(d) / t_0;
	} else {
		tmp = (((((D_m * M_m) * M_m) * (Math.abs(d) * h)) / (((d * d) * l) * t_0)) * -0.125) * D_m;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((l * h))
	tmp = 0
	if M_m <= 2.8e-31:
		tmp = math.fabs(d) / t_0
	else:
		tmp = (((((D_m * M_m) * M_m) * (math.fabs(d) * h)) / (((d * d) * l) * t_0)) * -0.125) * D_m
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(l * h))
	tmp = 0.0
	if (M_m <= 2.8e-31)
		tmp = Float64(abs(d) / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(D_m * M_m) * M_m) * Float64(abs(d) * h)) / Float64(Float64(Float64(d * d) * l) * t_0)) * -0.125) * D_m);
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((l * h));
	tmp = 0.0;
	if (M_m <= 2.8e-31)
		tmp = abs(d) / t_0;
	else
		tmp = (((((D_m * M_m) * M_m) * (abs(d) * h)) / (((d * d) * l) * t_0)) * -0.125) * D_m;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M$95$m, 2.8e-31], N[(N[Abs[d], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * N[(N[Abs[d], $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(d * d), $MachinePrecision] * l), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * D$95$m), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < 2.7999999999999999e-31

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 21.1

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   4. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. sqrt-undiv N/A

         \[\leadsto \frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      2. pow2 N/A

         \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}} \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      7. lift-fabs.f64 42.7

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      10. lower-*.f64 42.7

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   7. Applied rewrites 42.7%

      \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   if 2.7999999999999999e-31 < M

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left({D}^{2} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{{D}^{2}}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}\right) \cdot \color{blue}{{D}^{2}} \]

      4. unpow2 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}\right) \cdot \left(D \cdot \color{blue}{D}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}\right) \cdot D\right) \cdot \color{blue}{D} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}\right) \cdot D\right) \cdot \color{blue}{D} \]

   4. Applied rewrites 26.7%

      \[\leadsto \color{blue}{\left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot -0.125\right) \cdot D\right) \cdot D} \]

   5. Taylor expanded in d around 0

      \[\leadsto \left(\frac{-1}{8} \cdot \frac{D \cdot \left({M}^{2} \cdot \left(h \cdot \left|d\right|\right)\right)}{{d}^{2} \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)}\right) \cdot D \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{D \cdot \left({M}^{2} \cdot \left(h \cdot \left|d\right|\right)\right)}{{d}^{2} \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}\right) \cdot D \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{D \cdot \left({M}^{2} \cdot \left(h \cdot \left|d\right|\right)\right)}{{d}^{2} \cdot \left(\ell \cdot \sqrt{h \cdot \ell}\right)} \cdot \frac{-1}{8}\right) \cdot D \]

   7. Applied rewrites 23.6%

      \[\leadsto \left(\frac{\left(\left(D \cdot M\right) \cdot M\right) \cdot \left(\left|d\right| \cdot h\right)}{\left(\left(d \cdot d\right) \cdot \ell\right) \cdot \sqrt{\ell \cdot h}} \cdot -0.125\right) \cdot D \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 52.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -2 \cdot 10^{-181}:\\ \;\;\;\;\left(\frac{\left(\left(D\_m \cdot M\_m\right) \cdot M\_m\right) \cdot \left|d\right|}{\left(\sqrt{\frac{\ell}{h}} \cdot \ell\right) \cdot \left(d \cdot d\right)} \cdot -0.125\right) \cdot D\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
       (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l))))
      -2e-181)
   (*
    (*
     (/ (* (* (* D_m M_m) M_m) (fabs d)) (* (* (sqrt (/ l h)) l) (* d d)))
     -0.125)
    D_m)
   (/ (fabs d) (sqrt (* l h)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -2e-181) {
		tmp = (((((D_m * M_m) * M_m) * fabs(d)) / ((sqrt((l / h)) * l) * (d * d))) * -0.125) * D_m;
	} else {
		tmp = fabs(d) / sqrt((l * h));
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (((((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m_m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= (-2d-181)) then
        tmp = (((((d_m * m_m) * m_m) * abs(d)) / ((sqrt((l / h)) * l) * (d * d))) * (-0.125d0)) * d_m
    else
        tmp = abs(d) / sqrt((l * h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (((Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -2e-181) {
		tmp = (((((D_m * M_m) * M_m) * Math.abs(d)) / ((Math.sqrt((l / h)) * l) * (d * d))) * -0.125) * D_m;
	} else {
		tmp = Math.abs(d) / Math.sqrt((l * h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if ((math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -2e-181:
		tmp = (((((D_m * M_m) * M_m) * math.fabs(d)) / ((math.sqrt((l / h)) * l) * (d * d))) * -0.125) * D_m
	else:
		tmp = math.fabs(d) / math.sqrt((l * h))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -2e-181)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(D_m * M_m) * M_m) * abs(d)) / Float64(Float64(sqrt(Float64(l / h)) * l) * Float64(d * d))) * -0.125) * D_m);
	else
		tmp = Float64(abs(d) / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (((((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -2e-181)
		tmp = (((((D_m * M_m) * M_m) * abs(d)) / ((sqrt((l / h)) * l) * (d * d))) * -0.125) * D_m;
	else
		tmp = abs(d) / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-181], N[(N[(N[(N[(N[(N[(D$95$m * M$95$m), $MachinePrecision] * M$95$m), $MachinePrecision] * N[Abs[d], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sqrt[N[(l / h), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision] * N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision] * D$95$m), $MachinePrecision], N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -2.00000000000000009e-181

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in d around 0

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell}} \]
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{-1}{8} \cdot \left({D}^{2} \cdot \color{blue}{\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{8} \cdot \left(\frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell} \cdot \color{blue}{{D}^{2}}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}\right) \cdot \color{blue}{{D}^{2}} \]

      4. unpow2 N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}\right) \cdot \left(D \cdot \color{blue}{D}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}\right) \cdot D\right) \cdot \color{blue}{D} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{-1}{8} \cdot \frac{{M}^{2} \cdot \left(h \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right)}{{d}^{2} \cdot \ell}\right) \cdot D\right) \cdot \color{blue}{D} \]

   4. Applied rewrites 26.7%

      \[\leadsto \color{blue}{\left(\left(\left(\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \cdot h\right) \cdot \left(M \cdot \frac{M}{\left(d \cdot d\right) \cdot \ell}\right)\right) \cdot -0.125\right) \cdot D\right) \cdot D} \]

   5. Taylor expanded in h around inf

      \[\leadsto \left(\frac{-1}{8} \cdot \frac{D \cdot \left({M}^{2} \cdot \left|d\right|\right)}{{d}^{2} \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)}\right) \cdot D \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{D \cdot \left({M}^{2} \cdot \left|d\right|\right)}{{d}^{2} \cdot \left(\ell \cdot \sqrt{\frac{\ell}{h}}\right)} \cdot \frac{-1}{8}\right) \cdot D \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{D \cdot \left({M}^{2} \cdot \left|d\right|\right)}{{d}^{2} \cdot \left(\sqrt{\frac{\ell}{h}} \cdot \ell\right)} \cdot \frac{-1}{8}\right) \cdot D \]

      3. associate-*l* N/A

         \[\leadsto \left(\frac{\left(D \cdot {M}^{2}\right) \cdot \left|d\right|}{{d}^{2} \cdot \left(\sqrt{\frac{\ell}{h}} \cdot \ell\right)} \cdot \frac{-1}{8}\right) \cdot D \]

      4. pow2 N/A

         \[\leadsto \left(\frac{\left(D \cdot \left(M \cdot M\right)\right) \cdot \left|d\right|}{{d}^{2} \cdot \left(\sqrt{\frac{\ell}{h}} \cdot \ell\right)} \cdot \frac{-1}{8}\right) \cdot D \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{\left(D \cdot \left(M \cdot M\right)\right) \cdot \left|d\right|}{\left(\sqrt{\frac{\ell}{h}} \cdot \ell\right) \cdot {d}^{2}} \cdot \frac{-1}{8}\right) \cdot D \]

      6. pow2 N/A

         \[\leadsto \left(\frac{\left(D \cdot \left(M \cdot M\right)\right) \cdot \left|d\right|}{\left(\sqrt{\frac{\ell}{h}} \cdot \ell\right) \cdot \left(d \cdot d\right)} \cdot \frac{-1}{8}\right) \cdot D \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{\left(D \cdot \left(M \cdot M\right)\right) \cdot \left|d\right|}{\left(\sqrt{\frac{\ell}{h}} \cdot \ell\right) \cdot \left(d \cdot d\right)} \cdot \frac{-1}{8}\right) \cdot D \]

   7. Applied rewrites 14.9%

      \[\leadsto \left(\frac{\left(\left(D \cdot M\right) \cdot M\right) \cdot \left|d\right|}{\left(\sqrt{\frac{\ell}{h}} \cdot \ell\right) \cdot \left(d \cdot d\right)} \cdot -0.125\right) \cdot D \]

   if -2.00000000000000009e-181 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 21.1

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   4. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. sqrt-undiv N/A

         \[\leadsto \frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      2. pow2 N/A

         \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}} \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      7. lift-fabs.f64 42.7

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      10. lower-*.f64 42.7

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   7. Applied rewrites 42.7%

      \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 51.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -2 \cdot 10^{-214}:\\ \;\;\;\;\left(\left(D\_m \cdot D\_m\right) \cdot \frac{\sqrt{\frac{h}{\ell}} \cdot \left(M\_m \cdot M\_m\right)}{\ell \cdot d}\right) \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<=
      (*
       (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
       (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l))))
      -2e-214)
   (* (* (* D_m D_m) (/ (* (sqrt (/ h l)) (* M_m M_m)) (* l d))) -0.125)
   (/ (fabs d) (sqrt (* l h)))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -2e-214) {
		tmp = ((D_m * D_m) * ((sqrt((h / l)) * (M_m * M_m)) / (l * d))) * -0.125;
	} else {
		tmp = fabs(d) / sqrt((l * h));
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (((((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m_m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= (-2d-214)) then
        tmp = ((d_m * d_m) * ((sqrt((h / l)) * (m_m * m_m)) / (l * d))) * (-0.125d0)
    else
        tmp = abs(d) / sqrt((l * h))
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (((Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -2e-214) {
		tmp = ((D_m * D_m) * ((Math.sqrt((h / l)) * (M_m * M_m)) / (l * d))) * -0.125;
	} else {
		tmp = Math.abs(d) / Math.sqrt((l * h));
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if ((math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -2e-214:
		tmp = ((D_m * D_m) * ((math.sqrt((h / l)) * (M_m * M_m)) / (l * d))) * -0.125
	else:
		tmp = math.fabs(d) / math.sqrt((l * h))
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -2e-214)
		tmp = Float64(Float64(Float64(D_m * D_m) * Float64(Float64(sqrt(Float64(h / l)) * Float64(M_m * M_m)) / Float64(l * d))) * -0.125);
	else
		tmp = Float64(abs(d) / sqrt(Float64(l * h)));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (((((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -2e-214)
		tmp = ((D_m * D_m) * ((sqrt((h / l)) * (M_m * M_m)) / (l * d))) * -0.125;
	else
		tmp = abs(d) / sqrt((l * h));
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-214], N[(N[(N[(D$95$m * D$95$m), $MachinePrecision] * N[(N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(M$95$m * M$95$m), $MachinePrecision]), $MachinePrecision] / N[(l * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125), $MachinePrecision], N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l)))) < -1.99999999999999983e-214

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   2. Taylor expanded in h around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot \left(h \cdot \left(\sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}\right)\right)\right)}{{d}^{2} \cdot \ell} + \sqrt{d \cdot h} \cdot \sqrt{\frac{d}{\ell}}}{h}} \]

   4. Applied rewrites 17.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(\left(\left(\left(\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}} \cdot h\right) \cdot M\right) \cdot M\right) \cdot D\right) \cdot D}{\left(d \cdot d\right) \cdot \ell}, -0.125, \sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}\right)}{h}} \]

   5. Taylor expanded in d around 0

      \[\leadsto \frac{-1}{8} \cdot \color{blue}{\frac{{D}^{2} \cdot \left({M}^{2} \cdot \sqrt{\frac{h}{\ell}}\right)}{d \cdot \ell}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot \sqrt{\frac{h}{\ell}}\right)}{d \cdot \ell} \cdot \frac{-1}{8} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{D}^{2} \cdot \left({M}^{2} \cdot \sqrt{\frac{h}{\ell}}\right)}{d \cdot \ell} \cdot \frac{-1}{8} \]

   7. Applied rewrites 34.2%

      \[\leadsto \left(\left(D \cdot D\right) \cdot \frac{\sqrt{\frac{h}{\ell}} \cdot \left(M \cdot M\right)}{\ell \cdot d}\right) \cdot \color{blue}{-0.125} \]

   if -1.99999999999999983e-214 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 21.1

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   4. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. sqrt-undiv N/A

         \[\leadsto \frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      2. pow2 N/A

         \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}} \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      7. lift-fabs.f64 42.7

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      10. lower-*.f64 42.7

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   7. Applied rewrites 42.7%

      \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 48.8% accurate, 4.3× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;h \leq -2 \cdot 10^{+196}:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{elif}\;h \leq 2.25 \cdot 10^{+57}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|d\right|}{t\_0 \cdot \ell}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l))))
   (if (<= h -2e+196)
     (/ (* t_0 d) h)
     (if (<= h 2.25e+57) (/ (fabs d) (sqrt (* l h))) (/ (fabs d) (* t_0 l))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((h / l));
	double tmp;
	if (h <= -2e+196) {
		tmp = (t_0 * d) / h;
	} else if (h <= 2.25e+57) {
		tmp = fabs(d) / sqrt((l * h));
	} else {
		tmp = fabs(d) / (t_0 * l);
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((h / l))
    if (h <= (-2d+196)) then
        tmp = (t_0 * d) / h
    else if (h <= 2.25d+57) then
        tmp = abs(d) / sqrt((l * h))
    else
        tmp = abs(d) / (t_0 * l)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((h / l));
	double tmp;
	if (h <= -2e+196) {
		tmp = (t_0 * d) / h;
	} else if (h <= 2.25e+57) {
		tmp = Math.abs(d) / Math.sqrt((l * h));
	} else {
		tmp = Math.abs(d) / (t_0 * l);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((h / l))
	tmp = 0
	if h <= -2e+196:
		tmp = (t_0 * d) / h
	elif h <= 2.25e+57:
		tmp = math.fabs(d) / math.sqrt((l * h))
	else:
		tmp = math.fabs(d) / (t_0 * l)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(h / l))
	tmp = 0.0
	if (h <= -2e+196)
		tmp = Float64(Float64(t_0 * d) / h);
	elseif (h <= 2.25e+57)
		tmp = Float64(abs(d) / sqrt(Float64(l * h)));
	else
		tmp = Float64(abs(d) / Float64(t_0 * l));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((h / l));
	tmp = 0.0;
	if (h <= -2e+196)
		tmp = (t_0 * d) / h;
	elseif (h <= 2.25e+57)
		tmp = abs(d) / sqrt((l * h));
	else
		tmp = abs(d) / (t_0 * l);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -2e+196], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[h, 2.25e+57], N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[d], $MachinePrecision] / N[(t$95$0 * l), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.9999999999999999e196

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 21.1

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   4. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in d around 0

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      4. lift-/.f64 37.2

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   7. Applied rewrites 37.2%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   if -1.9999999999999999e196 < h < 2.24999999999999998e57

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 21.1

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   4. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. sqrt-undiv N/A

         \[\leadsto \frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      2. pow2 N/A

         \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}} \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      7. lift-fabs.f64 42.7

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      10. lower-*.f64 42.7

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   7. Applied rewrites 42.7%

      \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   if 2.24999999999999998e57 < h

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   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. frac-times N/A

          \[\leadsto \sqrt{\color{blue}{\frac{d \cdot d}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\frac{\color{blue}{{d}^{2}}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      12. fabs-pow2-rev N/A

          \[\leadsto \sqrt{\frac{\color{blue}{\left|{d}^{2}\right|}}{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      13. sqrt-div-sound-left N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\left|{d}^{2}\right|}}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      14. fabs-pow2-rev N/A

          \[\leadsto \frac{\sqrt{\color{blue}{{d}^{2}}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{d \cdot d}}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      16. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      17. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      18. lower-fabs.f64 N/A

          \[\leadsto \frac{\color{blue}{\left|d\right|}}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      19. lower-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

      20. lower-*.f64 69.8

          \[\leadsto \frac{\left|d\right|}{\sqrt{\color{blue}{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{h \cdot \ell}}} \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   4. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\ell \cdot \sqrt{\frac{h}{\ell}}}} \]
   5. Step-by-step derivation

      1. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\left|d\right|}{\ell \cdot \sqrt{\frac{h}{\ell}}} \]

      2. pow2 N/A

         \[\leadsto \frac{\left|d\right|}{\ell \cdot \sqrt{\frac{h}{\ell}}} \]

      3. sqrt-undiv N/A

         \[\leadsto \frac{\left|\color{blue}{d}\right|}{\ell \cdot \sqrt{\frac{h}{\ell}}} \]

      4. pow1/2 N/A

         \[\leadsto \frac{\left|\color{blue}{d}\right|}{\ell \cdot \sqrt{\frac{h}{\ell}}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\left|d\right|}{\ell \cdot \sqrt{\frac{h}{\ell}}} \]

      6. pow2 N/A

         \[\leadsto \frac{\left|d\right|}{\ell \cdot \sqrt{\frac{h}{\ell}}} \]

      7. frac-times N/A

         \[\leadsto \frac{\left|d\right|}{\ell \cdot \sqrt{\frac{h}{\ell}}} \]

      8. pow-prod-down N/A

         \[\leadsto \frac{\left|\color{blue}{d}\right|}{\ell \cdot \sqrt{\frac{h}{\ell}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\color{blue}{\ell \cdot \sqrt{\frac{h}{\ell}}}} \]

      10. lift-fabs.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\color{blue}{\ell} \cdot \sqrt{\frac{h}{\ell}}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\frac{h}{\ell}} \cdot \color{blue}{\ell}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\frac{h}{\ell}} \cdot \color{blue}{\ell}} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \frac{\left|d\right|}{\sqrt{\frac{h}{\ell}} \cdot \ell} \]

      14. lift-/.f64 23.0

          \[\leadsto \frac{\left|d\right|}{\sqrt{\frac{h}{\ell}} \cdot \ell} \]

   6. Applied rewrites 23.0%

      \[\leadsto \color{blue}{\frac{\left|d\right|}{\sqrt{\frac{h}{\ell}} \cdot \ell}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 44.1% accurate, 4.6× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{h}{\ell}}\\ \mathbf{if}\;h \leq -2 \cdot 10^{+196}:\\ \;\;\;\;\frac{t\_0 \cdot d}{h}\\ \mathbf{elif}\;h \leq 5.4 \cdot 10^{+69}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{d}{h}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ h l))))
   (if (<= h -2e+196)
     (/ (* t_0 d) h)
     (if (<= h 5.4e+69) (/ (fabs d) (sqrt (* l h))) (* t_0 (/ d h))))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = sqrt((h / l));
	double tmp;
	if (h <= -2e+196) {
		tmp = (t_0 * d) / h;
	} else if (h <= 5.4e+69) {
		tmp = fabs(d) / sqrt((l * h));
	} else {
		tmp = t_0 * (d / h);
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((h / l))
    if (h <= (-2d+196)) then
        tmp = (t_0 * d) / h
    else if (h <= 5.4d+69) then
        tmp = abs(d) / sqrt((l * h))
    else
        tmp = t_0 * (d / h)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.sqrt((h / l));
	double tmp;
	if (h <= -2e+196) {
		tmp = (t_0 * d) / h;
	} else if (h <= 5.4e+69) {
		tmp = Math.abs(d) / Math.sqrt((l * h));
	} else {
		tmp = t_0 * (d / h);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.sqrt((h / l))
	tmp = 0
	if h <= -2e+196:
		tmp = (t_0 * d) / h
	elif h <= 5.4e+69:
		tmp = math.fabs(d) / math.sqrt((l * h))
	else:
		tmp = t_0 * (d / h)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = sqrt(Float64(h / l))
	tmp = 0.0
	if (h <= -2e+196)
		tmp = Float64(Float64(t_0 * d) / h);
	elseif (h <= 5.4e+69)
		tmp = Float64(abs(d) / sqrt(Float64(l * h)));
	else
		tmp = Float64(t_0 * Float64(d / h));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = sqrt((h / l));
	tmp = 0.0;
	if (h <= -2e+196)
		tmp = (t_0 * d) / h;
	elseif (h <= 5.4e+69)
		tmp = abs(d) / sqrt((l * h));
	else
		tmp = t_0 * (d / h);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[h, -2e+196], N[(N[(t$95$0 * d), $MachinePrecision] / h), $MachinePrecision], If[LessEqual[h, 5.4e+69], N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(d / h), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if h < -1.9999999999999999e196

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 21.1

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   4. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in d around 0

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{h} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      4. lift-/.f64 37.2

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   7. Applied rewrites 37.2%

      \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

   if -1.9999999999999999e196 < h < 5.3999999999999996e69

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 21.1

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   4. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. sqrt-undiv N/A

         \[\leadsto \frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      2. pow2 N/A

         \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}} \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      7. lift-fabs.f64 42.7

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      10. lower-*.f64 42.7

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   7. Applied rewrites 42.7%

      \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   if 5.3999999999999996e69 < h

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 21.1

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   4. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. sqrt-undiv N/A

         \[\leadsto \frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      2. pow2 N/A

         \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}} \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      7. lift-fabs.f64 42.7

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      10. lower-*.f64 42.7

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   7. Applied rewrites 42.7%

      \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   8. Taylor expanded in d around 0

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{\frac{h}{\ell}} \cdot \frac{d}{\color{blue}{h}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{h}{\ell}} \cdot \frac{d}{\color{blue}{h}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{h}{\ell}} \cdot \frac{d}{h} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{h}{\ell}} \cdot \frac{d}{h} \]

      6. lower-/.f64 36.9

         \[\leadsto \sqrt{\frac{h}{\ell}} \cdot \frac{d}{h} \]

   10. Applied rewrites 36.9%

       \[\leadsto \sqrt{\frac{h}{\ell}} \cdot \color{blue}{\frac{d}{h}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 43.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} \mathbf{if}\;h \leq 5.4 \cdot 10^{+69}:\\ \;\;\;\;\frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{h}{\ell}} \cdot \frac{d}{h}\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (if (<= h 5.4e+69) (/ (fabs d) (sqrt (* l h))) (* (sqrt (/ h l)) (/ d h))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= 5.4e+69) {
		tmp = fabs(d) / sqrt((l * h));
	} else {
		tmp = sqrt((h / l)) * (d / h);
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: tmp
    if (h <= 5.4d+69) then
        tmp = abs(d) / sqrt((l * h))
    else
        tmp = sqrt((h / l)) * (d / h)
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double tmp;
	if (h <= 5.4e+69) {
		tmp = Math.abs(d) / Math.sqrt((l * h));
	} else {
		tmp = Math.sqrt((h / l)) * (d / h);
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	tmp = 0
	if h <= 5.4e+69:
		tmp = math.fabs(d) / math.sqrt((l * h))
	else:
		tmp = math.sqrt((h / l)) * (d / h)
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	tmp = 0.0
	if (h <= 5.4e+69)
		tmp = Float64(abs(d) / sqrt(Float64(l * h)));
	else
		tmp = Float64(sqrt(Float64(h / l)) * Float64(d / h));
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	tmp = 0.0;
	if (h <= 5.4e+69)
		tmp = abs(d) / sqrt((l * h));
	else
		tmp = sqrt((h / l)) * (d / h);
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := If[LessEqual[h, 5.4e+69], N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision] * N[(d / h), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if h < 5.3999999999999996e69

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 21.1

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   4. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. sqrt-undiv N/A

         \[\leadsto \frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      2. pow2 N/A

         \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}} \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      7. lift-fabs.f64 42.7

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      10. lower-*.f64 42.7

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   7. Applied rewrites 42.7%

      \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   if 5.3999999999999996e69 < h

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 21.1

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   4. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. sqrt-undiv N/A

         \[\leadsto \frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      2. pow2 N/A

         \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}} \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      7. lift-fabs.f64 42.7

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      10. lower-*.f64 42.7

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   7. Applied rewrites 42.7%

      \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \]

   8. Taylor expanded in d around 0

      \[\leadsto \frac{d \cdot \sqrt{\frac{h}{\ell}}}{\color{blue}{h}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{h}{\ell}} \cdot d}{h} \]

      2. associate-/l* N/A

         \[\leadsto \sqrt{\frac{h}{\ell}} \cdot \frac{d}{\color{blue}{h}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{h}{\ell}} \cdot \frac{d}{\color{blue}{h}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{h}{\ell}} \cdot \frac{d}{h} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{h}{\ell}} \cdot \frac{d}{h} \]

      6. lower-/.f64 36.9

         \[\leadsto \sqrt{\frac{h}{\ell}} \cdot \frac{d}{h} \]

   10. Applied rewrites 36.9%

       \[\leadsto \sqrt{\frac{h}{\ell}} \cdot \color{blue}{\frac{d}{h}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 43.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \begin{array}{l} t_0 := \frac{\left|d\right|}{\sqrt{\ell \cdot h}}\\ \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M\_m \cdot D\_m}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -2 \cdot 10^{-214}:\\ \;\;\;\;-t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m)
 :precision binary64
 (let* ((t_0 (/ (fabs d) (sqrt (* l h)))))
   (if (<=
        (*
         (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
         (-
          1.0
          (* (* (/ 1.0 2.0) (pow (/ (* M_m D_m) (* 2.0 d)) 2.0)) (/ h l))))
        -2e-214)
     (- t_0)
     t_0)))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = fabs(d) / sqrt((l * h));
	double tmp;
	if (((pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -2e-214) {
		tmp = -t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs(d) / sqrt((l * h))
    if (((((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m_m * d_m) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))) <= (-2d-214)) then
        tmp = -t_0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	double t_0 = Math.abs(d) / Math.sqrt((l * h));
	double tmp;
	if (((Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -2e-214) {
		tmp = -t_0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	t_0 = math.fabs(d) / math.sqrt((l * h))
	tmp = 0
	if ((math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M_m * D_m) / (2.0 * d)), 2.0)) * (h / l)))) <= -2e-214:
		tmp = -t_0
	else:
		tmp = t_0
	return tmp

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	t_0 = Float64(abs(d) / sqrt(Float64(l * h)))
	tmp = 0.0
	if (Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M_m * D_m) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l)))) <= -2e-214)
		tmp = Float64(-t_0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp_2 = code(d, h, l, M_m, D_m)
	t_0 = abs(d) / sqrt((l * h));
	tmp = 0.0;
	if (((((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M_m * D_m) / (2.0 * d)) ^ 2.0)) * (h / l)))) <= -2e-214)
		tmp = -t_0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := Block[{t$95$0 = N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M$95$m * D$95$m), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-214], (-t$95$0), t$95$0]]

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)))) < -1.99999999999999983e-214

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 21.1

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   4. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   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. sqrt-undiv N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}}\right) \]

      3. pow2 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}}\right) \]

      4. rem-sqrt-square-rev N/A

         \[\leadsto \mathsf{neg}\left(\frac{\left|d\right|}{\sqrt{h \cdot \ell}}\right) \]

      5. lower-neg.f64 N/A

         \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      7. lift-*.f64 N/A

         \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      8. lift-/.f64 N/A

         \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      9. lift-fabs.f64 10.1

         \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      10. lift-*.f64 N/A

          \[\leadsto -\frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      11. *-commutative N/A

          \[\leadsto -\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      12. lower-*.f64 10.1

          \[\leadsto -\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   7. Applied rewrites 10.1%

      \[\leadsto -\frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   if -1.99999999999999983e-214 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))) (pow.f64 (/.f64 d l) (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))) (-.f64 #s(literal 1 binary64) (*.f64 (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)) (pow.f64 (/.f64 (*.f64 M D) (*.f64 #s(literal 2 binary64) d)) #s(literal 2 binary64))) (/.f64 h l))))

   1. Initial program 66.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   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 21.1

         \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

   4. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

   5. Taylor expanded in h around inf

      \[\leadsto \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]
   6. Step-by-step derivation

      1. sqrt-undiv N/A

         \[\leadsto \frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

      2. pow2 N/A

         \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}} \]

      3. rem-sqrt-square-rev N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      7. lift-fabs.f64 42.7

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

      9. *-commutative N/A

         \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

      10. lower-*.f64 42.7

          \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   7. Applied rewrites 42.7%

      \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 42.7% accurate, 9.0× speedup

Math

\[\begin{array}{l} M_m = \left|M\right| \\ D_m = \left|D\right| \\ [d, h, l, M_m, D_m] = \mathsf{sort}([d, h, l, M_m, D_m])\\ \\ \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

M_m = (fabs.f64 M)
D_m = (fabs.f64 D)
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
(FPCore (d h l M_m D_m) :precision binary64 (/ (fabs d) (sqrt (* l h))))

C

M_m = fabs(M);
D_m = fabs(D);
assert(d < h && h < l && l < M_m && M_m < D_m);
double code(double d, double h, double l, double M_m, double D_m) {
	return fabs(d) / sqrt((l * h));
}

Fortran

M_m =     private
D_m =     private
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(d, h, l, m_m, d_m)
use fmin_fmax_functions
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m_m
    real(8), intent (in) :: d_m
    code = abs(d) / sqrt((l * h))
end function

Java

M_m = Math.abs(M);
D_m = Math.abs(D);
assert d < h && h < l && l < M_m && M_m < D_m;
public static double code(double d, double h, double l, double M_m, double D_m) {
	return Math.abs(d) / Math.sqrt((l * h));
}

Python

M_m = math.fabs(M)
D_m = math.fabs(D)
[d, h, l, M_m, D_m] = sort([d, h, l, M_m, D_m])
def code(d, h, l, M_m, D_m):
	return math.fabs(d) / math.sqrt((l * h))

Julia

M_m = abs(M)
D_m = abs(D)
d, h, l, M_m, D_m = sort([d, h, l, M_m, D_m])
function code(d, h, l, M_m, D_m)
	return Float64(abs(d) / sqrt(Float64(l * h)))
end

MATLAB

M_m = abs(M);
D_m = abs(D);
d, h, l, M_m, D_m = num2cell(sort([d, h, l, M_m, D_m])){:}
function tmp = code(d, h, l, M_m, D_m)
	tmp = abs(d) / sqrt((l * h));
end

Wolfram

M_m = N[Abs[M], $MachinePrecision]
D_m = N[Abs[D], $MachinePrecision]
NOTE: d, h, l, M_m, and D_m should be sorted in increasing order before calling this function.
code[d_, h_, l_, M$95$m_, D$95$m_] := N[(N[Abs[d], $MachinePrecision] / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.9%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

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 21.1

      \[\leadsto \frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h} \]

4. Applied rewrites 21.1%

   \[\leadsto \color{blue}{\frac{\sqrt{\left(h \cdot d\right) \cdot \frac{d}{\ell}}}{h}} \]

5. Taylor expanded in h around inf

   \[\leadsto \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]
6. Step-by-step derivation

   1. sqrt-undiv N/A

      \[\leadsto \frac{\sqrt{{d}^{2}}}{\sqrt{h \cdot \ell}} \]

   2. pow2 N/A

      \[\leadsto \frac{\sqrt{d \cdot d}}{\sqrt{h \cdot \ell}} \]

   3. rem-sqrt-square-rev N/A

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

   4. lift-sqrt.f64 N/A

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

   6. lift-/.f64 N/A

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

   7. lift-fabs.f64 42.7

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\left|d\right|}{\sqrt{h \cdot \ell}} \]

   9. *-commutative N/A

      \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

   10. lower-*.f64 42.7

       \[\leadsto \frac{\left|d\right|}{\sqrt{\ell \cdot h}} \]

7. Applied rewrites 42.7%

   \[\leadsto \frac{\left|d\right|}{\color{blue}{\sqrt{\ell \cdot h}}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025135 
(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)))))